Foundations of the Theory of Algebraic Invariants

G. B. Gurevich FOUNDATIONS OF THE THEORY OF ALGEBRAIC INVARIANTS G. B. GUREVICH FOUNDATIONS OF THE THEORY OF ALGEBRA...

Author: G.B. Gurevich; Translators: J.R.M. Radok and A.J.M. Spencer

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G. B. Gurevich

FOUNDATIONS OF THE THEORY OF ALGEBRAIC INVARIANTS

G. B. GUREVICH

FOUNDATIONS OF THE THEORY OF ALGEBRAIC IN VARIANTS

Translated by

J. R. M. RADOK and

A. J. M. SPENCER

1964

P. NOORDHOI-1= L'I'D - GRONINGEN THE NETHERLANDS

© Copyright 1964 by P. Noordholf Ltd. Groningen, The Netherlands. This book or parts thereof may not be reproduced in any form without written permission of the publishers. Printed in The Netherlands.

FOREWORD

The theory of algebraic invariants has found insufficient attention in Russian mathematical literature. The book by Alekseev [I I]*, written in 1899, is largely out of date, while individual chapters in certain text-

books on algebra (Sushkevich, Bocher, etc.) written later, give only the beginnings of the theory. The present book is to fill this gap. Its essential special feature is wide utilization of classical methods as well as of the basic concepts and notation of tensor algebra; this makes it possible to present all problems at once in as general a form as possible. In addition, the Author believes that only by this means can one succeed in bringing full clarity to the problem of Aronhold's symbolic (§ 18). Throughout I have attempted not to leave out of sight the close link

which exists between invariant theory and geometry; the extensive geometric introduction of Chapter 1 also serves this purpose. Chapter 11 which is concerned with the foundations of tensor algebra also bears

an introductory character. The general propositions of the theory of algebraic invariants are given in Chapters III and IV, the most important particular results in Chapters V-VII; among these Chapter V is more classical in spirit. At the end of each section, I have given exercises of which there are altQgether approximately 500. They serve a double purpose in that some

explain the preceding work, others consider problems treated insufficiently in the main text. At times, a set of problems will present some substantial section of theory, for example, the exercises following §§ 3, 10-14, 25 contain the classification of binary forms of fourth order in the real domain. Certain problems relate to the subsequent text; in that case they have been provided with asterisks. In collecting the exercises, I have made use of the references [3, 11, 13, 14, 16, 19] as well as of H. Beck's ,Koordinaten Geometric". However, a substantial number of the problems have been published here for the first time.

In conclusion, I wish to express my gratitude to Ia. S. Dubnov, who has studied the manuscript of Chapter I and made a number of valuable comments, and to M. G. Freidinii for his painstaking editorial work.

G. B. Gurevich. * Numbers in square brackets refer to the list of references at the end of the text.

CONTENTS CHAPTER 1. Geometric introduction §

I The invariants of transformations of Cartesian coordinates

§ § § § § §

2 The group of motions . . . . . . . . . . . . . . . . . . . . . . . 3 The group of motions and reflections; the principal group . 4 Affine transformations of the plane . . . . . . . . . . . . . . 5 Projective transformations of a straight line . . . . . . . . . . 6 Projective transformations of the plane . . . . . . . . . . . . . 7 The reduction of affine invariants to projective invariants and of

§

metric invariants to affine and projective invariants . 8 Multi-dimensional space . . . . . . . . . . . . .

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39 51

68 85

CHAPTER 11. The foundations of tensor algebra

9 Tensors and operations on tensors .

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99

§ 10 The geometric meaning of tensors; the projective point of view

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115

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126 133

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§ 11 The geometric meaning of tensors; the affine point of view § 12 Relative tensors and their operations . . . . . . . . . . .

CHAPTER Ill. Invariants and concomitants of tensors and their simplest properties

§ 13 Invariants and concomitants of tensors . . § 14 Examples of invariants and concomitants . § 15 The simplest properties of invariants . . .

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139 143 159

CHAPTER IV. The fundamental theorem of the theory of invariants and its consequences

§ 16 Tensors with constant components . . . . § 17 Proof of the fundamental theorem . . . § 18 The symbolic method in invariant theory . § 19 Fundamental identities . . . . . . . . § 20 Invariant processes . . . . . . . . § 21 Complete systems of invariants . . .

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189

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209 223 235

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CHAPTER V. Binary forms

§ 22 Linear and quadratic binary forms . . . . . . . . . . . . . . 246 § 23 Cubic binary forms . . . . . . . . . . . . . . . . . 261 § 24 Gordan's method for finding complete systems of concomitants of .

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binary forms. Gordan-C'lebsch series .

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272

§ 25 Binary forms of fourth order . . . . . . . . . . . . . . . . . § 26 Differential equations for invariants and semi-invariants of binary forms . . . . . . . . . . . . . . . . . . . . . . . .

285 294

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VIII

CONTENTS

CHAPTER V1. Ternary forms. Second order tensors . . . . . . . . . . . . . 308 § 27 The quadratic ternary form . . . § 28 Geometric interpretations of the simplest concomitants of ternary forms of order r . . . . . . . . . . . . . . . . . . . . . . . 317 § 29 The quadratic n-ary form . . 326 .

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§ 30 Mixed second order tensors (affinors)

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338

CHAPTER VII. Polyvectors

§ 31 Certain general properties of polyvectors . . . . . . . . . . . . § 32 Geometric interpretation of polyvectors. Weitzenbock's complex

353

symbolic notation . . . . . . . . . . . . . . . . . . . . . § 33 Condition for divisibility of a polyvector by one or several vectors § 34 The bi-vector . . . . . . . . . . . . . . . . . . . . . . § 35 The tri-vector . . . . . . . . . . . . . . . . . . . . . . .

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363 374

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381

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390

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Answers and hints to exercises

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396

Literature .

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418

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419

Index

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CHAPTER I GEOMETRIC INTRODUCTION

The idea of an invariant, alongside those of number, set, function, transformation, etc., represents one of the most general and basic concepts in mathematics. The term invariant means unchanging; by this term we denote every quantity which, while related in a definite manner

to a mathematical object under consideration, remains unchanged under certain transformations. Our objective is the study of the foundations of the special theory of algebraic invariants of linear transformations. This theory arises in connection with a number of problems of algebra and geometry and was first formulated as a new subdivision of algebra in the work of Cayley (1821-1895) and Sylvester (1814-1897). The present chapter gives a preliminary presentation of the simplest

concepts of this theory based on their geometric background; the rigorous definitions will be given in Chapters II and III.

§ I. The invariants of transformations of Cartesian coordinates 1.1 We already find it necessary to study the invariants of linear transformations in connection with the simplest problems of analytic geometry. For the sake of simplicity, we will restrict ourselves here to two-dimensional geometry. In order to investigate the properties of geometric figures in a plane by algebraic methods we select in an arbitrary manner a system of orthogonal

Cartesian coordinates *) (Fig. 1); in this manner we establish a (1, 1) correspondence between the points of the plane and the pairs of numbers (x, y) - the coordinates of the points. The choice of the coordinate system

is quite arbitrary and in the majority of cases the coordinate axes will *) We will always assume the system of Cartesian rec'ilinear coordinates to be righthanded, i.e., we will measure the angle between the axes OX and OY: 1,C (OX, OY) = +90° in the counter-clockwise direction. Since we will not encounter oblique coor-

dinates, the word orthogonal will be omitted often.

2

CHAP. I

GEOMETRIC INTRODUCTION

lie outside the figure under consideration. As a consequence, an equation interrelating the coordinates of the points of the figure will, in general,

describe a property of the figure in relation to the external coordinate axes; an arbitrary function of the coordinates of the points of the figure will be a certain quantity which involves its disposition relative to the coordinate axes. There arises quite naturally the problem of the search for functions of th.- coordinates of the points which are equal to quantities

inherent in the geometric figure and not related to the coordinate axes. Y Y

k

.+90

X

O Fig.

I

h

X Fig. 2

It is easily seen that it is necessary and sufficient for this purpose that these functions can be expressed in the same manner in any coordinate system, i.e., that they remain invariant (unchanged) for all transformations of the Cartesian orthogonal coordinates. The same applies to those laws which-express properties of the figure which are independent of its disposition with respect to the coordinate axes. We will illustrate these statements by means of a number of simple examples. A transformation of Cartesian orthogonal coordinates is known to be given by the formulae x = x' cos w - y' sin w + h,

y = x' sin w + y' cos w + k,

(l.])

where x, y are the coordinates of points referred to the old coordinate system OXY and x', y' are the coordinates of the same points referred to the new coordinate system OX'Y' (Fig. 2). We will denote the coordinates of the points Ml, M2, M3, ... by MI(xi, yi), M2(x2, Y2), M3(x3, y3), .... The equality y, = 0 indicates then that the point M, lies on the x-axis; the relation x, = x2 means that the straight line M, M2 is parallel to the y-axis. Both these equalities

3

TRANSFORMATIONS OF CARTESIAN COOR13INATFS

1.1

are not invariant with respect to the transformation (1.1) and express properties of the points .411 and M2 with respect to the coordinate axes. Similarly, the expressions x2-x1, Jxi + yI are not invariant with respect

to coordinate transformations; therefore they are equal to quantities which are related to the dispositions of the points M1 and M2 with respect

to the coordinate axes: the first is the projection of the vector M1 At, onto the x-axis, the second is the distance of the point M1 from the origin

of coordinates 0. In contrast, the function of the coordinates of two points

(x2-X1) 2+(Y2-YI)2

(1.2)

is invariant with respect to the transformation (1.1). In fact, it follows from (1.1) that

x2 -XI = (x2 -x,) cos

sin w,

Y2 -Y1 = (x2-x'1) sin w+(y'2- y'1) cos w, whence (x2-x1)2+(Y2-YI)2 = (x2-xi)2+(Y2-YI)2.

As a consequence, (1.2) must represent a quantity which is only related to the points M1 and M2; actually, (1.2) is the square of the distance

between the points MI and M2. It is readily verified that the function of the coordinates of three points I X,

YI

X2

Y2

X3

Y3

1!

is likewise invariant under the transformation of Cartesian orthogonal coordinates: X1

YI

1

X2

y2 y3

1

x3

1

xi COs w-y, sin w+h = ` x2 cos w-y2 sin w+h j x3 cos w-y3 sin to+h

xI sin w+y', cos w+k x' sin w+y2 cos w+k x'3 sin w+y3 cos w+k

xi cos w-y, sin co x, sin w+y, cos co

1

x2cosw-v2sinw x2sin w+y2cose

1

x3 cos w-y3 sin w x3 sin w+y3 cos co

1

xl

.v;

1

X2

Y2

1

X3

Y3

1

COs co

sin w 0

-sine) cosw 0

0

X,

Yi

01,

X2

Y2

1

X3

Y3

1

1 1

11 1

1

4


CHAP. I

Therefore (1.3) represents some quantity which is determined by the

three points M,, M2, M3. In order to elucidate the meaning of the invariance of the expression (1.3), we can proceed in the following man-

ner. We displace the origin of coordinates to the point M, and direct the x-axis from MI to M2; then (Fig. 3) the coordinates of the points

X M3 Fig. 3

MI, M2, M3 assume the values: MI(0, 0), M2(x2, 0), x2 > 0, M3(x3,y3) and the expression (1.3) equals (y2y3)/2. We arrive at a well-known result: if the points MI, M2, M3 do not lie on a straight line, the expression

(1.3) is equal to the area of the triangle MI M2 M3 with a plus (+) sign if the rotation from the vector M, M2 to the vector M, M3 (through the smaller angle) occurs in the positive direction, and a minus (-) sign if this rotation is in the negative direction. If the points M1, M2 and M3 lie on a straight line, the quantity (1.3) is equal to zero. We will say that (1.3) is equal to the orientated area of the triangle M1 M2M3 (where the

order of the vertices is essential). Consider next the equations x3 = x1+Ax2,

Y3 = Y1+AY2,

(1.4)

relating the coordinates of the points M1, M2, M3; if we subject them to the transformation (1.1), the equalities (1.4) will be violated. Hence, if the relations (1.4) apply, the points M1, M2, M3 are related in a definite

manner to the coordinate axes (cf. Exercise 1). On the other hand, the equations

x_ 3

X1+ AX2

1+A

Y3

__

YI +_Y2

1+A

1.1-1.2

5

TRANSFORMATIONS OF CARTESIAN COORDINATES

are readily verified to be invariant with respect to the transformation (1.1); thus, if the equations (1.5) apply in one orthogonal Cartesian coordinate system, they will hold in any other such system. Therefore the equations express some geometric property of the points M,, M2, M3, namely, that these points lie on some straight line and that the point M3 divides the segment M, M2 in the ratio A. In 1.1, we have only considered functions of the coordinates+of points. The reasoning of 1.1 can be extended also to other geometric quantities. 1.2

Let there be given certain vectors a,, a2, ... with components

(XI, YI), (X2, Y2), .... Under a transformation of the system of Cartesian orthogonal coordinates, the components X, Y of the vectors change

in accordance with the formulae X = X' cos co - Y' sin w, Y = X' sin w+ Y' cos co.

[These formulae could have been derived from (1.1) by noting that the components of a vector are equal to the differences between the coordinates of its ends]. It follows from (1.6) that X2+Y2 = X,2+Y,2 I

I

I

1

X, X2+Y1 Y2 = Xi Xz+Y1'Y;,

X, jX2

Y,

Xi cos w- Yi sin w X, sin w+ Y, cos w

Y21

X2cosw-Y2sinw X2sinw+Yzcosw X,

YI

X2

YY

sin co

X It

Yi

- sin w cos co

XZ

Yi

cos co

Thus, the expressions

Xi+Yi ,

X, X2+Y1 Y2,

X,Y2-X2Y,

(1.7)

are invariants of the vectors under transformation of the Cartesian coordinates. Their geometric meanings are well-known: they are the square of the length of the vector a,, the scalar product a, - a2 of the vec-

tors a, and a2 and their vector product a, x a2; the last is equal to the orientated area of the parallelogram constructed on the mutual origin

6


CHAP. I

of the vectors aI and a2. All these results can also be established by a special choice of coordinate axes, as has been done for the function (1.3). 1.3

In analytic geometry, a straight line g is given by the equation

Ax+By+C = 0.

(1.8)

If in here we substitute for x, y their expressions (1.1), we see that in the

new coordinate system X'O' Y' the equation of the same straight line

g is A'x'+B'y'+C' = 0, where

A' = A cos w+B sin co,

B' = -A sin w+B cos w, C' = Ah+Bk+C. The equation of a straight line, and hence also the straight line itself, is

completely defined by the specification of the coefficients A, B, C on the left-hand side of (1.8). Therefore A, B, C are called the homogeneous (orthogonal Cartesian) coordinates of the straight line: homogeneous, because multiplication of all the three coefficients A, B, C by the same non-zero number ) does not change the straight line g; one has also to keep in mind that at least one of the numbers A or B must be different from zero. We may now call (1.9) the formulae of transformation of the orthogonal Cartesian coordinates of a straight line. We consider now the fact that in (1.9) the new coordinates A', B' are expressed only in terms of the old coordinates A, B, and, in fact, by the same rule as that for transformation of the components of a vector [in (1.6), the old coordinates are expressed in terms of the new, while in (1.9) the new are expressed in terms of the old; this corresponds to a replacement of co by -w]. This corresponds in all respects to the fact that A, B are the components of a vector which is perpendicular to the straight line g. As a consequence, the results of 1.2 give us immediately the three invariants of the transformation (1.9):

AI+Bi,

AI A2+B, B2,

AI B2-A2B,,

(1.10)

where A,, BI and A2, B2 are the coordinates of the two straight lines g, and g2. However, we might think mistakenly that the invariants

1.3


7

(1.10) are equal to certain geometric quantities connected with the straight lines g, and g2; the fact is that by multiplying all the three coordinates of the straight line g, by one and the same number A, and the coordinates of 92 by A.2 we do not alter these lines, but the values of the

invariants (1.10) change. As a consequence of this fact, a geometric meaning can be given to these invariants only if they are equal to zero. In fact, the first of the equalities A, B2 - A2 B, = 0,

A, A2 + B, B2 = 0,

Ai+BI = 0

means that the straight lines g, and 92 are perpendicular, the second, that

these lines are parallel and the third, that the line g, is isotropic. In order that an invariant of the transformation (1.9) can have a geometric meaning, it is necessary (and, obviously, sufficient) that it remain unchanged on multiplication of all three coordinates of the straight line by an arbitrary number A. In other words, it must be homogeneous of degree zero with respect to the coordinates of each of the straight lines which enter into it. Examples of such invariants are the expressions

A, B2-A2 81 At A2+B, B2

A, A2+B, B2

42+Bi AZ+B2-

where the symbols denote the same quantities as in (1.10); their geometric meanings are well-known:

A, B2-A2B, A, A2+B, B2

= t g (P,

- -A, A2+B, B2

= cos cp,

(1 . 11)

(1.12)

A2 +B2 V/A2+B2 v/A2+B2 I

where cp is the angle between the straight lines g, and g2 (cf. Exercise 4).

Consider next the expression Axo + Byo + C,

(1.13)

involving the coordinates A, B, C of a straight line g and the coordinates

of a point M(xo, y,). By the same calculations which led to (1.9), it is clear that

Axo+Byo+C: = A'xo+B'yo+C';


8

CHAP.

I

the expression (1.13) is a joint invariant of the straight line g and the point

M. For the reason stated above this invariant cannot have a geometric significance; only when it is set equal to zero will it have a meaning, namely that the point M lies on the straight line g. The joint invariant of the straight line g and the point M

Ax0+Byo+C

(1.14)

v/A2+B2

is homogeneous of degree zero with respect to the coordinates A, B, C of the straight line g; hence it represents a geometric quantity determined by the point M and the straight line g. In order to explain its meaning, we direct the x-axis along g in such a manner that the direction of the vector a, perpendicular to g and having in the former coordinate system the components A, B, coincides with the direction of the y-axis; then, in the new coordinate system, A' = 0, B' > 0, C' = 0, and (1.14) is equal to yo. Thus, the invariant (1.14) is equal to the distance of the point M from g with a plus (+) or (-) sign. We now draw the vector a (A, B), perpendicular to the straight line g, at some point of g. The invariant (1.14) gives the distance between M and g, with a plus (+) sign if the point M is on that side of g towards which the vector a points, and with

a minus (-) sign if it lies on the opposite side. We also note that the homogeneity of the invariant (1.14) is incomplete; it does not change its value when all three coordinates of the straight line g are multiplied by % only if i > 0; if i. < 0, it changes its sign (the root

in the denominator always being given a positive sign). This property also elucidates the fact that the geometric significance of the invariant (1.14) is not determined completely by specification of the point M and the straight line g; one has still to give a positive sign to one of the halfplanes into which the plane is divided by the straight line g. 1.4

A second order curve r is given by the equation

Axe+2Bxy+Cy'+2Dx+2Ey+F = 0;

(1.15)

the numbers A, B, C, D, E, F can be called the orthogonal Cartesian coordinates of the second order curve F. Just as in the case of the straight

line, they will he homogeneous coordinates, i.e., on multiplying all the six coordinates by the same non-zero number (and only if it is non-zero), the curve F will not change. In addition, one of the numbers A, B, C

1.4-1.5

9


must still be different from zero. For a transformation of the coordinates of points by the formulae (1.1), the coordinates of the curve are readily

seen to change in the following manner: A' = A cos' w+2B sin to cos w+C sin' w, B' -A sin w cos w+B(cos' w-sin ' w)+C sin co cos co, C' = Asin 2to -2B sin to cOsto +C cos' w, D' (Ah+Bk+D) cos w+(Bh+Ck+E) sin to,

(1.16)

E' = -(Ah+Bk+D) sin w+(Bh+Ck+E) cos co, F' = Ah2+2Bhk+Ck2+2Dh+2Ek+F. The transformation (1.16) is known to leave the following functions of the coordinates of the curve F invariant:

A B D rI=A+C, b,=AC-B', h3= B C E D E F

(1.17)

In view of the homogeneity of the coordinates A, B, ..., F, a geometric meaning attaches to these invariants only if they are equal to zero:

n = 0 means that the asymptotic directions of F are perpendicular, b2 = 0, that F is a parabola, 63 = 0, that r degenerates into two straight lines.

The invariants n and 63 are of odd degree in A, . . ., F; multiplying all the coordinates of F by the same number ) < 0, changes their signs. In contrast, the invariant d2, which is of even degree in A, . . ., F, maintains its sign when the coordinates A, ..., F are multiplied by an arbitrary number. Therefore the sign of the invariant 62 likewise has a geometric meaning: For 6, < 0, r is an ellipse, for 62 > 0, a hyperbola. The invariant 703I/6z is homogeneous of degree zero in the coordinates A, . . ., F of the curve F; its value is real only for curves of the elliptic type (or for (53 = 0). For an ellipse, it is equal-to the area bounded by the ellipse; this can be verified by reducing the equation of the ellipse to its canonical form. 1.5

We see from the above that the study of all the geometric figures under consideration by analytic methods with the aid of Cartesian coor-

in


CHAP. I

dinates reduces to the following: allotting to a geometric figure a certain system of numbers, its orthogonal Cartesian coordinates, we construct first of all formulae which describe the law of transformation of these coordinates for the transition from one system of coordinates to another. Then we select those functions of the coordinates of the figure in question and those equations interrelating these coordinates which are invariant with respect to the stated transformations. This gives us, under suitable conditions, the geometrical properties of the figure under consideration. As a generalization of this method we will now introduce the concept of the geometric object (Veblen, 1926). *)

Let there be given N variables which are functions of a Cartesian system of coordinates; this means that for any such specified coordinate system these variables assume definite numerical values. On a change of the coordinate system, these numbers, generally speaking, vary; we will demand that the values of the variables, corresponding to the new system of coordinates. are themselves only functions of their values corresponding to the old system and of the parameters of the coordinate transformation. Then these variables define a geometric object with respect to transformation of Cartesian coordinates or, more briefly, a Cartesian geometric object. The values of the variables which correspond to a given system of Cartesian coordinates are called the coordinates of the geometric object in this coordinate system. The number N of coordinates, the law of their transformation for transformations of Cartesian coordinates and likewise the conditions under which two geometric objects are assumed to be identical determine the nature of a geometric object. It is, important to note that for transition from a first coordinate system

to a second, and then from the second to a third, one must obtain, of course, the same result as for direct transition from the first to the third system. This requirement imposes a very strong limitation on the choice

of transformation laws of the coordinates of a geometric object. (Cf. Exercise 12). The study of geometry by analytical methods reduces to the search for functions of the coordinates of different types of geometric objects which *) Cf. 0. Veblen aiid J. H. C. Whitehead: The foundations of differential geometry, Cambridge Univ. Press, 1932: J. A. Schouten and van Dantzig: Was ist Geometrie? (Proceedings of the seminar on vector and tensor analysis, No. 1--111, 1935). We give here a definition of a geometric object in a somewhat narrow form, allowing for the needs of the theory of algebraic invariants.

1.5

TRANSFORMATIONS 'OF CAttFEMAN COORDINATES

II

are invariant with respect to coordinate transformations and for relations

between these coordinates which also have this property. From this point of view, a vector, a straight line, a system consisting of a straight line and a point, a curve of second order, etc., are geometric objects. In considering the transformation law of the coordinates of the straight line [cf. (1.9)], we have selected a new geometric object with only two coordinates A and B which transformed in the same manner as the components of a vector. Since these coordinates are homogeneous, they do not determine, in essence, a vector, but the direction of a vector, and thereby the direction of a straight line (where the term direction has a somewhat different meaning than in other cases: two collinear vectors are

assumed to have the same direction). In studying the problem of the angle between two straight lines and the conditions for their being parallel

and perpendicular, we are concerned with this geometric object. Similarly, in studying the formulae (1.16), we selected three geometric objects. The first is determined by the three coordinates A, B, C (where the

new coordinates A', B', C' are expressed only in terms of A, B, C and the parameters of the coordinate transformation). This object corresponds to the set of all those second order curves in the equations of which the three coefficients A, B, C are the same. In the general theory of second order curves, we study the properties of exactly this geometrical object, investigating, for example, the problem of the asymptotic directions of second order curves and of their principal directions, since all the corresponding formulae involve only the coefficients A, B, C. The invariants

I and 62 are the invariants of this first geometric object. The second geometric object connected with second order curves is determined by the five coordinates A, B, C, D, E; it corresponds to the set of all second order curves whose equations differ from each other only by constant terms. In the theory of the centres, diameters, and asymptotes of second order curves, we are dealing with this second geometric object. Finally, the third geometric object, determined by the six coordinates A, B, C, D, E, F is the second order curve itself. Therefore in studying the first and second geometric objects, we study also the third geometric

object. In its pure form, the third geometric object is considered, for example, in the theory of tangents and poles.

CHAP. I


12

Exercises

1. The equations (1.4) are invariant with respect to those transformations (1.1) for which h = k = 0 (i.e., for rotation of the coordinate axes about the origin 0). Consequently, the equations (1.4) express a property of the points M,, M,, M3 with respect to the origin of coordinates; what is this property? 2. X and Y are orthogonal Cartesian components of the vector a; in what manner does the expression X+iYchange under transitions from one system of coordinates to another? Does the equality X+iY = 0 express a geometric property of the vector a? 3. Let A, B, C be coordinates of the straight line g and (x,, y,) and (x,, y2) the coordi-

nates of the points M, and M:. What is the geometric meaning of the invariant Ax, I-By,+C By, (- C

Ax2

4. How must one measure the angle tp between the straight lines g, and g, in order that (1.11) and (1.12) will apply? 5. Find the geometric meanings of the invariants ()2/(712-46,) and [cf. (1.17)]. 6. For the cases of the ellipse and hyperbola find the geometric meanings of the invariants 6,2/6,3 and (rib,)/b2s [cf. (1.17)]. 7. Given the circle

A(x2±y2)+2Dx -2Fv+ F

0.

A

O,

(1.18)

the expression A(xo2-}-yo2)-t-2Dxo+2Eyo4 F

'q == 2A

11/ 2

is a joint invariant of the circle (1.18) and the points M(xo, yo) which is homogeneous of degree zero with respect to the coordinates of the circle. What is its geometric interpretation?

8. Let A, B, . . ., F be the orthogonal Cartesian coordinates of the second order curve r (§ 1.4). What are the changes under coordinate transformations in the expressions

p - Ax-1-Bv , D,

q = Bx I Cy+E,

(1.19)

where x, y are orthogonal Cartesian coordinates of the point M? 9. Let 2., a be the coordinates of the vector u; by the results of Exercise 8, Ap+,uq (cf. (1.19)] will be a joint invariant of the vector u, the point M and the line T. What is the geometric meaning of the equality 2.p+yq = 0? 10. What is the geometric interpretation of the invariant )s 2

622(p2.4 q2)

[cf. (1.17) and (1.19)]? 11. Let A, B, ..., F be the orthogonal Cartesian coordinates of the second order curve F; what are the changes in A -C-t 2Bi for a transformation of the coordinates? What is the geometric meaning of A -C+2Bi - 0 for real A, B, C? Cf. § 7, Exercise 15. 12. A Cartesian geometric object has only one coordinate 1. which changes in accordance with the law 2.' (p)., where q- is a continuous function of the parameters of the transformation. What must be the form of this function?

2.1

GROUP OF MOTIONS

13

§ 2. The group of motions

The formulae (1.1) can be studied from another point of view which is deeper and leads to important generalizations; it is convenient for this purpose to interchange the dashed and undashed symbols so that one obtains 2.1

x' = x cos w-y sin w+h, y' = x sin w+y cos w+k.

(2.1)

We will assume that in the formulae (2.1) (x, y) and (x', y') are the coordinates of two different points M and M' referred to the same Cartesian coordinate system. Then (2.1) relates to every point M of the plane another point M' of the same plane (where we will write briefly: M-+M'). This correspondence is single-valued and invertible: given the coordinates (x, y) of a point M, we find from (2.1) in a unique manner the coordinates

(x', y') of the point M'; conversely, from the given coordinates (x', y') of the point M' we can find by use of the same formulae in a unique manner the coordinates (x, y) of the point M, since the value of the determinant

cosw -sinw Isin w

coswI - 1

is non-zero. Thus, in this new interpretation, the formulae (2.1) determine a single-valued and invertible transformation of the plane.

Fig. 4

If we again introduce a new system of Cartesian coordinates X'O'Y' (Fig. 4) into consideration and compare the formulae (2.1) with the for-


14

CHAP.

I

mulae (1.1), we note that the point M' has the same coordinates (x, y) with respect to the new system as the point M has with respect to the old system. It follows from this that the transformation of the plane, given by (2.1), can be described in the following manner. We imagine that the plane, considered as a rigid body, slips along itself and that as a result

of its motion the coordinate system XOY comes into coincidence with

X'O'Y'. Then after this motion every point of the plane M(x,y) is matched to a point M', related to it by (2.1). For this reason the transformations (2.1) are called motions; in these geometric motions, only the

initial and final positions of the moving plane are taken into account, its intermediate positions being inessential from the point of view of geometry.

Now, it is obvious that the motions (2.1) have the group property: if S, and S2 are two motions, corresponding to the parameter values w, h, k and w', h', k', the resultant transformation S3 = S2S which is obtained by performing on the plane the transformation S1 and then on the transformed plane the transformation S2, will likewise be a motion. In order to verify this, one need only substitute into the equations of the

motion S2

x" = x' cos w' - y' sin w' + h', y" = x' sin w'+ y' cos w' + k', the values x' and y' from the formulae (2.1) for the transformation S1. This analytical verification shows that the motion S3 = S2 S1 corresponds to the parameter values co"

= w+w'h" = h Cos w'- k sin w'+ h', k" = h sin w'+k cos w'+k'.

If S. is a motion, then also the inverse transformation S`', by which every point M' corresponds to the point M which the motion S transforms

into M', is likewise a motion; if w, h, k are the valus of the parameters for S, the motion S-1 will be given by the parameter values

co' = -w,

h' = -h cos w-k sin co,

k' = h sin w-k cos w.

(2.3)

This is verified by solving (2. 1) for x and y. The identity transformation E

(i.e., the transformation for which M - M) is likewise a motion (with w = h = k = 0). There is no need to verify the associative property, since it is true for any three transformations: if one has by Sl that M-+M',

2.1-2.2

GROUP OF MOTIONS

15

by S2 that M' -- M", by S3 that M" -+ M"', both of the transformations S3(S2S1) and (S3S2)S1 will relate M to M"', so that S3(S2 S1) = (S3 S2)S1 .

(2.4)

Thus, the system of transformations (2.1) satisfies all the postulates for a group and is therefore a group; it is called the group of motions of the plane. The transformation (2.1) depends on three parameters co, h, k, so that the group of motions of the plane is three parametric.

The equations (2.1) determine the motions in a given coordinate system; what are the changes in the equations defining a given motion S for transition from one coordinate system to another? Let the motion S, transforming the point M(x, y) into the point M'(l;, q), be obtained in the coordinate system XOY for parameter values co, h, k; the transition from the coordinate system XOY to the system X'O'Y' is determined by (1.1) with the parameters cu, h, k; the motion with these parameter values will be denoted by T. Let (x', y') -+ (x, y) for the transformation T, (x, y) - q) for S, q) q') for T-'; consequently, (x', y') -* (c', q') for T-1ST. Thus, the transformation Sin the new coordinates is

given by the same formulae as the transformation T 'ST in the old coordinates. One and the same motion in different Cartesian coordinate systems is described by the same formulae (2.1), with diferent values of the parameters. It is clear that this result is related to the group character of the trans-

formation (2.1), and likewise to the fact that the formulae for the transition from one Cartesian coordinate system to another are identical to the transformation formulae for the group of motions. 2.2 The formulae (1.9) can also be interpreted in a different manner. Exchange the dashed and undashed symbols; then in these formulae A', B', C' will be the coordinates of the straight line into which the

straight line with coordinates A, B, C is transformed by the motion (2.1). It is readily seen that such transformations of straight lines also form a group. Thus, if the transformation S of points and the transformation S of straight lines correspond to the same values of the parameters co, h, k, and S3 = S2 SI , then also S 3 = S2 S1; in other words, these two groups are homomorphic. The same type of reasoning can also be applied to the formulae (1.6) and (1.16). The results of § I can now be interpreted in a different manner: all the


16

CHAP. I

invariants and invariant relations considered there will obviously remain

unchanged for the transformations (2.1); therefore they are called invariants of the group of motions and invariant relations with respect to this group. Thus, we see that the transformations of the group of motions do

not change the distance between two points, the orientated area of a triangle, the scalar and vector products of two vectors, the angle between two straight lines; they also transform parallel lines into parallel lines, points lying on one straight line into points with this same property, etc. Just as for straight lines, vectors and second order curves, the formulae for coordinate transformations of any Cartesian geometric object under

a transformation of the coordinate system can be interpreted as transformation formulae for its coordinates as a consequence of a motion; therefore we will call these objects geometric objects with respect to the group of motions. The remark made in 1.5 with regard to the limitations to be imposed on the equations for the transformations of the coordinates of an object

can now be given a clearer form: the coordinate transformations of a Cartesian geometric object must form a group which is homomorphic to

the group of transformations (1.1). Exercises 1. Show that the transformation which relates every point of the plane to the point symmetric to it with respect to the point C(xo, yo) is a motion, and find the corresponding values of the parameters. 2. Construct the formulae for the transformation of the parameters w, h, k of the motion S under transition from one Cartesian coordinate system to another. 3. Find the points which remain fixed for the motion (2.1) (the fixed points of the transformation). 4. Show that every motion is either a parallel translation or a rotation. 5. Verify by direct calculation that each of the groups (1.6) and (1.9) is homomorphic

to the group of transformations (1.1). 6. Find vectors, directions (in the sense of § 1.5) and straight lines which remain fixed under the motions (2.1). 7. Find the second order curves which remain fixed under the motions (2.1). 8. A motion also represents a Cartesian geometric object; verify that the transformations of this object for transition from one Cartesian coordinate system to another (ef. Exercise 2) form a group, homomorphic to the group (l.1).

§ 3. The group of motions and reflections; the principal group We have seen in § 2 that the fundamental quantities and geometric relations studied in elementary (Euclidean) geometry are invariants with 3.1

2.2-3.1

GROUP OF MOTIONS AND REFLECTIONS. PRINCIPAL GROUP

17

respect to the group of motions. Two other groups are also very essential

for Euclidean geometry; we will now proceed to their study limiting ourselves again, for the sake of simplicity, to the plane case. In many problems of elementary geometry in two dimensions the concept of symmetry with respect to a straight tine is basic; this concept is linked in an essential manner to the transformation relating every point M of the plane to a point M', symmetric with respect to a given straight line g. This transformation is called a reflection with respect to the straight line g, and the straight line is called the axis of reflection. A reflection with

respect to the x-axis is determined by the formulae

V:xx' y = -Y.

(3.1)

The formulae for a reflection with respect to an arbitrary straight line in Cartesian coordinates are readily found (Exercise 2). Reflections on their own do not form a group, since the product of two reflections will not be a reflection. This can be verified by means of very simple examples (Exercise 1). However, we obtain a group if we combine motions and reflections, i.e., if we consider the system of transformations each of which

is a motion or a reflection or a product of a reflection by a motion. This group consists of transformations of the form

x' = x cos w-Ey sin co+h, y' = x sin w+Ey cos co +k, E = ±1, and is called the group of motions and reflections (Exercise 3). For

(3.2)

1,

the formulae (3.2) define a motion [cf. (2.1)]; for e = -1, they define the product of a reflection V with respect to the x-axis [cf. (3.1)] and a motion. Thus, the group of motions and reflections contains the group of motions as a subgroup. The invariants of the group of motions and reflections are called metric invariants. It follows from the above that every metric invariant will be an invariant of the group of motions. An invariant of the group of motions will be a metric invariant if and only if it remains unchanged for the transformations (3.1); this follows from the fact that every transformation (3.2) is either a motion or a product of a transformation (3.1) and a motion. Now, it is not difficult to establish which of the invariants of the group


18

CHAP. I

of motions studied in §§ 1, 2 are metric invariants. It is sufficient for this

purpose to demonstrate the changes in the coordinates of a vector, a straight line and a second order curve for the transformation (3.1). Reasoning as in § 1, we see that under the transformation (3.1) the component X of a vector remains unchanged, while Y only changes sign;

for a straight line, the coordinates A and C do not change, while the coordinate B only changes sign; for a second order curve, all coordinates remain unchanged except for the coordinates B and E which only change their signs. Starting from these observations we discover that the distance between two points, the ratio in which an interval is divided by a point,

the length of a vector, the scalar product of two vectors, the angle between two straight lines determined by (1.12), and the invariants rl, (S2, b, of a second order curve are metric invariants; the disposition of three points on a single straight line and the parallelism and orthogonality of two straight lines are invariant relations with respect to the group of

motions and reflections. In contrast, the orientated area of a triangle [cf. (1.3)] is not a metric invariant, since it changes sign for the transformation (3.1); the expression (1.3) is a metric invariant if we take its absolute value, i.e., if we proceed from the orientated area to the usual area without sign. The same is also true for the vector product of two vectors and for the orientated angle between two straight lines defined by (1.11). We can say that on transition from the group of motions to the group of motions and reflections we lose the invariance of those properties of figures which are related to the concept of the orientation of the plane. 3.2 The second of the groups mentioned above is the principal group.

It consists of the transformations x' = p(x cos w-ey sin w)+h, y' = p(x sin w+Fy cos w)+k,

F=±1, p00.

(3.3)

The simplest of the transformations (3.3) is obtained for e = 1,

w=h=k=0:

To:

x' = px, y' = py;

p00.

It represents a homothetic transformation with centre at the origin of coordinates and with coefficient of homothety p. Each of the transforma-

3.1-3.2

GROUP OF MOTIONS AND REFLECTIONS. PRINCIPAL GROUP

19

tions (3.3) has the form

U=STo,

(3.5)

where S is one of the transformations (3.2), i.e., it is a motion or the product of a reflection V and a motion. Hence we conclude that the transformations (3.3) are similarity transformations. Taking into consideration the results of § I and (3.5), we see that it is sufficient for a study of the effects of the transformations of the principal group on the expressions and relations of §§ 1, 2 to investigate the changes in them due to the transformation (3.4). For this purpose we must take into account the readily established fact that the components of a vector change under the homothetic transformation TP in the same manner as the coordinates of points; the first two coordinates A and B of a straight

line, as a result of the transformations (3.4), are multiplied by p-', while the third coordinate C remains unchanged. The first three coordinates A, B, C of second order curves are multiplied by p-2; the following two, D, E,' by p-; and the last, F, does not change. In this manner, we discover that the expression (1.2) is multiplied by p2 under the transformation, while the expression (1.3) gains the factor ep2. An expression which for any transformation acquires nothing else but a multiplier which only depends on the parameters of the transformation is called a relative invariant of the transformation; an expression

which does not change at all under the transformation is in contrast called an absolute invariant. Thus, with respect to the principal group, the length of a segment and the area of a triangle will not be absolute invariants, but only relative invariants; we obtain absolute invariants, if we select the ratio of the lengths of two segments and the ratio of two areas.

One can proceed in an analogous manner with the expressions (1.7), (1.10), (1.14) and (1.17); all of these are relative invariants of the principal

group; only the expression (1.12) is an absolute invariant. The relation (1,5) and the expressions (1.7), (1.10) and (1.17), when they are set equal to zero, retain their invariant character with respect to the principal group also. Thus, the similarity transformations do not preserve lengths and areas, but only the ratios of lengths and areas; the rectilinear disposition of points, parallelism of straight lines and the angle between two straight lines are concepts which are invariant with respect to the principal group.

20


CHAP. I

3.3 The results of §§ 2, 3 lead us to the study of Euclidean geometry from the point of view of group theory *). Its basis is the three groups stated

above: the group of motions, the group of motions and reflections, the principal group. Euclidean geometry studies those relations between figures and those quantities connected with them which are invariant with respect to one of these groups. The fundamental concepts of Euclidean

geometry are equality and similarity which are closely related to these groups. Equality is equivalence with respect to the group of motions and reflections: two figures are equal if there exists a transformation of the form (3.2) which transforms one into the other. Similarity is equivalence

with respect to the principal group: two figures are similar, if there exists a transformation which belongs to the principal group and trans-

forms the one into the other. Starting from this point of view of Euclidean geometry, F. Klein states

in his Erlangen Program a concept which leads to a broad view of the development of geometry: by means of geometric studies, one can establish any group of transformations and create the geometry of that group of transformations. A study of very simple examples realizes this general idea and establishes the objectives of further investigations. Exercises

!. The transformation S, is a reflection with respect to the straight line y = 0, the transformation S, a reflection with respect to y = b. Find the equations of the transformations S, S, and SI S, . 2. Construct the formulae for reflection with respect to the straight line Ax+By+C = 0 in Cartesian coordinates. 3. Verify by direct computations that the system of transformations (3.2) represents a group. 4. Show that the product of any reflection and any motion, and likewise the product

of any motion and any reflection, is always a transformation of the form (3.2). 5. Find the points, vectors, directions (in the sense of § 1.5) and straight lines which remain unchanged under the transformations (3.2) for e I (the case e = I has been considered in § 2, Exercises 3 and 6). 6. Prove that for e = -1 each transformation (3.2) is either a reflection or a reflection combined with a parallel translation the direction of which is parallel to the axis of

reflection (cf. § 2, Exercise 4). 7. Find the second order curves which remain unchanged under the transformations

(3.2) for e = -1 (cf. § 2, Exercise 7). 8. Verify that the system of transformations (3.3) is a group. *) The first person to adopt such a point of view was F. Klein (1849-1925) in his so-called Erlangen Program 12).

3.3-4.1

AFFINE TRANSFORMATIONS OF THE PLANE

21

9. Verify that the set of all the transformations (3.3) with F = I likewise forms a group. 10. If p is real in the equations (3.3), it can always be assumed to be positive (otherwise this may be achieved by replacing w by w 1-n); find the fixed points of the trans-

formations (3.3) for p > 0, p # 1. 11. Show that for real p . I each transformation of the principal group is either homothetic or the product of a homothetism and a rotation about the centre of homothety (centre of transformation) or the product of a reflection with respect to a straight line (axis of transformation) and a homothetism with centre on the axis of the transformation. 12. Find the vectors, straight lines, directions (in the sense of § 1.5) and second order curves which remain unchanged under the transformations (3.4), if p # I is real. 13. What are the changes in the metric invariants of § 1, Exercises 5 and 6 for transformations of the principal group? 14. As is shown by the result of § 1, Example 5, b, _ e2 1

(3.6)

e4

712--46s

where e is the eccentricity of a second order curve; if 6, < 0, then (3.6) gives two values for e. Explain the reason for this ambiguity. 15. Show that all homothetisms (with different centres and different coefficients of homothety) form, together with parallel translations, a group.

§ 4. Affine transformations of the plane 4.1 An affine transformation S of the plane is defined by the equations S:

x' = px+qy+h, y = rx+sy+k,

(4.1)

where it is assumed that the determinant As = p r

q

# 0.

(4.2)

s

In the equations (4.1), (x, y) and (x', y') are the coordinates of two points M and M' in the same Cartesian coordinate system. The affine transformation S relates the point M to the point M'; by (4.2), the correspondence established by (4.1) is mutually single-valued. Let T be a second affine transformation given by

x" = p'x'+q'y'+h', = r'x' +s'y'+k',

y

p'

q'

(4.3) r' s' # 0. Substituting the values x' and y' from (4.1) into the equations (4.3), we obtain the equations for the transformation TS:

TS:

AT

x=

Y" = r"x + s"y + k",

I

(4.4)

CHAP.


22

I

where

p" = pp'+rq', r" = pr'+rs',

q" = qp'+sq', s" = qr'+ss',

h" = hp'+kq'+h', k" = hr'+ks'+k'.

It follows from (4.5) that ATS =

pp'+rq' pr'+rs'

qp'+sq' _ qr'+ss'

p

q

r

s

so that Ills 0 0, and TS is likewise an affine transformation. The identity

transformation E is obtained from (4.1) by setting p = s = 1, q = h = r = k = 0. Solving (4.1) for x and y, we verify that together with S, S -1 is also an affine transformation, where AS

'JS-

= 1.

Consequently,

Theorem 4.1; The set of all affine transformations of the plane is a six parameter group which is called the affine group of the plane. We note that the groups considered in §§ 2, 3 are obviously subgroups of the affine group. A simple calculation shows that the relations (1.5) are invariant with respect to the affine group: if the coordinates of the three points M1, M, , M3 are interrelated by the equations (1.5), the coordinates of the points M1, MZ, M3 which correspond to them by the transformations (4.1), will be interrelated by the same equations. Thus, three points M1, M,, M31 lying on a straight line, are transformed by an affine transformation into points M', M2, M3 which likewise lie on a straight line, and also the ratio A in which M3 divides the segment M1 M2 remains unaltered. Since the sign of the number A shows whether the point M3 lies inside or outside the segment M1 M2, we see that under an affine transformation of the plane the points of a straight line g are transformed to points lying on a certain straight line g', and points of the segment M1 M2 to points

of the segment M1 M. Consequently, for an affine transformation, a vector transforms into a vector. Since the components of a vector are equal to the differences between the coordinates of its end points, the formulae for the transformation of the components of a vector differ in the case of an affine transformation of the plane from the formulae (4.1) only by the absence of the free terms h and k.

4.1-4.2


23

The formulae defining a given affine transformation depend on the choice of the Cartesian coordinate system; it is readily seen what changes take place in them for a transition to a new Cartesian coordinate system. This transition is accomplished by means of the equations (1.1): if we denote the affine transformation determined by these formulae (with x, y and x', y' interchanged) by T, we verify by reasoning analogous to that at the end of 2.1 that in the new coordinates the transformation S is given by the same formulae which in the old coordinates defined the transformation T` ST. By the group property this is likewise an affine transformation. Thus, on transition to a new Cartesian coordinate system, the formulae (4.1) for the transformation S retain their form, i.e., only the values of the parameters p, q, h, r, s, k change. As a consequence the study of the properties of affine transformations can proceed along identical lines in any Cartesian coordinate system. Further, it is seen from this reasoning that nothing will be changed if 4.2

we use complete arbitrariness in the choice of the parameters of the transformation T and call the coordinates of a point any independent, linear, nonhomogeneous functions x', x2 of the Cartesian coordinates x, y of the point (here the superscript is not a power index, but the order number of the coordinate); x', x2 are called the affine coordinates of the point. To every point of the plane there corresponds a definite pair of

numbers, its affine coordinates; conversely, a point is determined by given affine coordinates which, by virtue of the independence of the linear functions, express x', x2 in terms of x, y, its Cartesian coordinates, and consequently by the point itself. The study of the affine properties of a figure, i.e., of the properties which are invariant with respect to affine transformations, is more conveniently performed in affine coordinates,

since under these conditions the equations of affine transformations retain their form and the arbitrariness in the choice of the coordinate system is considerably wider. It is not difficult to explain the geometric meaning of affine transforma-

tions. We will assume that the point x' = x2 = 0 has been selected as origin 0 of the Cartesian coordinate system (such an assumption obvious-

ly does not reduce the generality); then x' and x2 will be independent linear homogeneous functions of the Cartesian coordinates x, y and, conversely, x, y will be linear homogeneous functions of x', x2 X

x.CI +/1x2,

Y = Y-x' +Sx2,

(4.8)

24


CHAP. I

which are likewise independent, so that the determinant a

#

Y

b

(4.9)

# 0.

-4The radius vector OM of an arbitrary point M of the plane is

OM = r = xi+yj, where i, j are the unit vectors of the Cartesian coordinate system; or, by (4.8),

r = (ax'+flx2)i+(yx'+bx2)j = (ai+yj)xI+(fli+8j)x2.

(4.10)

We now introduce the vectors

el = ai+yj,

e2 = fii+Sj,

which, by (4.9), are not collinear. Then (4.10) assumes the form 2

r = x'e,+x2e2 = Ex°e,.

(4.11)

a=1

Thus, x' and x2 are the coefficients in the decomposition of the radius vector of the point M with respect to the vectors e, and e2. The affine coordinate system is determined by the choice of a certain point 0 as origin of coordinates and by the two vectors e, and e2 as coordinate vectors; in contrast to the case of Cartesian coordinate systems, where the

coordinate vectors necessarily have unit length and are orthogonal, in this case these vectors are quite arbitrary [except that they must not be collinear; cf. (4.9)]. The affine components of a vector a are determined in an analogous manner; they are the coefficients in the decomposition of the vector a with respect to the vectors e,,e2: 2

a = a'e,+a2e2 =

a=I

a°e ;

(4.12)

it is readily seen that the affine components of a vector are also equal to the differences between the affine coordinates of its ends. Following Einstein, in expressions of the form (4.11) and (4.12) we will omit the summation signs, i.e., we will follow in future the following convention: if in a product any index appears twice, once as subscript, a second time as superscript, such an index will always imply summation

4.2-4.3


25

over all its possible values. The formulae (4.11) and (4.12) may then be

written in the form

r=x%,

a=1,2.

a=a°e2,

(4.13)

4.3 We will now construct the formulae for the transition from one system of affine coordinates Oe1 e2 to another system O'e1, e2. (Fig. 5).

Fig. 5

The position of the new system with respect to the old system is deter-

mined by knowledge of the vector 00' = h and the expansions of the new coordinate vectors in terms of the old ones: 2

1

Cl, = p1 e1

e2, = p .e1

+P22

'e2 ,

or, using Einstein's summation convention,

e; = p;.e;,

i = 1, 2,

i' = 1', 2'.

(4.14)

Since the vectors e1 and e2, are linearly independent (and not collinear),

the determinant d

FPM3

2

1

pI.

1

Pr

PI, P2'

2

=I Pi'I00.

(4.15)

We have from Fig. 5 that

r=r'+h, i.e., if x', x2 are the coordinates of a point with respect to the old system x2. of affine coordinates and x'" the coordinates of the same point with


26

CHAP. I

respect to the new system, we have x' e; = x'le,,+h

(4.16)

[cf. (4.13)]. Decomposing the vector h with respect to the vectors el, e2 and replacing the vectors e;- by their expressions (4.14), we obtain x'e, = x' p;. e;+h'e;;

(4.17)

comparing the coefficients of the linearly independent vectors e, and e2 on both sides of (4.17), we arrive finally at the relation

x' = p. x`+ h',

i = 1, 2,

i' = 1', 2'.

(4.18)

The reader's attention will now be directed to the fact that in the transition from (4.16) to (4.17), as a result of the application of the Einstein notation, the brackets were removed without special mention. We recommend for the sake of more clarity that the reader perform these calculations in detail without use of the Einstein convention. Analogous arguments lead to the formulae for the transformation of the components of a vector

a' =

'l

p;. a

,

i = 1, 2,

i' = 1', 2',

(4.19)

where a', a2 are the old, a", a2' the new components of the same vector a. The formulae (4.19) can also be derived from (4.18) by noting that the components of a vector are equal to the differences between the corre-

sponding coordinates of its ends. It is very important to compare the formulae (4.14) and (4.19): (4.14) expresses the new components of a vector in terms of the old components, and (4.19) the old ones in terms of the new ones; in addition, in the transition from the right-hand side of (4.14) to the right-hand side of

(4.19), rows become columns (this is immediately clear on expanding these equations). It may be said that the transformations of coordinate vectors and of the components of a vector are, in a known sense, opposites; therefore, in affine geometry, the usual vectors (directed segments) are called vectors (i.e., changing in an opposite way). If we wish to have expressions for the new components of a vector in

terms of the old, we must solve equations (4.19) with respect to a' and a2: a' = q; as,

i = 1, 2,

i' _ 1', 2',

(4.20)

4.3-4.4

27


where t

qt

P2'=e.,

1' q2 =- P2'

PI' =--d,

2'

d,

qt

q2 =

Pill

(4.21)

A

pi', p2., pZ. in the are the reduced minors corresponding to the elements determinant (4.15) *) (here we have to draw attention to the transposition of the indices; the minor corresponding to the element pi. has been denoted by qf). It is readily verified that the matrices IIP;'II and Ilgf II are the inverses of each other; their product is equal to the unit matrix. It follows

from this [or directly from (4.21)] that the determinant

In an analogous manner, solving equations (4.18) for x", x2', we have

x' = q x' + ht ,

i = 1, 2,

i' = 1', 2',

(4.22)

where

h" = -qf'h, it is easily seen that ht',

h2.

i = 1, 2,

i' = 1', 2';

(4.23)

represent the components in the expansion of

the vector O'O = -h in terms of the vectors et-,

e2'-

4.4 As has been stated in 4.2 affine transformations are determined

in affine coordinates by equations of the same type as in Cartesian coordinates. In correspondence with the formulae for transformations of affine coordinates, one usually writes i, a = 1, 2;

x' = Paxa+h',

(4.24)

the determinant

d=IPi Pi 00 P2

(4.25)

P2

is called the determinant of the transformation. We deduce from (4.24) [cf. (4.18), (4.22)): x` = q' xa+h',

i, a = 1, 2,

(4.26)

where qa, h' are determined by formulae analogous to (4.21) and (4.23). *) The reduced minor of an element of a determinant is the factor of that element (the algebraic complement) in the expansion of the determinant.

28

CHAP. I


In (4.24) and (4.26), x', x2 are the affine coordinates of the original point, x', z2 the coordinates of the point obtained as a result of the transformation. Both points are referred to the same system of affine coordinates.

In an analogous manner, if by the affine transformation (4.24) the vector a - a, then the components (a', a2) and (a', a2) of these vectors arc interrelated by the equations a' = pMaa,

i,

1, 2,

(4.27)

a' = qa act,

i, at = 1, 2.

(4.28)

We see immediately from (4.28) that under an affine transformation: 1) equal vectors become equal vectors, 2) collinear vectors become colli-

near vectors whose ratio is preserved: if b = .la, then also b = .la; 3) the sum of vectors corresponds to the sum of the transformed vectors:

if a -+ a*, b - b*, then a+ b - a* + V. It follows from the fact that affine transformations preserve the colinearity of vectors that they also preserve parallelism of straight lines. Consider next the expression

[ab] =

a1

a2

b'

b2

(4.29)

composed of the components a', a2 and b', b2 of two contravariant vec-

tors a, b; we have at a 1

2

a

2

1

a

qa

a

q1

ba

[a b]

2

qa a

a

qa ba

__

2

R1

qI

a at

a

q2

q2 1

bt

b2

2

= d -' [ab].

Thus: Theorem 4.2: The expression (4.29) formed from the components of two

contravariant vectors a, b is a relative invariant of the affine group with

weight -1. This statement means that under an affine transformation the expression (4.29) only acquires a factor which is equal to the determinant of the

transformation to the power (- 1). The ratio of two such expressions will be an absolute invariant of affine transformations. The equality

4.4-4.5


29

[ab] = 0 is invariant for affine transformations; its geometric interpretation is that the vectors a and b are collinear. By (4.8), a straight line is determined in affine coordinates by equations which are linear in the coordinates of a point x', x2 which lies on it. Let a straight line g be given by the equation 4.5

ui x`+h = 0, i = 1, 2.

(4.30)

Under a transformation of the affine coordinates by means of the formulae (4.18), equation (4.30) assumes the form

uip;.x"+uih'+h = 0. Letting

ui. = P;. ui,

i = 1, 2,

h'=u,h'+h;

i' = 1', 2',

i=1,2,

(4.31) (4.32)

we reduce the equation of the straight line to the form

ui.x'l+h' = 0;

i' = 1', 2'.

The formulae (4.31) and (4.32) give the law of transformation of the coordinates u, , u2 and h of a straight line for an affine transformation of the plane. The equations (4.31) show that the first two coordinates ul and u2 of the straight line represent independent geometric objects whose coordinates transform in the same manner as the components of a vector [cf. (4.14)]. Therefore the geometric object ui (i = 1, 2) is called a covariant vector (covariant means here changing in unison). In this context, the components ul, u2 of a covariant vector, in contrast to those of a straight line, are assumed to be non-homogeneous: two covariant vectors

whose components are not equal, but proportional, are different. In order to distinguish covariant and contravariant vectors, we will agree to write the indices in the first case as subscripts, in the second case as superscripts. Similar reasoning leads to the following convention regarding indices: for an affine transformation (4.24), a covariant vector u with coordinates ui is transformed into a covariant vector with coordinates

ui = P;ua,

i, a = 1, 2.

(4.33)

Let a be a contravariant, u a covariant vector. Comparing (4.27) and


30

CHAP. 1

(4.33), we have aù; = a'p.u,, = a°tu.,

i.e.,

Theorem 4.3:

The expression

a'u, = (au) = (ua),

(4.34)

called the scalar product of a contravariant vector a by a covariant vector u, is an affine invariant.

It is not difficult to explain the geometric meaning of a covariant vector is. The straight line g itself with the equation (4.30) does not determine the covariant vector is completely, since the coordinates ul, u2, h of

the straight line are homogeneous and the components of is are nonhomogeneous. Consider vectors which start from an arbitrary point A(x', x2) of the straight line g and which satisfy the invariant relation

(iAM) = 1

(4.35)

(Fig. 6). If the coordinates of the point M are (y', y2), then [cf. (4.34)]

K

Fig. 6

the equality (4.35) assumes the form

u,yl-u,x' = 1, or, since by (4.30) u,x' = -h, u, y'+ h = 1.

(4.36)

Thus, the ends of the vectors AM lie on the straight line g' which is determined by (4.36) and is parallel to the straight line g. To a covariant vector correspond two parallel straight lines g and g', where the order of these straight lines is essential: on one of these straight lines, the vectors


4.5

31

AM begin, on the other, they end. Therefore, in the literature, covariant vectors are also called doublets (after the French word double); g is the starting line of the doublet, g' the finishing line. In our figures, the starting line of the doublet is marked by a broken line, the finishing line by a solid line. By changing h in equations (4.30) and (4.36), we obtain other pairs of

straight lines which likewise correspond to the covariant vector u; such pairs of straight lines are derived from each other by parallel translations. Thus the equality of covariant vectors has been defined. The equality (4.35) will not be violated if we multiply both components u1, u2 of the covariant vector u by an arbitrary number A and divide the

vector AM by this number. The covariant vector with components u1 A, u2 A is called the product of the covariant vector u and the number A.

In accordance with the preceding observation, the covariant vector uA = Au is constructed in the following manner: leaving the initial straight line g of the covariant vector u unchanged, we draw from an arbitrary point A of this straight line a vector AM, where M is some point

on the finishing line g' of the covariant vector u. The straight line g", parallel to g' and passing through the point N, determined by

AN =AM --, A

will be the finishing line of the covariant vector uA. Reverting to the scalar product of the covariant and contravariant vectors u and a, we will now establish its geometric significance. Draw the vector AK = a (Fig. 6) and construct through the point A the starting line g of the covariant vector u. Let the finishing line g of the covariant

vector u intersect AK at the point M and let AK

AM Then, by (4.35),

(uAM) = I,

(u AK) A

=

I;

(u K) = 1,.


32

CHAP. I

Thus

AK

(4.37)

AM This geometric interpretation of the scalar product of the covariant and contravariant vectors u and a loses its meaning if au = 0. However, the vector a is then parallel to the straight lines of the covariant vector: if the coordinates of the origin of the vector a satisfy the equation (4.30), the coordinates of its ends also satisfy this equation.

Fig. 7

We call the sum of two covariant vectors u; and v; a covariant vector with components u;+v;; the invariance of this expression with respect to the transformations (4.33) is obvious. It is not difficult to give the geometric construction for a sum of covariant vectors (for the case when their starting and finishing lines are not parallel). By the above result

(uOA) = 1;

(vOA) = 0,

so that

((u+v)OA) _ (u0A)+(vOA) = 1; in a similar manner we can show that ((u+v)OB) = 1. Consequently, if the starting line of the covariant vector u+v passes

4.3.4.6


33

through the point 0, its finishing line must pass through the points A and B. 4.6 Since Cartesian coordinates are expressed in a linear manner in terms of affine coordinates, second order curves F are determined in affine coordinates also by second order equations in x1 and x2: aI J(x,)2+2a,2 x'x2+a22(x2)2+2a1 x' +2a2 x2+h = 0,

or, in abbreviated form,

aijx'xj+2aixi+h = 0,

aij = aji

(4.38)

(since on summation the indices i and j assume independently from each other the values 1, 2, both the terms a, 2 x' x2 and a21 x2 x' will occur and, by the equality a, 2 = a21, give together the term 2a12 x' x2). The affine transformation (4.24) transforms the curve f into the curve r with the equation aij pa pp'xaXa+ aij pa xahj+ aij hjpa

za

+aijhhj+2aipaza+2aih'+h = 0. On the left-hand side of this equation, the second and third terms are identical: obviously, it is of no consequence what symbols are used for the indices with respect to which summation occurs; hence

aijh'p'xa = ajihjp2xa = aijpaXahj,

(since ail = aji). If we now let i

j

aaB = pa pp ai j ,

.

.

a9 = asa

as = pa hjai j ±- pa ai, h

(4.39)

= aijh'hj+2a,h1+h,

the equation of the transformed curve assumes the form

aap 1T +2aaza+h = 0; for an affine transformation of the plane, a second order curve goes over into a second order curve. The equations (4.39) show the changes in the six affine coordinates all, a12 = a21, a22, a,, a2, h of the second order curve resulting from the affine transformation (4.24). According to the first equation (4.39), the first three coordinates a11, a12 , a22 define an independent geometric

CHAP. I


34

object; another geometric object is defined by the first five coordinates

ail, a12, a22, al, a2 (cf. 1.5). Among the three metric invariants of a second order curve, the first, a,, +a22, ceases completely to be an invariant; this is verified by the example of the very simple transformation: z' = 2x', 12 = x2 which gives a +a22 = 2611 +622; the other two invariants all

a12

a21

a22

laijl,

b2 =

63 =

all

a12

al

a21

a22 a2

a2

a,

(4.40)

h

remain relative invariants of weight 2 in the case of transition to an affine transformation. In fact (multiplying the determinants by the rule:

row by column), one has

d2=laijl= aijp,Al aij Pz pi Pi

P1 2 I I P2 P2

I

Pi

ail p,

a;1 Pj

P2

P2

ai2 p1

ai2 P2

P1

all

a12

P1

P2

a21

a22

P1

P2

1

2

(4.41)

aijPi h'+aiPi aijp' h'+aiPi aij p2 Pi aij p2 P2 aijPi h'+aip, a,,p2h'+aiPi aijhihj +2aih'+h aijPi

153=

pi

I

aij pl P2 aijPi P'2

aijPi P2

a,1P2

i ailh+a,

ai2

ai2 P2

ai2

ai p, i

ai p2

ai h' +h

I

i

0 0

ailPl

1

Pi

0

all

a12

a1

Pi

P2

P2 h2

0

a21

a2

Pi

P2

1

a,

a22 a2

It

0

0

P;1

h'+a2

h'

h2 1

Since 62 and S 3 are relative invariants and the coordinates a , a, 2, a22 , a,, a2, h of a second order curve are homogeneous, geometric meanings

can only be attached to the equalities S2 = 0 and S3 = 0. The first of these denotes that the second order curve T is of the parabolic type, the second that I' represents two straight lines. In the case of real coordinates

a , al 2, ... , h and real affine transformations, the inequalities S2 > 0 and S2 < 0 also have geometric significance (cf. 2.4): if a2 > 0, r is of the elliptic type, if 62 < 0, it is of the hyperbolic type (cf. Exercise 13).

4.6-4.7


35

Following the ideas of F. Klein (3.3) we define now an affine geometry. i.e., the geometry of the affine group of transformations. We call an affine property of a geometric figure a property which is 4.7

invariant with respect to affine transformations; we call a quantity which is connected with a geometric figure affine if it is invariant with respect to

the transformations of the affine group. In the plane case considered here, the following properties have been shown to be affine: points lying on a straight line, a point M3 lying between two points Ml and M2 (i.e., on the segment M, M2), two (contravariant) vectors being equal or two such vectors being collinear, a second order curve being of the parabolic type, a second order curve representing a pair of straight lines, etc. Two straight lines being perpendicular and the equality of two nonparallel lines are examples of properties which are not affine. The ratios

of the lengths of two parallel segments and of two collinear vectors, the scalar product of a covariant and a contravariant vector, the ratio of the orientated areas of two parallelograms (Exercise 5) are affine quantities. The length of a segment, the angle between two straight lines, the scalar product of two contravariant vectors are not invariant with respect to affine transformations (Exercise 1) and therefore they are not affine quantities. Affine geometry studies the affine properties of figures and acne quantities which are related to them. All the concepts of affIne geometry must refer to invariant character with respect to the affine group; only such ideas may play a role in the affine geometry proposed here. On the basis of the above statements, the following concepts will be affine: the straight

line, the segment of a straight line, the centre of a segment, parallel straight lines, the triangle, the quadrangle, the parallelogram, the trapezoid, the sum of two vectors, parallel translation, the centre of a second order curve and its diameter, the conjugate to a given direction (since these two ideas are defined by the concepts of the centre of a segment and of parallel straight lines), the asymptotic direction of a second order curve (Exercise 12), etc. Equality of non-parallel segments, perpen-

dicularity of two straight lines, the property of the sides of a triangle being equal, the right-angled triangle, the rhombus, the rectangle, the circle, rotation about a point, the axes of a second order curve are concepts which are not affine, since they are not invariant with respect to the affine group; in affine geometry, this type of concept loses its meaning. As a very simple example of a proposition of affine geometry one has

36


CHAP. I

the theorem that the medians of a triangle intersect in a point and are subdivided by this point in the ratio 2 : 1, since this formulation involves

only affine concepts. If KLIIBC (Fig. 8), then it is known that AK

AL

AB

AC

This is an affine proposition: each part of the equality involves the ratios of parallel segments, i.e., affinc concepts. However, if we write the same

relation in the form AK

AB

AL

AC

it loses its affine character, since in affine geometry the ratio of two segments which are not parallel has no meaning.

Fig. 8

The concepts of equality and similarity in elementary Euclidean geometry are replaced in affine geometry by the concept of affine equivalence. Two geometric figures are called equivalent in an affine sense, if there

exists an affine transformation which transfers the one figure into the other; from the point of view of affine geometry, such figures do not differ

from each other: all their properties (meaning, of course, the affine properties) are identical. It is readily shown (Exercises 2, 3) that all triangles are affinely equivalent; the same statement holds true for all parallelograms.


4.7

37

In order to give these results a purely analytic form, we will introduce

the concept of an affine geometric object (cf. 1.5): N functions of an affine coordinate system define an affine geometric object, if in the case of transformation of the affine coordinate system their values corresponding to the new system are expressed only in terms of their values corresponding to the old system and of the parameters of the transformation. The numerical values of these functions, corresponding to a given affine

coordinate system, are called the coordinates of the geometric object in this coordinate system. It is readily seen that for all possible values of the parameters of the transformations the coordinates of the object must form a group which is homomorphic to the affine group. In the case of an affine transformation, the coordinates of an affine geometric object must vary by the same law as in the case of a transformation of affine coordinates. In affine geometry, one is concerned with the study of functions of the coordinates of affine geometric objects and of relations between these coordinates which are invariant with respect to affine transformations.

So far we have adopted the point of view of affine geometry in the real domain, i.e., we have assumed that all the coordinates of the geometric

objects as well as the parameters of the affine transformations are real. If we follow a procedure frequently adopted and assume the coordinates of the geometric objects and the parameters of the affine transformations to be complex, we will be dealing with affine geometry in the complex domain, which differs in certain respects from the geometry in the real

domain. For example, in complex affine geometry, the distinction between second order curves of the elliptic and hyperbolic types loses it meaning (since it is impossible to speak of the sign of S2); only the differ-

ence between central and non-central curves retains its significance (Exercise 14). Exercises 1. Let there be given in Cartesian coordinates the transformation: x' = x, y'=x+ y; under this transformation, what correspond to the points M1(0,0). M,(l, --! ), to the vectors a (1, 1) and b (1, -1) and to the straight lines g, and g, with the equa-

tions x = 0, x+y= 0? Find the distance M1M,, the scalar product ab and the angle between the straight lines g, and g, and compare them with the corresponding quantities for the transformed points, vectors and straight lines. 2. Show that there exists one and only one affine transformation of the plane for which three points A, B, C go over into three given points A', B', C', where the arbitrariness in the choice of these six points is only limited by the requirement that the points

A, B, C as well as the points A', B', C' must not lie on a straight line.

38

GEOMETRIC INTRODucrION

CHAP. I

3. Show that any two parallelograms are affinely equivalent; explain the conditions under which two trapezoids will be affinely equivalent; answer the same question for the case of any two quadrangles ABCD and A'B'C'D'. 4. Show that the determinant A . ip,,j of a transformation of affine coordinates (cf. (4.18)) is equal to the ratio of the orientated areas of the new and old coordinate parallelograms: coordinate parallelograms are parallelograms based on the coordinate vectors e, and e2 starting from the origin of coordinates. 5. Show that the expression [ab], formed from the components of two contravariant vectors a, b [cf. (4.29] is equal to the ratio of the orientated area of the parallelogram

constructed on the vectors a, b drawn from one point, to the orientated area of the coordinate parallelogram. Hence it follows that the ratio of two such expressions is equal to the ratio of the orientated areas of two parallelograms. 6. Show that the expression S'

x2 y2 Z2

(4.42) 1

t

composed of the affine coordinates of the three points x, y, z [(cf. (1.3)] is a relative invariant of weight -- I and that it is equal to the ratio of the orientated areas of the triangle xyz and the coordinate triangle. What is the geometric meaning of the vanishing

of the expression (4.42)? 7. The covariant vectors e', e2, related to the coordinate vectors e,, e2 by

(e1e') -- (e2e2) = 1, (e,e2) _ (e2e') = 0 are called covariant coordinate vectors; what are their components? Indicate the covariant coordinate vectors on a sketch showing the coordinate vectors starting from the

origin of coordinates. Show that the covariant vector u with components ur, u2 is equal to u,e'+ u2e2. 8. Show that the expression u,

[UV]

i

r1

u, ! "2 1

constructed from the components of the two covariant vectors u and v is a relative invariant of weight -F I and that it is equal to the ratio of the orientated areas of the

coordinate parallelogram and the parallelogram bounded by the intial and final straight lines of the two covariant vectors. 9. For the case of an ellipse [O, 0, 6, > 0, cf. (4.40)] show that the expression 1 = nj821/b! is equal to the ratio of the area of the ellipse to the area of the coordinate parallelogram. 10. Let a;, be the first three coordinates of a second order curve and g, .] two contravariant vectors. Show that it, ,1' is an invariant for affine transformations. 11. For the conditions of Exercise 10 prove the identity ati si;i a4l a2

(4.43)

where r`= is the first of the invariants (4.40). 12. A straight line passes through the point x0 and is parallel to a vector ;; study the problem of its intersection with the second degree curve given by (4.38). On the basis of these results find the equations for the determination of the asymptotic directions and the coordinates of its centre, the equations of the diameters, and for 8, A 0

the condition for conjugacy of two diameters. 13. In the case of non-degenerate second degree curves (b, = 0) derive the affine

classification in the real domain.

5.1

PROJECTIVE TRANSFORMATIONS OF A STRAIGHT LINE

39

14. For non-degenerate second degree curves (h3:0) derive the affine classification in the complex domain.

15. Show that the transformations of oblique coordinates (with fixed origin) sin (8-a) sin sin (0-fl) sin f z r +y Y sin 0 -- z sin 0+ Y sin 0 sin 0 -

for all possible values of the parameters 0, a, fi do not form a group.

§ 5. Projective transformations of a straight line

5.1 We now proceed to projective transformations, beginning with the case of the straight line, which makes it possible to demonstrate analogies between projective transformations of the straight line and affine transformations of the plane. A point on a straight line is determined by one of its Cartesian coordi-

nates, say ; a transformation of Cartesian coordinates is given by The same equation also describes motion on the straight line; thus, in this case, motion coincides with parallel translation. Affine transforma-

tions, defined by equations of the form

'=p5+h,

p

0,

coincide with homothetic transformations and consequently the difference between the metric (Euclidean) and affine geometries disappears.

We recognize the role of the straight line as the limit of Euclidean geometry when we study the geometry of the projective group. This group consists of the projective transformations of the straight line each of which is determined by an equation of the form (5.1)

where Ip

q

Ir

s

# 0.

(5.2)

The fact that the transformations (5.1) form a group has still to be verified. However, first of all, one must rid the transformations (5.1) of the fundamental defect that (for r 0 0) they do not have the property of mutual single-valuedness; the point = -s/r of the transformation (5.1) does not refer to any point (since the denominator then vanishes).

CHAP. I


40

On the other hand, it is readily verified that we do not obtain a value of

for ' = p/r. In order to remove this defect of projective transformations we first of all introduce homogeneous Cartesian coordinates for the points of the straight line (cf. the homogeneous coordinates of a straight line and second

order curve of 1.3, 1.4) by letting (5.3)

t * 0.

0 corresponds to a point of the straight line; the numbers Ax, dt (A 0 0) correspond to the same point of the straight line. Conversely, a point of the straight line corresponds to an Each pair of numbers x, t for t

infinite set of pairs of numbers x, t which are proportional to each other; for all these pairs of numbers t : 0. In homogeneous coordinates, equa-

tion (5.1) assumes the form X'

px + q t

x'

t'

t'

rx+st

px+qt

rx+st

or, since x', t' are determined exactly apart from a common multiplier,

x' = px+qt, (5.4) I' = rx +st. For the point = -s/r, one may assume that x = s, t = -r; (5.4) then gives x' = ps-qr :o 0 [cf. (5.2)], t' = 0. Conversely, for x # 0, t = 0, we find from (5.4): x' = px, t' = rx, ' = x'/t' = p/r. Thus, in order to make projective transformations mutually singlevalued, one has to complete the straight line by a single point with homogeneous coordinates: x 96 0, t = 0 (where, as a consequence of the homogeneity, we do not alter the point by changing x). This point (x, 0) with x 0 0 is called an improper point of the straight line. If we let t - 0 in (5.3) with x constant, we find S - o' ; hence the improper point of the straight line is also often called the point at infinity. Thus, in order to construct a geometry of the projective group on the straight line, i.e., the projective geometry of the straight line, we must change the structure of the straight line by adding to it a new, improper point; the set of points which is obtained in this manner is called the projective straight line. On the projective straight line, every pair of homo-

geneous coordinates (x, t) of which at least one is different from zero


5.1

41

corresponds to a point; if two pairs of coordinates are proportional, they correspond to the same point. If pairs of coordinates are not proportional, the corresponding points differ. Now the goal has been attained; the projective transformations have actually acquired the property of mutual single-valuedness. Every point

P(x, t) of the transformation (5.4) refers to a unique point P'(x', t'); if both coordinates .r, t are multiplied by the same number A :p, 0, then x', t' are multiplied by the same number, and as a consequence the point

P' does not change. By (5.2), x' = t' = 0 only occurs for x = t = 0. Conversely, on the basis of the same relation (5.2), every point P'(x', t')

corresponds to one and only one point P(x, t). We still have to verify that the transformations (5.4) form a group; this can be achieved as in 4.1 and will be left to the reader. The projective

group of the straight line is three-parametric: as a result of the hdmogeneity of the coordinates, the transformation does not change if all parameters are multiplied by the same number; hence there are only three essential parameters. The following theorems will now be proved:

Theorem 5.1: There exists one and only one projective transformation of a straight line under which there correspond to the points P', Q', R' three arbitarily chosen different points P, Q, R. The points P', Q', R' must likewise be different and this is the only limitation on the arbitrariness oj' their choice.

First, we assume that we have selected for the points P, Q, R the im-

proper point M, (l, 0), the origin of coordinates 0(0, 1) and the point E(1, 1); let the coordinates of the points P', Q', R' be (x1, t1), ('N'2, 1-1) and (x3, t3). As M, - P', 0 Q', one has [cf. (5.4)] Px1 = P,

Pt1 = r, whence we obtain from E - R'

vx2 = q,

[32]

T,

(5.5)

T13 = Pt1+Qt2,

TX3 = P.r1+ax2,

P=

at2 = s,

CT =

L12]

[131 T,

[12]

where for the sake of conciseness

[12] = Y1

X-)

tI

t2

(5.6)

CHAP. I


42

etc. Since the points P', Q', R' are different, the values of the three deter-

0, a # 0, if T # 0.

minants [12], [13], [32] differ from zero and p

The formulae (5.5) and (5.6) define the coefficients p, q, r, s exactly apart

from a general multiplier T : 0, i.e., the projective transformation is otherwise defined uniquely; the determinant p

ql - po[12] 9- o.

r

s

It is easy to proceed from the particular case under consideration to the general case. In fact, if in a projective transformation S we let the points

M. - P, 0 - Q, E -* R, and in a projective transformation T the points M. -+ P', 0 -+ Q', E -+ R', we will have in the transformation TS -', which is likewise projective, P - P', Q -+ Q', R -+ R'. Conversely,

if for the projective transformation E we have P - P', Q - Q', R -i R', we find for the transformation ES that M. - P', 0 Q', E - R', i.e., ES = T, E = TS-'. The theorem has thus been proved completely. 5.2 Reasoning as in 4.2, we verify that the equations of the projective

transformations preserve their form if they are written in terms of the projective coordinates x', x2 which themselves represent two arbitrary independent linear homogeneous functions of the homogeneous Cartesian

coordinates x, t of a point. The projective coordinates of a point are likewise homogeneous: two pairs of coordinates, proportional to each other, determine the same point. Obviously, in the case of a transition from one system of projective coordinates to another, the equations will likewise be linear and homogeneous; in correspondence to (4.18), we will write them in the form

x' = p,

xV,

i = 1, 2,

i' = 1', 2',

Ip

I = A # 0,

(5.7)

where x' are the old and x" the new projective coordinates of the same point. Solving the equations (5.7) for the new coordinates [cf. (4.19), (4.20)] we obtain

x' = q; x', i = 1, 2,

i' = 1', 2', Iq; 1 = d -'

0,

(5.8)

where of are given in terms of the p;- by (4.21). We will assume that in (5.7) the old coordinates are homogeneous Cartesian coordinates. Reasoning which is completely analogous to that in the proof of Theorem 5.1 shows that the coefficients p are determined exactly apart from a multiplier, if we are given the Cartesian coordinates

5.1-5.2


43

of the distinct points P, Q, R which in the new system of projective coordinates have the coordinates (1, 0), (0, 1), (1, 1). The points with the coordinates (1, 0), (0, 1), (1, 1) are called the first, second and unit coordinate points, respectively; thus, a projective coordinate system is determined by the specification of three coordinate points, where one may select for this role any three distinct points of a straight line. As has been stated above, in projective coordinates projective transformations are also expressed by linear homogeneous equations; in order to preserve the analogy with (5.7) and (5.8), we will write them in the form (5.9) i, x = 1, 2, x' = p; z', i,

` = qQ x2,

1, 2,

(5.10)

where xI are the coordinates of the transformed point x and . ` are the coordinates, in the same system of projective coordinates, of the point

which corresponds to it as the result of the transformation. The determinant Pa =

Pi

pi

Pi

Pi

=d

(5.11)

is called the determinant of the projective transformation. We note immediately the complete agreement between the formulae

(5.9) and (5.10) and the formulae (4.27) and (4.28): the projective coordinates of a point in the case of a projective transformation transform just like the affine coordinates of a contravariant vector in the case of'an affine transformation of the plane. A certain difference is due to the fact that the projective coordinates of points on a straight line are homogeneous and the affine coordinates of a vector in a plane are not homogeneous. As a consequence there exists a complete correspondence between points of a projective straight line and directions (in the sense of 1.5) in an

affine plane. If we allow the direction from a fixed point to vary in a continuous manner, we may return to the initial direction; correspondingly, we must assume that a projective straight line, in contrast to an ordinary straight line, is closed. We see that the addition of the single improper

point to the straight line changes its properties essentially. Neglecting the above difference, we will assume in the projective geometry of the straight line that the concepts of a point and a contravariant

vector coincide. It is now natural to pose the question of what corre-


44

CHAP. I

sponds in projective geometry of the straight line to a covariant vector u with components ul and u2 which change under a projective transforma-

tion in accordance with the law

u; = P;ut,

i,

or = 1, 2;

(5.12)

the coordinates ul and u2 will also be assumed to be homogeneous. It follows from (5.9) and (5.12) that (ux) = u;x' is an invariant of projective transformations (cf. Theorem 4.3); as a consequence of the homogeneity of the coordinates, a geometric meaning can only attach to the equation i = 1, 2,

u; x' = 0,

which for given ul and u2 determines completely the ratio x' :.V2. Thus, in the case of the straight line, a covariant vector also corresponds to a point. In one-dimensional projective space, the difference between covariant and contravariant vectors is inessential (we will see later that

for a larger number of dimensions this is not the case). The analytic reason for this fact is that the difference between the formulae of the transformations (5.9) and (5.12) is also inessential: if u, and u2 transform in accordance with (5.12), then x' = -u2 and x 2= uI change in accordance with (5.9) [as can be verified using (4.21)]. 5.3

From (5.9) [(cf. Theorem 4.2)] follows

Theorem 5.2:

The expression xI

[xy) =

IyI

y2

,

(5.13)

involving the coordinates x', x2 and y', y2 of two points x, y is a relative projective invariant of weight - 1. As a consequence of the homogeneity of the coordinates and the fact that the invariant is relative, a geometric meaning can only be attached

to the equality [xy] = 0: the points then coincide. With the aid of invariants of the type (5.13) we will now construct an

absolute projective invariant which, in addition, is homogeneous of degree zero with respect to the coordinates of the points. For three points x, z, t, the ratio [xz] : [xt] will be an absolute invariant; it will be homogeneous of degree zero only with respect to the coordinates of the point r,

but not with regard to the coordinates of the points z, t. In order to achieve complete homogeneity of degree zero, one has to introduce yet

5.2-5.3


45

another point y and construct the expression xyzt

[XZ]

[xt]

}

_ [xz][Yt] [yt] [xt][Yz]

[YZ]

(5.14)

The invariant {xyzt} is absolute and homogeneous of degree zero with

respect to the coordinates of each of the points x, y, z, t; it represents therefore a certain projective geometric quantity which is related to these points. The invariant {xyzt} is called the cross-ratio of the four points x, y, z, t. However, the order of the points is essential here. It follows directly from (5.14) that {yxzt} = {x1-

_,

{xytz} =

{xyzt}'

{xyzt} = {ztxy},

(5.15)

whence one obtains

{xyzt} = {yxtz} = {ztxy} = {tzyx}.

(5.16)

In particular, we note the cases in which some of the points x, y, z, t coincide. For the case when only three of these points are different, one finds

{xxzt} = 1, {.xyxt} = 0,

{xyzx} = oo;

(5.17)

{xyyx} = oo;

(5.18)

if the points coincide in pairs, one has {xxyy} = 1,

{xyxy} = 0,

in all other possible cases of coincidence of the points, the cross-ratio loses its meaning. The fact that the simplest projective geometric quantity has been found to be related to four points is not an accident: in projective geometry of the straight line, three points cannot be related to a geometric quantity (c4pable of assuming different values), since by Theorem 5.1 any three points can be teduced to any other three points by a projective transformation. Since the group of motions + h is a subgroup of the group (5.1), an absolute invariant of'the projective group will also be an invariant of the group of motions; therefore the cross-ratio of four points must be expressible in terms of quantities of Euclidean geometry. In order to find this expression for the case when none of the points are improper, we assume the coordinates to be homogeneous Cartesian and let x2 = y2

46


CHAP. I

= z2 = t2 = 1, so that x', y', z', t' become non-homogeneous Cartesian coordinates of the points x, y, z, t. For this choice of coordinates, the formula (5.14) becomes X-z' y' ' {xyzt} = rt _ tt Y

i.e., {xyzt} is equal to the ratio in which the point x divides the segment zt divided by the ratio in which the point y divides the same segment. The case when one of the points is improper will be left to the reader

as an exercise (Exercise 4). This proposition indicates the following: if {xyzt} < 0, one of the points x, y lies inside the segment z, t, the other outside it; in this case one says that the pairs of points x, y and z, t divide each other. If {xyzt} > 0,

both points x, y lie either inside or outside the segment zt: the pairs of points x, y and z, t do not divide each other (cf. Exercise 5). From the projective invariance of the cross-ratio follows Theorem 5.3: The property that two pairs of points divide or do not divide each other is invariant with respect to projective transformations. The concept of the cross-ratio makes it possible to establish a geometric

meaning for projective coordinates. In (5.14), let x, y, z be the first, second and unit coordinate points, respectively; then x2 = y' = 0,

x'=y2=z'=z2=1, and {xyzt} _

t 2

Theorem 5.4: The4,ratio t' : t2 of the projective coordinates of the point t is equal to the cross-ratio of four points: the first, second and unit coordinate point and the point t. From Theorem 5.4 follows Theorem 5.5: Specification of three points x, y, z and of the crossratio {xyzt} defines uniquely a point t. 5.4 The cross-ratio of four points x, y, z, t depends on their order; by (5.16), we can, without changing the value of the cross-ratio, make any of these points, for example x, into the first. For the remaining three points, there are six different possible orders; consequently, the crossratio cannot have more than six distinct values. We will find these values, restricting ourselves to the case when not

5.3-5.4

47


less than three of the points x, y, z, t are different; then, without violating generality, we can assume that the points x, y, z are different and that t

does not coincide with x. We select x, y, z as first, second and unit 0 coordinate points; then, in accordance with the last proposition, t2 and we can denote the coordinates of the point t by (A, 1), whence (5.14) gives

aI = {xyzt} =

x2 = {xytz} =

A,

x3 = {xzyt} = 1-A, (5.19)

a4 = {xzty} =

11

)(s = {xtyz}_

2-1

x6 T {xtzy}

A

In general, these six values will be different. We will explain for what values of ). they may turn out to be identical. In view of the fact that any

of the six values (5.19) can be selected as the initial value, i.e., as a,, one may limit oneself to a study of those cases when at, is equal to one of the other values. Comparing a, with each of the other a;, we find that the a f can assume the same value only for the following values of a :

A = ±1, 0, 2, 1, -w, -(0 2,

where co = (- 1 +i``3)/2 is one of the cube roots of unity. If a = 0, either [xz] or [yt] will be equal to zero [cf. (5.14)]; some of the points x, y, z, t coincide. If A = 1, then a3 = a5 = 0, i.e., we again have coinciding points. For both these values of A, three of the crossratios (5.19) are different: 1, 0, co. Incidentally, we have recognised that these three values of the cross-ratios are only possible if only three of the four points x, y, z, t are different or if they coincide in pairs (cf. 5.3). If A = -1, again only three of the cross-ratios (5.19) will be different, namely -1, 2, 1/2; in the cases A = 1/2 and A = 2, we obtain the same values. Thus, a particular case of distribution of four points on a straight line has been isolated. If {xyzt} = -1, one says that the pairs of points x, y and z, t, divide each other harmonically. The order of the points in each pair and the order of the pairs is of no consequence, since here [cf. (5.15), (5.16)] {xyzt} = {yxzt} = {xytz} = {yxtz} = {ztxy} _ {ztyx}

{tzxy} _ {tzyx} (cf. Exercise 6). If the coordinates of the points of the straight line are also allowed to

48


CHAP. I

assume complex values, an additional special case presents itself: for 2 = -co or for a = -co', only two of the a; in (5.19) will be different, namely -w, -w2. In that case it is said that the set of four points x, y, z, t is equianharmonic. 5.5

On the projective straight line the role of the second order curve

is played by the set of points whose coordinates satisfy the equation a11(x1)2+2x12 x'x2+a22(x2)2 = 0,

or, in abbreviated form (aid = ali):

ai,x'x' = 0,

i,j = 1,2.

(5.20)

In the case of the projective transformation (5.9) we obtain nit x'x' = ai; pp., z°z11; setting 4411 = P01 p'ie aii

3 11 = 41101 ,

(5.21)

we reduce the equation to the form a'11 x01Y11 = '0.

Since the equation (5.20) is a quadratic in the ratio x' : x2, it determines a pair of points (the coordinates of which may be complex). Thus, a pair of points corresponds to a geometric object ai j with the three homogeneous coordinates all, a21 = a12 , a22; the equation (5.21) is the law of transformation of the coordinates of this object in the case

of a projective transformation of the straight line [cf. the first of the formulae (4.39)].

It is seen from the same derivation of the formulae (5.21) that the expression

ai1x'Y' = a11(x')2+2a12x'x2+a22(x2)2

(5.22)

called the binary quadratic form in the variables x', x2 (the form is a homogeneous polynomial and it is binary since it involves two variables), is a joint projective invariant of the geometric object a. and the point x; when it is equal to zero, this means that x is one of the points correspond-

ing to the object air. Since the geometric object a;, and the quadratic form (5.22) completely determine each other, we will speak sometimes of the quadratic form ail.

5.5


49

The agreement between the formula (5.21) and the first of the relations

(4.39) proves immediately [cf. (4.41)] that the determinant

D=laijl=j all

j azl

a12 a 22 1

= all a22-(a,2)2

(5.23)

is a relative projective invariant of weight 2; it is called the discriminant of the quadratic form (5.22). The equality D = 0 denotes that the qua-

dratic equation (5.20) has equal roots, i.e., that the two points corresponding to the quadratic form a;1 coincide. Next, we consider two quadractic forms ail x'x',

b; j x'x';

(5.24)

as has been proved above, the discriminant of the form (a+1+2b;j)xix', given by (a,1+2b,1)(a22+Ab22)-(a12+AbI2)2,

is a relative invariant of projective transformations of weight 2. This statement will now be written in the.form of an equality; since 2 is quite arbitrary, the coefficients of 2 on each side of the above equality must be equal. Comparing them we find that the expression a,, b22+a22 b,, -2a12 b,2

(5.25)

is a joint relative invariant of weight 2 of the two quadratic forms a;1 and bl j [cf. Exercise 12). As a consequence of the homogeneity of the coordinates a;j and b,1, a geometric meaning can only be attached to the case when the invariant (5.25) is equal to zero. In order to explain the geometric significance of the invariant equality all b22+a22 bl1-2a12 b12 = 0

(5.26)

(under the assumption that the discriminants of both forms are different from zero), we select the points p, q, determined by the quadratic form al j, as first and second coordinate points. Then a, I = a22 = 0, a12 # 0 and (5.26) assumes the form b,2= 0. As a consequence the points r, s corresponding to the form b11 will have coordinates (Al, 22), (Al' -22), where 21, 22 is one of the solutions of the equation bl1().1)-+622(22)2

2122 # 0. 0, b11 b22 = 0, Using (5.14), a calculation gives { pgrs} = -1. Thus, one arrives at

50


CHAP.

I

Theorem 5.6: If the invariant (5.25) of two quadratic forms (5.24) is equal to zero, the pairs of points determined by these quadratic -forms divide each other harmonically. (cf. Exercise 13). One has to keep in mind that the result is only true in such a general form if the coordinates of the points can assume complex values. Exercises 1. The position of a new system of projective coordinates is determined completely with respect to an old system if one is given the old coordinates of the new coordinate points: the first - (1171, i'), the second - - (q', tl') and the unit point (Z', '). Express the

coefficients of the coordinate transformation in terms of these numbers. 2. Show that every non-identical, projective transformation of a straight line with real parameters has two fixed (double) points which are real and different, real and coincident, or complex conjugate; in the first case, the projective transformation is called hyperbolic, in the second case parabolic, in the third case elliptic. 3. A non-identical projective transformation S is called involutionary, if S-' = S. Find the conditions which must be satisfied by the parameters of a projective transfor-

mation in order that it will be involutionary. Can an involution be parabolic? (cf. Exercise 2). 4. Establish the Euclidean geometric significance of the cross-ratio {xyzr}, if t is an

improper point of the straight line. 5. Give the definition of the concept of pairs of points which divide or do not divide each other for the case when one of the points is improper, in such a way that the rule

of (5.3) for the sign of the cross-ratio is retained. 6. In order to generalize the concept of harmonic pairs of points to the case when the points in one or both pairs coincide, the condition {xyzr} _ - I must be rewritten in a form which does not involve a denominator. How is the harmonic property defined in this case? Write the condition for pairs of points to be harmonic in the stated form

under the assumption that the points z and t are the first and second coordinate points. 7. Prove that every projective invariant of several points is a function of the crossratios of these points, taken four at a time; it should be assumed that at least three of

the points must be different. '8. Show that a,lx'y' is a joint absolute invariant of the quadratic form a and the two points x, v. What is the geometric meaning of the equality a,,x'y' -- 0? 9. The equality a,;x'v' = 0 (a,). - a;,) establishes a correspondence between every point x of the straight line and another of its points y; show that this correspondence is involutionary (if the discriminant of the form ail is different from zero; cf. Exercise 3). What are the double points of this involution? Prove that the involution is determined by two pairs of points which correspond to each other.

10. On a straight line, three pairs of points are given: p and x = q-r, q and y = r-p, r and z = p-q t); prove that there exists always an involution in which in each of these three pairs the points correspond to each other (in other words. these three pairs of points are found to be in involution). t) The symbol x = q -- r denotes that the coordinates of the points s, q, r are related

by x' = q' -r'.

5.5-6.1

PROJECTIVE TRANSFORMATIONS OF THE PLANE

51

11. Interpret the result of Exercise 10 in terms of the affine geometry of the plane (cf. § 4, Exercise 12). 12. Verify the invariance of the expression (5.25) by direct computations. 13. What is the geometric significance of the equality (5.26), if in one or both of the

quadratic forms (5.24) the discriminant is equal to zero? 14. What is the geometric meaning of the vanishing of the joint invariant (all

(5.27)

of the two quadratic forms (5.24)? 15. Show that the invariant (a,,b::-i alibi,

-2a:b,t)2

(5.28)

of the two quadratic forms (5.24) is absolute. This invariant is homogeneous of degree zero with respect to a,, and be,; what is its geometric significance?

§ 6. Projective transformations of the plane

6.1. In order to construct a geometry of the projective group in the plane, i.e., the projective geometry of the plane, we must, as in the case of the straight line (cf. 5.1), complete the plane by new elements. First of all we introduce homogeneous coordinates x, y, t for the points of the plane

=

ty, t#0,

(6.1)

where , , are the Cartesian coordinates of a point. Now, we add to the points of the plane new, improper points for which one of the coordinates

x, y is non-zero and

t = 0.

(6.2)

The plane which is augmented by these improper points is called the projective plane t). Every set of three numbers (x, y, t) of which even only

one is non-zero corresponds to a point of the projective plane; proportional sets of three numbers correspond to the same point, those which are not proportional to different points. In homogeneous Cartesian coordinates, the equation of the straight line has the form

Ax+By+Ct = 0;

(6.3)

t) More exactly, the set obtained in this manner becomes the projective plane after the introduction of projective coordinates and projective transformations (cf. 6.2).


52

CHAP. I

the straight line is the geometric locus of points whose homogeneous coordinates satisfy a homogeneous linear equation. Since the set of all improper points of the plane is given by (6.2), which is also homogeneous

and linear, it is natural to assume that this set also represents a straight line. The set of all straight lines in the plane must be augmented by the single improper, straight line defined by (6.2). The coefficients A, B, C in (6.3) are the homogeneous coordinates of the straight line (1.3); the improper straight line has the coordinates A = B = 0, C = 1. Thus, we can omit the requirement that at least one of the numbers A, B is to be non-zero. In the projective plane, every set of three numbers A, B, C at least one of which is non-zero corresponds

to a straight line: proportional sets of three points correspond to the same straight line, those that are not proportional to different straight lines.

We will assume that the straight line (6.3) is not improper so that we have among the numbers A, B at least one which is non-zero; setting t = 0 in the equation of the straignt line we find, in complete analogy to 5.1,

its only improper point with the coordinates x = -B, y = A. We note that by this result all parallel straight lines have the same improper point. This makes it possible to locate the improper points on the ordinary plane: an improper point is one which is common to all parallel straight lines; in other words, an improper point is nothing else but a direction in the sense of 1.5. 6.2

As in the projective geometry of the straight line (cf. 5.2), also in the case of the plane it is convenient to generalize the concept of the coordinates of a point. We will call projective coordinates of a point any three independent linear homogeneous functions x', x2, x3 of the Cartesian homogeneous coordinates of the point; as a consequence of the introduction of such coordinates we gain the same advantage as in § 5

in the case of the straight line. Next, we will define projective transformations of the plane; each such transformation is given by equations of the form x` = psxa, i, a = 1, 2, 3, (6.4) where za are the coordinates of the point corresponding in the transformation to the point with coordinates x'; the coordinates of both points are referred to the same system of projective coordinates. In that case

the determinant of the transformation

6.1-6.2


Pi

P1

Pi

4 = IPaI = P2

P2

P2

P3

Ps

Ps

53

must be non-zero. We will solve the equations (6.4) for 12; for this purpose, let Pa denote the cofactor of the element p3 in the determinant (6.5) and write P2

a

f, a = 1, 2, 3

4. = e >

(6.6)

(where qQ is the reduced cofactor*); one has to note here the transposition of the indices). Then, by a known theorem on the sum of the elements of the row of a determinant multiplied by the minors of the elements of the

same or another row, one finds 3

Pa

giP _ `-14

i, j, a = 1, 2, 3,

= S.1,

(6.7)

where 6.1 is the Kronecker Delta, defined by 1,

b

0,

a

if

j = a,

if j#a;

(6.8)

the equality (6.7) shows that the matrices J I I I and I Iq! I I are mutual inverses.

We now have from (6.4) and (6.7) (

qxi=

ia

q'p`1a=a.ila.

a

as a consequence of the definition (6.8) of 81 , only the term a = j in the sum 6.112 will be non-zero; hence this sum is equal to 11. Changing the

index notation, we obtain finally

1'=gixa,

i,a=1,2,3.

(6.9)

The above is, in essence, the traditional derivation of Cramer's rule; this derivation will be very useful below. By the multiplication rule for determinants IgaI

*) Cf. footnote p. 27.

IPxI = IgaP"a' I = Iaxl = 1,

54


CHAP. I

IgaI = d -' # 0.

(6.10)

whence

We will show that projective transformations form a group. Their mutual single-valuedness is obvious; the calculations above show that, if S is a

projective transformation, S' is also a projective transformation. The identical transformation follows from (6.4) for pa = S; [cf. (6.8)]. It has only still to be verified that together with Sand T the transformation

TS will also be projective; this is readily shown. Let the transformation S be given by (6.9) and the transformation T by

i,j= 1,2,3,

Iq;I #0.

Then the transformation TS is given by x1=yQx°,

j,

1,2,3,

(6.11)

where

i,j,a=1,2,3.

= gig',

Further, one has IgaI = Ig1ga1 = Ig1I

Jq,J # 0.

(6.12)

The relations (6.11) and (6.12) show that TS is a projective transformation. Next, we will investigate what happens to straight lines in the plane in the case of a projective transformation. In projective coordinates, a straight line is determined by an equation of the form

ui x' = 0,

i = 1, 2, 3.

(6.13)

The coordinates za of points corresponding to points of the straight line (6.13) under the transformation (6.4) will satisfy the equation ui pi x° = 0;

introducing the notation uQ = pa ui,

(6.14)

we find

uQxa=0, i.e.,

Theorem 6.1: In a projective transformation of the plane, points lying on a straight line go over into points which likewise lie on a straight line.

6.2


55

A projective transformation of points induces a projective transformation of straight lines, defined by (6.14), where uQ (a = 1, 2, 3) are the coordinates of the straight line which in this transformation corresponds to the straight line (u1, u2, u3). The formulae (6.4) and (6.14) show that in projective geometry of the plane a point can be identified with a contravariant vector (which is given exactly apart from a numerical factor) and a straight line with a covariant vector. It is clear from the derivation above that

(ux) = (xu) = ui x,

i = 1, 2, 3,

(6.15)

is an absolute projective invariant of the point x and the straight line u, as is verified directly by computation: uaXa

= u;paxa = u1x`;

the geometric significance of the equation (ux) = 0 is that the point x

lies on the straight line

u.

It is not difficult to express the coordinates of the straight line in terms of the coordinates of two distinct points which lie on it. If a straight line

u passes through the points x' and y', one has

u1x1+u2x2+u3x3 = 0, uly1+u2y2+u3y3 = 0, whence (exactly apart from a common multiplier) U1 = x2y3-x3y2,

u2 = x3y1-x1y3,

u3 = x1y2-x2y1;

(6.16)

these formulae can be written more briefly

u = [xY)

(6.17)

In a similar manner, one can express the coordinates of a point x in terms of the coordinates of two different straight lines u, v intersecting

atx:

x = [uv], i.e.,

X1 = u2v3-U3 V2,

X?

= U3Vk-u1 V3,

(6.18)

x3 = u1V2-u2v1.

We see that in the projective plane not only do two different points uniquely determine a straight line, but, conversely, two different straight lines always intersect in one and only one point.

CHAP. I


56

It follows from this that three points x, y, z lie on a straight line, if (uz) = 0, where u is determined by (6.17). With the notation ([xy]z) = [xyz] =

xI

x2

x3

YI

Y2

zI

z2

Y3 z3

(6.19)

we obtain the condition for three points to lie on a straight line in the form

[xyz] = 0.

(6.20)

This result leads to the result that [xyz] is a projective invariant; in fact, by (6.9), t o 2a 3 Sax Sax Sax i s 2 a 3 a

[xYz] = 9aY 4aY 4aY 1

a

2

qaz qaz

a

3a

4a'z

2

4

3

ql ql 2

3

= 92 92 42 2 3 93 q3 q3

x x2 x 3 1

Y Y2 1

z

I

z

2

Y

z

3I =

d-

1

[xyz].

3

Theorem 6.2: The expression [xyz] [cf. (6.19)] involving the coordinates of the three points x, y, z is a relative invariant of projective transformations of weight - 1. Analogous manipulations lead to Theorem 6.3:

The expression [uvw] =

U1

U2

U3

VI

V2

V3

WI

w2

w3

f

(6.21)

involving the coordinates of three straight lines u, v, w, represents a relative

invariant of projective invariants of weight +I We have now the geometric interpretation of the equality -

[uvw] = 0:

(6.22)

the straight lines u, v, w intersect in a single point. 6.3

Points on a straight line transform as a so-called range of points. Let x, y be two different points of a straight line; if z is a third point of the same line, we can express the projective coordinates z` of z linearly

in terms of the coordinates x', y1 of the points x and y, using (6.20):

z` = fix'+py',

i = 1, 2, 3.

(6.23)

Therefore three points lying on a straight line are often called dependent;

6.2-6.3


57

similarly, three points which do not lie on a straight line are called independent. In (6.23), the coefficients A, p are readily seen to remain unchanged if the coordinates are transformed by use of (6.9); however, they change if we multiply all the three coordinates of the point x by p, of the point y by a and of the point z by T (with pat # 0), although the points themselves do not change. The ratio p : A is then multiplied by p/a. The same applies to a fourth point of the range

t' = A'x`+p'y`,

i = 1, 2, 3,

(6.24)

whence it follows that the expression p

p

remains unchanged for any projective transformation of the plane as well as for a renormalisation of the coordinates of points (i.e., for multiplication of all coordinates of each of the points by any non-zero number) and therefore represents some projective geometric quantity which is related to the four points x, y, z, t; in this way there arises naturally

the proposition that this quantity can be expressed in terms of the crossratio of these points (5.3). In order to find this cross-ratio, we will assume that the straight line [.xy] coincides with the x-axis; in that case projective coordinates on it are themselves linear homogeneous functions of the two homogeneous Cartesian coordinates of the points on the straight

line, where these two functions are independent. Without restricting generality, we can assume that they are the first two coordinates; then these will be projective coordinates of a point of the straight line [xy] (5.2). By (5.14), using (6.23) and (6.24) as well as the fact that the determinant

#0 (since the points x, y are different), we find xl x2 yl y2

{xyzt} =

zt

tl xl Hence follows

z2

tl

t2

yl

y2

ZI

z2

x2 t2

1

pA, ).µ,


58

Theorem 6.4:

CHAP. I

The cross-ratio of four points belonging to a range, say

x, y, z = Ax+µy and t = A'x+µ'y (where x and y are different)

is

equal to Here the notation z = Ax + py is more convenient than that in (6.23). Theorem 6.4 makes it possible to establish whether two pairs of points

of a range divide each other or not. Straight lines of the plane which pass through one point x transform as

a pencil of straight lines; the point x is called the centre of the pencil.

Let u, v be two different straight lines of a pencil with coordinates u,, v,; the condition' (622) suggests that each straight line of a pencil will have coordinates of the form Au,+µv, (as a consequence of which three straight lines of the same pencil are called dependent). Reasoning as above in the case of a range of points, we see that is a projective geometric quantity which is related to the straight lines u, v, Au+µv, A'u+µ'v; it is called the cross-ratio of the four straight lines of the pencil.

Let us now intersect a pencil by a straight line w which does not pass through the centre of the pencil; the line w intersects the straight lines u, V. Au+µv, A' u+µ' v in the points [uw], [vw], A[uw]+p[vw],

A'[uw]+p'[vw].

(6.25)

Obviously, the points [uw] and [vw] are different; by Theorem 6.4, the cross-ratio of the points (6.25) is Thus we arrive at the important Theorem 6.5: The cross-ratio of the four points in which a straight line w, not passing through the centre of a pencil, intersects four lines u, v, Au +µv, A' u + p' v of this pencil does not depend on the choice of the straight line w and is equal to the cross-ratio (µA')/(Aµ') of the stated lines.

By determining the cross-ratio of four straight lines passing through one point we find an opportunity to introduce the concepts of division, non-division and harmonic separation of pairs of straight lines of a pencil and to relate them, on the basis of Theorem 6.5, to the corresponding concepts for points of a range (5.3, 5.4). In conclusion, the following theorem which will be required later will be stated: Theorem 6.6:

There exists one and only one projective transformation

6.3-6.4


59

of the plane under which four points x*, y*, z*, t* correspond to four arbitrary points x, y, z, t no three of which lie on the same straight line; no three of the points x*, y*, z*, t* may lie on one straight line and this is the only limitation on their otherwise arbitrary choice.

The proof of this theorem is quite analogous to that of Theorem 5.1

and will be left to the reader. 6.4 In 6.2 and 6.3, we have given attention to a situation in which a number of results could be obtained from each other by replacing

straight lines by points and vice versa. For example, such a relationship exists between (6.17), (6.18) and (6.20), (6.22); the same statement can be made with respect to the expressions for the cross-ratios of four points of a straight line and four straight lines of a pencil. It is not difficult to explain the reason for this effect: points as well as straight lines are determined in the projective plane by three homogeneous coordinates; the condition (6.13) which must be fulfilled in order that a given point will lie on a straight line involves the coordinates of the point and those of the straight line in a similar manner. Geometric properties of a figure are studied in relation to projective transformations which are determined for points and also for straight lines by linear homogeneous equations

which are mutually related: in order to obtain the matrix Q = Ilgall of the linear transformation (6.9) from the matrix P = llpJJI of the transformation (6.14), one must replace *) rows by columns in the

matrix P', the inverse of P: Q

It is not difficult to see that conversely

P=

(Q-1)*

(linear transformations which are interrelated in this manner are called contragredient, i.e., counteracting). As a consequence of these results, nothing is changed if we assume ui to be the coordinates of a point and x` the coordinates of a straight line; the only very essential difference will

consist of the fact that d is replaced by A', and as a consequence the weights of the invariants change sign (cf. Theorems 6.2, 6.3). We now arrive at the principle of duality in the projective geometry of the plane: *) In the matrix P, the number of the column is given by the superscript, in the matrix Q by the subscript.

CHAP. I


60

Theorem 6.7: If in any proposition of projective geometry of the plane we replace points by straight lines, and straight lines by points, we obtain a new statement which is also true for this geometry.

In the case of such a substitution, one must, of course, at times also change the terminology: thus, points on a straight line become straight lines passing through one point, etc. Two propositions which correspond

to each other by the principle of duality are called dual or correlative with respect to each other. The principle of duality plays a very important role in projective geometry, since it makes it possible to prove only one of two correlative theorems, the truth of the second then following directly from Theorem 6.7. Applying the duality principle to Theorems 6.5 and 6.6, we arrive at the following propositions:

Theorem 6.8: The cross-ratio of four straight lines joining a point z, not lying on a straight line g, to four points on g does not depend on the choice of the point z and is equal to the cross-ratio of these four points. Theorem 6.9: There exists one and only one projective transformation of the plane under which four straight lines a*, b*, c*, d* correspond to four arbitrary straight lines a, b, c, d no three of which pass through one point. Likewise, no three of the straight lines a*, b*, c*, d* may pass through one point; this is the only limitation imposed on the arbitrariness of their choice. 6.5

Next, we will dwell on the transformations of projective coordinates. In the case of the transition from one system of projective coordinates to another, the new coordinates are readily seen to be expressible in terms of the old coordinates and the old ones in terms of the new ones by means of linear homogeneous functiolts:

=r ' x'.=qix', '

ixpx, d 0 0,

Ig"I=d-I#b,

I = 1, 2, 3,

I=1,2,3,

(6.26)

where x' are the old, xr the new projective coordinates of the same point. Obviously, the relation between the p; and the q f is the same as between

the coefficients of the equations (6.4) and (6.9). The agreement between the formulae (6.26) and (6.4), (6.9) has the consequence that all statements regarding projective transformations may also be applied to transformations of projective coordinates. In particular, the invariants of projective transformations established above

6.4-6.5


61

will also be invariants of the transformations of projective coordinates. In studying Theorem 6.6 from this point of view, we see that we can select a projective system of coordinates in such a manner that any four given points no three of which lie on one straight line, will have the coordinates (1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 1, 1). The points with these coordinates are called the first, second, third and unit coordinate points (these are denoted in Fig.9 by 1, 2, 3, e, respectively). Following the duali-

ty principle, the straight lines with the coordinates (1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 1, 1) are called first, second, third and unit coordinate lines (denoted in Fig. 9 by I, 11, III, e, respectively). The triangle 123 is called

Fig. 9

the coordinate triangle; obviously, the first coordinate line passes through the second and third coordinate points, etc. Specification of the coordinate

points determines the coordinate lines and, conversely, the coordinate lines determine the coordinate points (Exercises 22, 23). Let x (Fig. 9) be an arbitrary point of the plane; we will join it by a straight line to the unit point a and find the point of intersection p of the straight line [xi] and the first coordinate line; then p = Ax+pi, where A and p must be determined from the condition p' = ,1x' +p = 0, whence

p : 2 = -x1.

In this manner, we find that the straight line [xe] intersects the second coordinate line at the point q = A' x+ p' e, and the third coordinate line

at the point r = A" x+ p" e, where p' : A' = - x2, p" : A" = - x3.

CHAP. I


62

By Theorem 6.4, 3

{xegr} = X3,

{xerp} = Xl,

{xepq} = X2;

(6.27)

the formulae (6.27) establish the geometric meaning of the projective coordinates of a point. In an analogous manner, using the duality principle, the geometric significance of the projective coordinates of a straight line can be elucidated.

6.6 We now proceed to the study of second order curves. In homogeneous Cartesian coordinates, the equation of such a curve will be homo-

geneous of degree two; the same will be'true in projective coordinates. Thus, the equation a11(X1)2+2a12x1x2+a22(x2)2+2a13xIx3+2a23x2x3+a33(X3)2

=0

or, more briefly, (6.28) i, j = 1, 2, 3 defines the second order curve T. The left-hand side of (6.28) is called a ternary quadratic form (because it involves three variables). After the projective transformation (6.4) the equation (6.28) assumes the form Oj j X X1 = 0,

alt = aji,

r alt papP

x = 0;

letting aQa = Pa p a;; ,

a, fi = 1, 2, 3,

a,p =

(6.29)

we can write this equation in the form aa6 kelp = 0,

a, f = 1, 2, 3,

i.e., we arrive at Theorem 6.10: Under a projective transformation, a second order curve becomes a second order curve. The equation (6.29) gives the law of transformation for the coordinates alt of a second order curve in the case of a projective transformation of the plane. The determinant all

a12

a13

D = (a1;l = a21

a22

a23

a31

a32

a33

(6.30)

6.5-6.6


63

is called the discriminant of the second order curve, or, more frequently, the discriminant o f the quadratic f o r m a i; x' x' . I t follows from (6.29) that

D = laapl = I p' i aql = Ipal . Iai; pel = l pal lal;l ' Ip1I = D 42 (where the multiplication of determinants is carried out according to the rule: first, rows by columns, and then, rows by rows). Theorem 6.11: The discriminant of a ternary quadratic form is a relative invariant of weight 2. 'As a consequence of the homogeneity of the coordinates ate, only the

equality D = 0 has geometric significance: the second order curve "degenerates" into two straight lines. Next, we consider the problem of the intersection of a second order curve F with a straight line, where for the sake of simplicity we restrict ourselves to the case of a curve which is non-degenerate (D # 0). Let x, y be two different points of the straight line; without restricting genera-

lity, we can assume that the point y does not lie on the curve, so that

a1jy`y'#0. If the coordinates .of the point of intersection are Ax'+py', we have a,j(Ax'+py')(AX'+pyf) = 0, or

d2

a, x'x'+2Apa,fx'y'+p2 - a,1 y'y' = 0.

(6.31)

The equation (6.31) is quadratic in p/A; consequently, every straight line intersects a second order curve which is not degenerate in two points which are real and different, real and coincident, or complex conjugate. In the second of these possible cases, the straight line is a tangent to the curve T. Assume now that also the point x does not lie on F:

a. x'x' 0 0;

(6.32)

we will require that the points x, y are divided harmonically by the points of intersection of the straight line [xy] and the curve T. In this case it is said that the points x, y are conjugate with respect to the curve F.

If pJ/AI and µ2/A2 are the roots of (6.31), this condition gives, on the basis of Theorem 6.4, the results

pI:pz=-1, A

,t2

Pi +p2=0, Al

22


64

CHAP. I

i.e., the sum of the roots of (6.3) must be equal to zero. Thus, the condition

aij xiyj = 0

(6.33)

is necessary and sufficient for the points x, y to be conjugate with respect

to F. By (6.33), the geometric locus of points conjugate to a given point x with respect to a curve F is the straight line u, yj = 0, where uj = a; j x',

i, j = 1, 2, 3;

(6.34)

this line is called the polar line of the point x with respect to the second order curve T.

If the point x lies on the line T, i.e., if a; j x'xj = 0,

(6.35)

both points of intersection of the straight line [xy] with the curve T must coincide at the point x, in order that (6.33) will be fulfilled; the straight line (6.34) is the tangent to I' at the point x. As a consequence, for a point x on the curve F, the polar line is the tangent at x. If we introduce the generalized definition of harnionic pdirs of points (cf. § 5, Exercise 6), this turns out to be in agreement with the above definition of the polar: all points of the tangent are conjugate to the point of contact. Comparing (6.34) with (6.35) and (6.32), we see that the two cases under consideration differ from each other by the fact that in the first the point x does not lie on the polar, whereas in the second case it lies on the polar. Since D = Ja1 jJ 96 0, the equations (6.34) can be solved for x'; denoting

the cofactor of the element aij in the determinant D by A'j, and the reduced minor by a'', we have A'j a'j=-. D

(6.36)

solving (6.34), using Cramer's rule, we find x' = a'juj,

(6.37)

where

laiji = D-' -A 0 [cf. (6.5) and (6.10)]. Thus, every straight line u corresponds to a point x for which it is a polar line; the point x is the pole of the straight line u.

6.6-6.7

PROJECTIVE TRANSFORMATIONS ON THE PLANE

65

The correspondence between points and straight lines, established by (6.34) and (6.37), is unique. Now, the condition for a straight line u to be tangential to the curve r is readily established; for this purpose it is necessary and sufficient that the pole of the straight line u, determined by (6.37), lies on F, and hence

a`'u;u, = 0,

a'' = a".

(6.38)

Thus we have derived the so called tangential equation of the curve F, i.e., the equation which is satisfied by the coordinates of all straight lines which touch r and of those straight lines only. The tangential equation of a curve is defined as the geometric locus of the straight lines touching it. If the tangential equation is of degree two in the coordinates of the straight lines, the line is called a curve of the second class. Thus, we have proved

Theorem 6.12: A second order curve which is not degenerate is a second class curve. The duality principle now permits us to assert that the preceding rea-

soning is reversible; thus one has Theorem 6.13: A second class curve (6.38), for which the discriminant Iai1I # 0, is a second order curve which is not degenerate. 6.7

Summarizing our results, we can now define the problems of projective geometry, following F. Klein. For this purpose we have only to repeat all the statements of 4.7, replacing there the word "affine" by the word "projective"; this is true for the definitions of projective property, projective quantity, projective geometric object, etc. In this context, one must keep in mind that in the propositions of projective geometry improper points need not receive special treatment: by Theorems 5.1 and 6.2, all points of the projective straight line and all points of the projective plane are projectively equivalent. The same result applies to the improper straight line. The following properties of geometric figures are projective: three points lying on a straight line (Theorem 6.1), two pairs of points of a straight line dividing each other harmonically (cf. 5.3, 5.4, 6.3), three pairs of points of a straight line being in involution [(6.22)], two points being conjugate with respect to a second order curve F, a straight line touching a curve F (cf. 6.6), etc. Two straight lines being parallel (Exercise 1) and a point being improper are examples of non-projective proper-

ties. The cross-ratio of four points of a straight but is a projective

GEOMETRIC INTRODUC77ON

66

CHAP. I

quantity; the ratio in which a point z divides a segment x, y (where the

points x, y, z lie on a straight line) is not a projective quantity (by Theorem 5.1). In the following exercises and in the later text, the reader will encounter further propositions of projective geometry. Exercises

1. A projective transformation is given in homogeneous Cartesian coordinates by the equations x' = t, y' = x, t' = y; show that in the case of this transformation the origin of coordinates becomes an improper point, and parallel straight lines x+2t = 0,

x-t = 0 become intersecting lines. 2. What form do (6.17) and (6.18) assume as the result of a projective transformation

of the plane? 3. The rank of the determinant [xyz] ]cf. (6.19)] is called the rank of the point system x, y, z; show that the rank r of three points is a projective invariant. What is the geome-

tric significance of the equalities r = 1, r = 2, r = 3? 4. A figure consisting of four points x, y, z, t, no three of which lie on a straight line, and of the six straight lines, joining these points in pairs, is called a perfect quadrangle; the points of intersection of the opposite sides [xy] and j xz] and [yt], [xt] and [yz] are called its diagonal points. Show that in a perfect qu rangle the two vertices x, y divide the diagonal points lying on the side [xyl and the points of intersection of these sides with the straight lines joining the remaining diagonal points harmonically.

5. Formulate the theorem which is the dual of the proposition of Exercise 4. 6. Show that in the equality (6.23) the quantities A and µ can be regarded as projective coordinates of the points of the straight line [xy]. What points in this coordinate system will be the first, second and unit coordinate points? 7. Find the point of intersection of a straight line u with the straight line joining two

points x, y. 8. Find the point of intersection of the straight lines joining the points x, y and the points z, t. 9. Formulate the dual results of the results of Exercises 7 and 8. 10. Let there be given five points x, y, z, s, t; find the cross-ratio of the four straight lines joining the point x to the points y, z, s, and t; it represents an absolute projective invariant I of the five points which is homogeneous of degree zero in the coordinates of each of these points. Construct the invariant dual to the invariant I, and state its geometric significance.

I1. What is the geometric meaning of the invariant (ux) (vy) (uy) (ux)

homogeneous of degree zero in the coordinates of each of the points x, y and each of the straight lines it, v?

12. What is the geometric meaning of the invariant

I_

[xyz] (ut)

[xytl (uz)

where x, y, z, t are points and u is a straight line?

6.7


67

13. Explain the geometric projective interpretation of the invariant of six straight

lines a, b, c, d, e, f. [abe] [cdf] [abf] [cde)

14. Let there be given a triangle with vertices x, y, z and two different straight lines u and v which do not coincide with any sides of the triangle. Let the straight line it intersect the sides [yz[, [zx), [xv) of the triangle at the points a, b, c, respectively, and let the side v intersect them at the poins p. q, r. Show that [yzap]

[zxbq]

[xycr] = + 1.

(6.39)

Conversely, if the relation (6.39) is true and the points a, b, c lie on a straight line,

then the points p, q, r lie on a straight line. 15. Let there be given a triangle with vertices x, y, z, a straight line v, which does not coincide with any of the sides of the triangle, and a point t which is not a vertex of the triangle. Let the straight line v intersect the sides of the triangle [yz], [zx] and [xy]

at the points a, b, c, respectively, and the straight lines [xt], [yt], [zt] at the points p, q, r. Prove that [yzap) [zxbq] [xycr] _ ---1. (6.40) Conversely, if the condition (6.40) is fulfilled and the points a, b, c lie on one straight line, then the straight lines [xp], [yq], [zr] intersect in one point. 16. Let there be given a triangle with vertices x, y, z and two points s, t which are not vertices of the triangle. Let the straight lines [xsl, [ys], [zs] intersect the sides of the triangle [yz], [zx], [xy) at the points a, b, c, respectively, and the straight lines [xt], [yt], [z1) at the points p, q, r, respectively; prove that

[xycr] = 1. (6.41) Conversely, if (6.41) is true and the straight lines [xal, [xb], [xc] intersect in a point, then the straight lines [xp], [xql, [xr] also pass through one point. 17. Formulate the propositions which correspond by the duality principle to the Exercises 14, 15, 16. 18. Let there be given a triangle with vertices x, y, z and three points p = Ay+µz, q = A'z+µ'x, r = A"x+µ"y on the sides of the triangle. Show that [yzap]

[zxbq]

is an invariant of projective transformations and explain its affine and projective geometric interpretations. 19. Prove Desargues' Theorem: Every straight line intersects three pairs of opposite sides of a perfect quadrangle (Exercise 4) in three pairs of points which lie in involution (§ 5, Exercise .10). 20. What points and what straight lines will be coordinate points and straight lines if a system of coordinates is Cartesian homogeneous? 21. Express the coefficients p;,, in the equations (6.26) for the case of transformation of projective coordinates in terms bf the old coordinates of the new coordinate points: the first (E', ', '), the second (!I', the third and the unit (01, 02, 0') coordinate point. 22. Let 1' denote the point of intersection of the straight line [111 with the straight line 1, etc. (cf. Fig. 9). Show that the straight lines 111, [1'2'] and e pass through one point. On the basis of this result, state the construction for the unit straight line e from given points 1, 2, 3, Z.


68

CHAP. I

23. Applying the duality principle to Exercise 22 state the construction of the unit point a from the straight lines 1, 11, Ill, e (Fig. 9). 24. If the polar of a point x passes through a point y, the polar of the point y passes through the point x; why? On the basis of this result establish the following:

1) if the point x moves along the straight line u, the polar of the point x rotates about the pole of the straight line u; 2) the polar of the point x joins the points of contact of the tangents to the

curve 1', drawn from the point x. 25. A triangle xyz is called the polar triangle of a second order curve 1' [cf. (6.28)], if its vertices are conjugate with respect to 1' in pairs. Show that for jail! # 0 there exists an infinite set of polar triangles of the curve r, where one may select as two vertices x, y any two points conjugate with respect to r which do not lie on P. What form does the equation of the curve ]'assume if one takes the vertices of a polar triangle as

first, second and third coordinate points? 26. Establish the projective classification of non-degenerate curves of second order in the complex and in the real domain. 27. Starting from (6.36), establish the law of transformation of the geometric object all in the case of a projective transformation of the plane. On the basis of this result prove the invariance of the left-hand side of (6.38). 28. Two straight lines are called conjugate with respect to a second order curve r, if one of them passes through the pole of the other; write down the condition under which straight lines u, v will be conjugate with respect to a'curve 1'. Show that this definition is equivalent to that which follows by the duality principle from the definition

of conjugate points (6.6). 29. Deduce the projective classification of second class curves [cf. (6.38)] for Ja"! L 0 in the complex and real regions. What is represented by the second class curve (6.38) if Ja1JJ :jr-, O?

30. A point r lies inside a triangle xyz; is this property projective? Is it afine?

§ 7. The reduction of affine invariants to projective invariants and of metric invariants to affine and projective invariants

We will now compare with each other the groups of transformations considered in §§ 3-6 and their related geometric systems, confining ourselves for the sake of clarity to the case of the plane. 7.1

The group of motions and reflections (cf. 3.1) is a subgroup of the principal group (cf. 3.2); the principal group is a subgroup of the affine group (cf. 4.1). Finally, the affine group may be regarded in a certain manner as a subgroup of the projective group (cf. 6.2): if we introduce affine homogeneous coordinates x', x2, x3 by letting 2

=

3,

X3 # 0,

(7.1)

X

where ', 2 are the non-homogeneous affine coordinates of a point, the equations of an affine transformation [cf. (4.24)] assume the form

7.1

REDUCTION OF AFFINE INVARIANTS TO PROJECTIVE ONES

69

(after a change in notation) 3'3 x=1,2,.3, d= 2'a 1'a X =Pax, X2 =Pax, X 3=P3x , 1

3 P1 Pt2p3#0. 1

(7.2)

P2 P2

For each of the groups under consideration, we will denote the number

of parameters by r. For the group of motions and reflections, we have r = 3, for the principal group, r = 4, for the affine group, r = 6, for the projective group, r = 8 (since its parameters pa are defined exactly apart from a common multiplier). Thus, the groups under consideration become successively more extensive, and the larger groups contain all the smaller groups. With the transition to a larger group, the concept of equivalence

of figures likewise becomes wider: figures which are equivalent in the geometry of the larger group, may not be equivalent with respect to the smaller group. With the growth of the group a number of properties of a figure lose significance, in the geometry corresponding to a larger group, one studies subtler and stronger properties of figures. We will elucidate these statements by a number of examples. With respect to the group of motions and reflections, there exists an invariant of two points in the distance between them (cf. 3.1). With respect

to the principal group, any two pairs of points are readily seen to be equivalent; an invariant exists now only for three points x, y, z (the ratio of the lengths of the segments xy and xz). In affine geometry, three points x, y, z only have an invariant when they lie on one straight line (it is the ratio in which the point z divides the segment xy); three points which do not lie on a straight line are equivalent to any other three independent points (cf. 4.2); for points any three of which are independent one only has an invariant for four points x, y, z, t; the ratio of the orien-

tated areas of the triangles xyz and xyt. In projective geometry, any three points of a straight line are equivalent to any three points on the same line (Theorem 5.1); an invariant exists only for four points of a shtaight line (the cross-ratio). Any four points, of which any three are independent, are equivalent with respect to the projective group to any other four points with the same property (Theorem 6.6); an invariant exists only for five points (if no three of them lie on the same straight line; § 6, Exercise 10).

In Euclidean geometry, triangles and quadrangles form infinite sets characterized by the lengths of their sides and the magnitudes of their angles. In affine geometry, all triangles are identical, since all of them are

70

GFOMFTRIC INTRODUCTION

CHAP.

I

equivalent with respect to the affine group (§ 4, Exercise 2). Not all quadrangles are equivalent (§ 4, Exercise 3); they may be grouped in special classes: e.g. parallelograms, trapezoids. In projective geometry, all quadrangles as well as all triangles are identical (Theorem 6.6). The left-hand side of the equation of a second order curve f has three invariants with respect to the group of motions and reflections: rl, 62 and

63. In affine geometry, only two remain: 62 and 63, of which both are relative. In projective geometry, the curve I' has only one relative invariant D = 63. In Euclidean geometry, non-degenerate second order curves (which are real, i.e., have at least one real point) can be classified as ellipses, hyperbolas or parabolas, all of which form infinite sets character-

ized by the magnitudes of the parameter p and the eccentricity e; one special class among the ellipses is formed by the circles. In affine geometry

(in the real plane), the subdivision into ellipse, parabola and hyperbola persists, but all ellipses are identical as well as all hyperbolas and parabolas (§ 4, Exercise 13); the concept of the circle loses its meaning. Finally, in real projective geometry, all real non-degenerate second order curves are equivalent (§ 6, Exercise 26); the distinction between ellipse, hyperbola and parabola vanishes.

7.2 We will now study the interrelationships between the groups of 7.1 in greater detail and will start with the affine and projective groups. In affine homogeneous coordinates, which are a particular case of projective coordinates *), affine transformations are defined by (7.2). We see immediately that affine transformations transform the improper straight line x3 = 0 into itself **). Conversely, also in affine homogeneous coordinates, projective transformations are given by (6.4); if we require that the straight line x3 = 0 transforms into itself, we obtain p; = p2 = 0.

Thus, affine transformations are represented by those and only those projective transformations in which the improper straight line corresponds

to itself. Now, in accordance with the aims of projective geometry, any straight line I can be taken as being improper. In fact, one can select a system of *) If in projective coordinates, the third coordinate x3 differs only by a constant factor from the third Cartesian homogeneous coordinate and the remaining coordinates are arbitrary linear functions of the Cartesian homogeneous coordinates, the projective coordinates are affine homogeneous. *0) By adding to the affine plane the points for which x3 = 0, we transform it into the projective plane.

7.1-7.2


71

projective coordinates such that the equation of the straight line 1 will be x3 = 0 (cf. 6.5). Projective transformations, which leave the straight line I unchanged, are now given in this system of projective coordinates by the equations (7.2). Further, if we exclude from the projective plane the points of the straight line I and introduce x'/x3 and x2/x3 as nonhomogeneous coordinates, the study of the properties of the group (7.2) and of its invariants will proceed just as in § 4. In any system of projective coordinates, the equation of the straight line / will be (7.3) 1; x' = 0, i = 1, 2, 3; projective transformations which leave the straight line unchanged are given by formulae which differ from (7.2): let S, be any projective transformation under which the equation x3 = 0 becomes the equation (7.3). Then every projective transformation which preserves the straight line (7.3), will have the form S, TS,-', (7.4) where T is one of the transformations (7.2). Clearly, the properties of the

transformation cannot differ from those which are found in any other coordinate system; consequently, the transformations (7.4) must form a group, and, in fact, one which is isomorphic to the affine group, as is readily verified by direct computations. Conversely, if one adds improper elements to the affine plane and introduces projective coordinates (which are different from affine homogeneous coordinates), the improper straight line has an equation of the form (7.3) and affine transformations are no longer described by (7.2),

but by the formulae (7.4). Thus, we have Theorem 7.1: The affine group for the plane represents the set of all those projective transformations of the projective plane which leave a certain line I of this plane unchanged. Theorem 7.1 defines affine transformations within projective geometry; it is natural to expect that one will arrive at the affine properties of a figure and at affine quantities as a consequence of this precise projective definition. In order to achieve this, we note first of all that affine geometric objects may, as a rule, be converted into projective ones by defining a law of transformation of their coordinates within a group which is wider than the affine group of projective transformations. In 6.2 and 6.6 this has been done for the cases of the straight line and the second order curve.

72


CHAP. I

We will consider here briefly the case of a contravariant vector. In affine homogeneous coordinates, i.e., when the equation of the improper straight line ( has the form

x' = 0, so that 1, - 1, = 0, 13 = 1, one has for the components of the vector a yI

a' -- y3 -

x'

at

X31

x=

ys y3 - X-3-1

where x', y' are the coordinates of the starting and end points of the vector. Since in this system of coordinates x3 = (Ix), it is natural to assume that in an arbitrary system of projective coordinates the vector a will have the non-homogeneous coordinates x,

V.

a

(Iv)

r= 1,2,3,

(Ix),

(7.5)

where 1, are the coordinates of the improper straight line; it follows from (7.5) that

((a) = 0.

(7.6)

The right-hand sides of (7.5) do not change during a renormalization of the coordinates of the points x, y; however, on multiplication of all 1, by p * 0, they are divided by p. Therefore one must fix the coordinates I, of the improper straight line, in order

that the components of the vector will remain fixed. The affine transformation p; must preserve the improper straight line; since the coordinates 1; are fixed, one must assume that

p;1,= 1

i,a -= 1,2,3.

(7.7)

The equations (7.7) normalize the parameters p' of the transformation, so that in the

formulae for the transformation of the components of the vector a'

p' = P;a

i, a ° 1, 2, 3;

(7.8)

the parameters pa are determined uniquely, and not only apart from a multiplier; this is in agreement with the non-homogeneity of the components a' of the vector a. If we have an arbitrary projective transformation, after the transformation the improper straight line will have other coordinates 1. which are related to the earlier coordinates by !z - pit, i, a = 1, 2, 3. (7.9) We must assume that in (7.9) the 1, as well as the 1. are fixed, and that the parameters

p.1 in the transformations (7.8) are again normalized. Thus, a contravariant vector, as a projective geometric object, in the case of fixed coordinates defines an improper straight line by three non-homogeneous components a', as a3, which are interrelated by (7.6). The law of transformation of these components for projective transformations is given by (7.8), where the parameters p.1 are normalized as a consequence of (7.9). If we assume that the a' are homogeneous components, they will determine a set of mutually collinear contravariant vectors, i.e., a direction in the sense of 1.5. On the other hand, by (7.6), the a' will be the coordinates of an improper point. Thus, we find it again confirmed that an improper point of the affine plane is the same thing as a direction in the sense of 1.5. It is necessary to emphasize that in affine geometry an

improper point must not be regarded as having the same rights as ordinary points: there does not exist an affine transformation which transforms an improper point into an ordinary point.

7.2


73

All affine geometric objects with which we will have to deal below will be defined in such a manner that the law of their transformation will be a consequence of the laws of transformation of the coordinates of points, straight lines, vectors; therefore, starting from the above results (cf. also Exercise 2), they may always be converted into projective geometric objects by stating the law of transformation of their coordinates in the case of projective transformations (and thereby also for transformations of projective coordinates). Now, let an affine geometric object a be defined by the coordinates a,, a2, ..., aN in an affine coordinate system, i.e., under the assumption that the improper straight line is x3 = 0, and let lp(a I

, a2,

.

, aN)

be an affine invariant of the geometric object a. Further, let S, be an arbitrary transformation of the projective coordinates under which the equation of the improper straight line I assumes the form (7.3). By converting a into a projective object, we may find its coordinates b1 , b2, ..., bN after the transformation S,, where

(p(a,.a2,...,aN) = 0(h,, h2,...,bN,11,12, 13) (since the parameters of the transformation will depend on the 1,). If T, is another transformation of projective coordinates for which also x3 = 0 l;x' = 0, then U = S,-'T, is a transformation of the form (7.2) and T, S, U; U leaves cp unaltered and S, transforms q into Iii. Thus, the function I(i depends only on the coordinates of the object and on the coordinates 1, of the improper straight line. Now, let S be an arbitrary transformation of projective coordinates, where in the old coordinate system the improper straight line has the equation (7.3) and in the new system the equation 1;, x' = 0.

Then S = S, transforms x3 = 0 into 1; x'= 0, S,-' transforms q1(b,,..., bN, 11,12,13) into cp(a,,..., aN) and S, transforms fp(al, ..., aN) into 11,, l2-, 13), where b1-, ..., bN are the coordinates of the object a in the new system of projective coordi-

nates. Thus

i.e., 0 represents a joint invariant of the object a and the improper line l;

74


CHAP. I

with respect to transformations of projective coordinates, and conseq uently also with respect to projective transformations. Conver-zely, every such invariant obviously becomes an affine invariant, if we set in it

11 = 12 = 0,13 = 1. Thus, one arrives at

Every affine invariant of a geometric object a is a joint projective invariant of the geometric object a and the improper Theorem 7.2:

straight line I. Analogous arguments can also be produced in the case of invariant equalities *).

Thus, in order to construct an affine geometry inside projective geometry, we must select in the projective plane some straight line 1 and exclude the points of this straight line from the points of the plane. The projective properties of the figure with respect to the straight line 1 will be affine properties, and projective quantities related to the figure under consideration and the straight line I will be affine quantities. We will illustrate these results by some examples. Examples:

1. In affine coordinates, the condition that two straight lines are parallel has the form Ul (7.10) V1

We will perform an arbitrary projective transformation St under which

the improper straight line x3 = 0 corresponds to the straight line lizi = 0 **). Without loss of generality, we can assume that 13 # 0; then we can take for S, the transformation X1 = X1,

x2 = z2,

X3 = It X1 + 12X2+- 13 X3,

d = 13 # 0,

(7.11)

Substituting the expression for x` from (7.11) into the equation of the straight line uix'=0, we find an equation which gives the new coordinates of the straight line as functions of the old coordinates; solving for the ') In the case when a property is expressed by several equalities which are together invariant, a difficulty arises the resolution of which would take us beyond the scope of this book (cf. [16], Abschnitt VI, § 9 and Abschnitt IX). ") We did not speak above about a projective transformation, but about a transformation of projective coordinates; obviously, this distinction is immaterial.

75


7.2

old coordinates, we find .

11

I2 .

.

U3 =

U2 = U2- - U3,

UI = U1- j U3,

U313

13

13

I1 *

The left-hand side of (7.10) now becomes U2-

I,

lI2

t!1

U1

U3

11

V3

12

V2 13

-

12 v3 13

13

1

V2

1'I

, V3

U1

U2

U3

VI

1)2

V3

11

I2

13

13

I

13

as a consequence of which (7.10) assumes the form (where asterisks have been omitted) [uvl] = 0. (7.12) The condition (7.12) means that the straight lines u, v intersect at a point which lies on the improper straight line I [cf. (6.22)].

Fig. 10

In defining projectively (with respect to the improper straight line 1) parallelism of straight lines, we can also define in the same manner equa-

lity of contravariant vectors; one has in Fig. 10: ABIICD, ACJIBD,

and hence AB = CD.

CHAP. I


76

2. The ratio in which a point z of the straight line [xy] divides the segment xy is equal to the cross-ratio {xyzt}, taken with opposite sign, where t is the point at which [xy] intersects the improper straight line I (§ 5, Exercise 4; on the straight line, affine and Euclidean geometries do not differ). 3. The condition for the non-degenerate second order curve

aijxx,

aij = aji,

i, j = 1, 2, 3,

(7.13)

to be a parabola is expressed in affine homogeneous coordinates by the equality

2

=

a33

=0

63

[cf. (6.36)1; since in such coordinates for the improper straight line 11 = /2 = 0, 13 = 1, this condition can be rewritten in the form a'j1i 1j = 0.

(7.14)

This equality means [cf. (6.38)] that the improper straight line 1 touches the curve F; as a consequence of its invariance (cf. § 6, Exercise 27), it will have the same form in any system of projective coordinates. The affine property of belonging to the class of parabolas has thus found a projective interpretation.

7.3 We will now proceed to a study of the relationships between the group of motions and reflections and the affine group: let us assume that the transformations of both groups have been written down in Cartesian non-homogeneous coordinates [cf. (3.2), (4.1)]. Obviously, the set of all transformations of the group of motions and reflections corresponds to a manifold of all circles of the plane with unit radius. It is readily shown that this property characterises the group of motions and reflections within

the affine group. The affine transformation

x = px'+qy'+h, y = rx'+sy'+k transforms the unit circle

(x-«)2+(y- fl)2-1 = 0

(7.15)

into the curve 0,

(7.16)

7.2-7.3


77

where a', fi' are the coordinates of the point corresponding to the point (a, f). The'curve (7.16) will be a circle with unit radius, if and only if

p2+r2 = 1,

q2+s2 = 1,

pq+rs = 0.

(7.17)

On the basis of the first two conditions (7.17), we may write p = cos w, v = -sin to, q = sin cp, s = cos cp; the third condition (7.17) then gives: sin ((p-w) = 0, p = w+mn, where m is an integer. The equations of the affine transformation now assume the form

x = x' cos w+ey' sin w+h, y = -x' sin w+ey' cos w+k,

e = (-1)m = ±1;

we thus have arrived at the equations (3.2) (with co replaced by -co, which corresponds to the replacement of x', y' by x, y and conversely). From the point of view of affine geometry, the concept of the circle has no meaning; in affine geometry, the equations (7.15) define a system of ellipses obtained from each other by parallel translations in terms of

the variables a, ft (note that parallel translation is an affine concept); in arbitrary affine coordinates, the ellipses of this system will have the equations 9if(x1-a1)(xJ-a1)-1

9i;=9;1,

= 0,

i,J = 1, 2,

(7.18)

g=I91;I>0,

where a', a2 are the coordinates of the centre of the ellipse. For convenience, we will call all curves which are obtainable from each other by parallel translation, a T-system of curves; the T-system of ellipses (7.18) is completely defined if the geometric object g1j with the three nonhomogeneous coordinates g1l, 912 = 921, 922 is specified. In the case of the affine transformation (4.24), these coordinates change according to the law [cf. (4.39)] 9.# = pa

9ii ,

i, .1, a, fl = 1, 2.

(7.19)

If we wish to study metric geometry in arbitrary affine -coordinates, the equatiofl of all unit circles likewise assumes the form (7.18). Thus, we arrive at Theorem 7.3: The group of motions and reflections of the plane represents the manifold of all those affine transformations of the plane which leave a given T-system of ellipses of this plane unaltered.

CHAP.


78

I

The geometric object g. of (7.19) determining the above T-system of ellipses is called a metric tensor (metric, because it defines a metric, i.e., a measure of lengths and angles in the plane; the term ,tensor" will be

defined in its full generality in § 9). Reasoning as in 7.2, one may prove Theorem 7.4: Each metric invariant of a geometric object a is a joint acne invariant of the object a and a T-system of ellipses G. An analogous result applies to the case of invariant equalities. Thus, in order to construct a metric (Euclidean) geometry within affine geometry, one must select in the plane some T-system of ellipses G and study the affine properties of a figure with respect to this T-system; these will be the metric properties of the figure. The affine quantities associated with this figure and the T-system G will be metric quantities. The following examples will serve to illustrate these statements. Examples:

1. The invariant n of a second order curve F [cf. (4.38)] is given in Cartesian coordinates by a,,+ a22; in the same coordinates, the metric tensor g;; will be: gl, = g22 = 1, 912 = 0. After an affine transformation S, we have, by (7.19), I

z

I

(7.20)

ga# = Ps Pp + Pa PB ,

and [cf. (4.41)] g=I9a6I=Ig1;I42=42,

4=IPaI.

(7.21)

Further [cf. (4.39)], we find

n = a11+a22 = giq,aa,+gigiaa#,

(7.22)

where the qi are the parameters of the inverse transformation S`

1

and are therefore related to the p; by the formulae [cf. (4.21)]

q11

P2

,

g2=I

P2

q1=- P i 2

q2=Pie, 2

as a consequence of which (7.22) assumes the form

{(Pi)2+(pi)2}a11-2(PiPi+PiPI)512+{(p )2+(Pi)2}a22 42


7.3

79

Using (7.20) and (7.21) and omitting asterisks, we obtain, finally, 922 all `2912 a12+911 a22 911922 _ (g 12)

7.23

The numerator on the right-hand side of (7.23) is a relative invariant for projective transformations of weight 2 [cf. (5.25)], so that, turns out to be a joint absolute affine invariant of the left-hand side of the equation

of the curve T and the metric tensor g,1. Obviously, the formula (7.23) for the invariant n also expresses it in Euclidean geometry if we introduce affine coordinates.

2. The scalar product ab of two contravariant vectors a and b in Cartesian coordinates is equal to a'b'+a2b2, i.e., (since g11 = g22 = 1. 912 = 0) ab = g,1 a'b1.

(7.24)

As a consequence of the invariance of the right-hand side of (7.24) with respect to affine transformations (§ 4, Exercise 10), it preserves its form after any affine transformation; thus the scalar product is defined as a joint affine invariant of the vectors a, b and the metric tensor g;1. On the basis of the definition of the scalar product as an affine invariant

with respect to a T-system of ellipses G (cf. Exercise 10), we can also obtain the definition of the square of the length of a vector (and consequently of the distance between two points of the angle between two vectors and of the angle between two straight lines). 3. The pseudo-scalar product of two vectors a x b in Cartesian coordinates is given by a'b2-a2b'; by Theorem 4.2, we have after an affine transformation [cf. (7.21)]: a1b2_a2bI 5'b2-a2b'

a'b2-a2b'

a

vg

Thus, in arbitrary affine coordinates, the pseudo-scalar product is given by

axb -- [ab] _

(7.25)

Jg

The formula (7.25) represents the pseudo-scalar product as a joint affine invariant of the vectors a, b and the tensor g,, . The presence of the square root in the denominator makes the invariance of the right-hand side of (7.25) incomplete; if the determinant of the affine transformation

CHAP. I


80

A < 0, this transformation changes the sign of the right-hand side of (7.25) (cf. Exercise 8). Obviously, this circumstance is linked to the fact that the pseudo-scalar product is an invariant of the group of motions,

but not an invariant for reflections (cf. 3.1). 7.4

In order to construct Euclidean geometry within projective

geometry, it is more convenient to start from the principal group. In Cartesian homogeneous coordinates, as a particular case of projective coordi-

nates, the equations of transformations of the principal group can be written in the form [cf. (3.3)]: z' = p(cos to - x' -E sin co x2)+hx3, 12 = p(sin w x' +E cos (0 x2)+kx3, E = +1' xs = x3

(7.26)

(where it will be more convenient to use index notation for Cartesian homogeneous coordinates also). The coordinates u; of the straight line are expressed in terms of the coordinates ui of the transformed straight line by a system of linear equations with a matrix which we know can be

obtained from the matrix of the system (7.26) by an interchange of columns and rows: ul = p(cos co u1 +sin co - uz), 112 = ep(-sin co u, +cos co u2), r = ± 1,

(7.27)

u3 = hul+kuz+u3. We obtain from the first two equations (7.27) (u1)2+(u2)2 = P2{(UI)2+(U2)2},

i.e., the second class curve with the equation (7.28)

(u1)2+(u2)2 = 0,

remains unaltered by any transformation of the principal group. Conversely, if we require that the projective transformation If

x = qa xa ,

ua = qa Ui , ;

1, a = 1, 2, 3

is to leave the curve (7.28) unaltered, we find

(q')' i+(qi)z 2

1)2 zz (9z+(qz) # 0,

q q3+q q3 = 0,

t

t

2

2

q1 qz+ql q2 = 0, 9293+g2 q 3 = 0, (93)2 x(93)2 = 0.

7.3-7.4

REDUCTION OF AFFINF INVARIANTS TO PROJECTIVE ONES

81

Let a2 be the value of both sides of the first of these equalities; reasoning [cf. (7.17)], we obtain (assuming the parameters of the transformation to be real) *)

as in (7.3)

qi = a cos w, qi = o sin co, q2 = -ea sin w,

q3=qs=O,

q2 = eo cos w,

= ±l.

With the notation p = a/q3 , and using the fact that the parameters q, are homogeneous, we reduce the equations of the transformation to the form (7.26). Writing (7.28) in the form

(iul+u2)(-iul+u2) = 0, we see that a straight line with coordinates which satisfy the equation (7.28) (such a line is called isotropic) passes through one of the points 1 and J with coordinates (i, 1, 0), (-1, 1, 0); for this reason one says that the equation (7.28) represents the pair of points I and J (the second class curve degenerates into two points). The points I and J are called circular points at infinity; thus, the transformations of the principal group

are characterized by the fact that they leave unaltered a manifold of circular points at infinity I and J (cf. Exercise 12), and hence also the improper straight line which joins them. From the point of view of projective geometry, we can formulate this result in Theorem 7.5: The principal group is the manifold of all projective transformations of the projective plane which leave a given straight line I and a pair of complex conjugate points I, J on this straight line unaltered.

Correspondingly, we also have Theorem 7.6: An invariant of a geometric object a with respect to the principal group is a joint projective invariant of the object a, the straight line 1 and the pair of points 1, J. Analogous results may be derived for invariant equalities. The straight line l together with the points I, J is called the absolute in the Euclidean

geometry of the plane. Thus, in Euclidean geometry, the projective properties of a figure are studied with respect to the absolute. The fact *) On the basis of the condition a # 0, it is readily shown that the result remains the same, if we permit the parameters qa to assume complex values.

CHAP. I


82

that one studies only invariants of the principal group is not material, since the equality of segments, and, consequently, of any figures can be interpreted as a projective property with respect to the absolute (Exercise 14). Examples :

1. In Cartesian homogeneous coordinates, a circle is given by an equa-

tion of the form a111(x1)2+(X2)2}+2a13X1X3+2a23X2x3+a33(x3)2

= 0, all 0 0;

direct computations show that every circle passes through both circular points at infinity. Conversely, if a non-degenerate second order curve (7.13) passes through both circular points at infinity, one has

all-2a12 i-a22 = 0,

all+2ai2 i-a22 = 0, whence

all ' a22,

a12 = 0,

i.e., the curve is a circle (a11 0 0, since otherwise the curve degenerates). The terminology of circular points at infinity follows from this charac-

teristic property of circles. 2. Let u, v be two intersecting straight lines; use their point of intersection 0 as origin and the straight line u as x-axis of a system of Cartesian homogeneous coordinates so that the coordinates of the line u are (0, 1, 0). Let cp be the orientated angle between the lines u, v; it is then easily seen that we may introduce for the line v the coordinates (sin cp, -cos gyp, 0) [cf. (1.11) and § 1, Exercise 4]. The isotropic straight lines p, q passing through the point 0 will have the coordinates (i, 1, 0) and (-i, 1, 0). It is readily seen that k = e19' ,

P = Au + µv,

A

q = A.'u +µ'v,

e

whence (by Theorem 6.5) the cross-ratio {uvpq} =

= e2i" N'

7.4.-7.5


83

and

?log {uvpq}.

(P = I

We have obtained the so called Laguerre formula determining the angle between two straight lines as a projective quantity related to the straight

lines u, v and the absolute plane. If we interchange the straight lines u and v [cf. (5.15)], the right-hand side of (7.29) changes sign, so that the angle 0 is actually orientated. For an interchange of p, q, the angle q likewise changes sign; transposition of p and q is equivalent to the replacement of I by J and of J by I. Thus, transposition of the circular points at infinity leads to reversion of the orientation of the plane.

The ideas studied in 7.2-7.4 were first formulated by F. Klein in the Erlangen Program [2]. Thus, we see that affine invariants may be reduced to projective ones, and metric invariants to affine, and hence to projective invariants; however, the invariants of the principal group reduce directly to projective ones. 7.5

This aspect of Euclidean and affine geometry points the way to a differ-

ent kind of generalization. In this, by introducing an absolute straight

line with two real points on it, we obtain Minkowskian geometry; in the case of a four-dimensional space, this represents the geometric foundation for the general theory of - relativity. If we take as the absolute a non-degenerate second class curve

uu = 0a'j=a11

a`j

la`'I

0,

(7.30)

or, what is the same thing (cf. Theorem 6.13), a non-degenerate second order curve I', we arrive at Non-Euclidean geometry. If we have real points on the curve and if in (7.30) all a'J are real, we are led to Lobachevskian geometry; however, if for real a'' the curve 1' has no real points,

we are confronted with Riemannian geometry. It is well known that F. Klein first proved in this way the absence of contradictions in Lobachev-

skian geometry. In conclusion, the following remark must be made: at the basis of all the reasoning in our present study lie eventually Cartesian coordinates whose definition is metric. As a consequence, the possibility of the construction of affine and Euclidean geometries on a purely projective foundation remains unproven.


84

CHAP. I

In order to remove this difficulty, we must establish our projective geometry without any utilisation of metric or affine properties of figures and such quantities. The usual approach to this task is a purely axiomatic design of projective geometry; the fundamental ideas of such a procedure were given by G. K. Ch. von Staudt *). In this case the principal goal is to prove that in the projective plane (or in projective space) one can introduce projective coordinates on the basis of axiomatic projective geometry only. The reader may become acquainted with this type of founda-

tion of projective geometry by reading, for example, N. V. Efimov's "Higher Geometry", Chapter V1, §§ 110-136. After such a basis has been adopted for the establishment of projective geometry, the above difficulty disappears completely. Exercises

1. Let the projective coordinates of the points A, B, C, D in Fig. 10 be x', y', z', t'

(i = 1, 2, 3), respectively. Show that the components of the vectors AB and CD computed by use of the rule (7.5) are equal to each other. 2. What coordinates must be used in a system of projective coordinates to specify a covariant vector (doublet)? 3. Derive from the theorems proved in § 6, Exercises 14 and 15, propositions of of lne

geometry, assuming the straight line v to be improper. As a result one obtains the theorem of Menelaus and Ceva (cf. § 6, Exercise 18). 4. Represent the ratio of the orientated areas of two triangles as a projective invariant with respect to the improper straight line and give its projective geometric interpretation. 5. Derive (7.14) by reasoning in the same manner as in the derivation of (7.12) in

Example I of 7.2. 6. Give projective (with respect to the improper straight line) interpretations of asymptotic direction, centre, diameter and conjugate diameter of a second order

curve r. 7. What are the geometric meanings of the numbers g,5 in Euclidean geometry, expressed in affine coordinates? 8. What sign must be given to the root in (7.25)?

9. Express the distance of the point (x', x') from the straight line (u,, u,, h) in affine coordinates. 10. Give the equalities

g($a'a' = A > 0, g,ra'b' = 0, gua'b' = u, i, j = 1, 2 (7.31) geometric interpretations with respect to a T-system of ellipses G, determined by the tensor gee, where a', b' (i = 1, 2) are the components of two vectors a, b. In the third of the equalities (7.31), the vector a is assumed to have unit length. *) G. K. Ch. von Staudt, Geometry der Lage, Ntlrnberg,1847; BeitrAge zur Geometric

der Lage, Ntirnberg, 1856-1860.

MULTI-DIMENSIONAL SPACE

7.5.8.1

85

11. Show that the condition rl = 0, i.e., g11au-2g11a11+8a,aji = 0 has the following affine geometric meaning: the asymptotic directions of the curve (4.38) arc parallel to two conjugate diameters of the T-system of ellipses (7.18). 12. Show that for e = I the transformations of the principal group leave each of the circular points 1, J unaltered, while for e = -1, 1 becomes J and J becomes I. 13. Starting from Laguerre's formula, explain the projective meaning of the perpendicularity of two straight lines. 14. Define the equality of segments as a projective property in relation to the absolute. 15. A second order curve 1' is given in Cartesian homogeneous coordinates by the

equation (7.13) with complex coefficients; what is the geometric significance of the 0? (cf. § 1, Exercise 11). equality

§ B.

Multi-dimensional space

8.1 We have shown above for the case of two-dimensional space that, guided by the idea of the invariants of a group of transformations, one may arrive by purely analytic means at all basic concepts of Euclidean, affine and projective geometries. With the aid of analogous arguments we may do the same thing in the case of three-dimensional space, where in many cases the only difference will be that the indices of the coordinates

will assume the values 1, 2, 3 and not 1, 2 (or, in the case of projective geometry; 1, 2, 3, 4 instead of 1, 2, 3). In all such questions there do not arise any new difficulties if we allow the number of dimensions of the space to be equal to any natural number; thus, we arrive at the idea of multi-dimensional space which proves to be of great use in geometry itself as well as for its applications, especially in mechanics and physics. In contemporary physics, the concept of multi-dimensional space lies at the basis of all important theories. In the case of the construction of a multi-dimensional geometry, where it is now impossible to be guided by visual methods, the leading role of the group theoretical point of view in geometry and invariant theory assumes a unique significance. In the following study, we will adopt the following conventions: in the case of Euclidean and affine spaces, the number of dimensions will always be denoted by n; the number of dimensions of projective space will be denoted by n -1. In all three cases, we will call the number n the order of the space. Thus, the order of an Euclidean or affine space is equal to its dimension; in the case of a projective space, the order of a space exceeds its number of dimensions by unity. In the absence of specific reservations, all indices will be assumed to run through the values 1, 2, ..., n, where is is the order of the space.

CHAP. I


86

8.2 In advancing to a detailed study of multi-dimensional spaces,

we will begin with projective space of order n. We consider a manifold P. each element of which consists of a system of n numbers xI, x2, . .., x" at least one of which must differ from zero. Conversely, let every such system of numbers correspond to an element of the set P"; if two systems of n numbers are proportional to each other, i.e., if each number of the first system can be obtained from the corresponding number of the second system by multiplication by one and the same number A:00, these systems correspond to the same element of P. However, systems of n numbers which are not proportional correspond to different elements of the manifold P. We will call each of the elements M of P. a point and the numbers , x2 , . . ., . , x", the components of the element

M, the coordinates of the point M. The system of coordinates in P", defined in this manner, is not the only

possible one. If we write x! = pi. x",

i = 1, 2, ... , n,

i' = 1', 2',

. .

.,

n',

(8.1)

where the numbers pI are the same for all points of the manifold P. and limited in their arbitrariness only by the condition that the determinant Ipf'I # 0,

(8.2)

then every point M of P. will be referred to a new system of numbers x1', x2',

..., x"" where the correspondence between the systems of

numbers x", x2', . . ., x" and the elements of P" retains its character. If in this manner we can exhaust all systems of coordinates with the above properties, we will call them systems of projective coordinates in P.

However, if the manifold P. admits also other coordinate systems with the same properties, we will select from all those systems some definite set such that each two systems of coordinates of this set will be interrelated by equations of the form (8.1) and we will call them a system of pro-

jective coordinates'). After specification of a projective system of coordinates inside the manifold P", it becomes a projective space of order n or of n -1 dimensions. In a projective space, all possible systems of projective coordinates

are completely equivalent; none of these systems receive any special preference. ) The first of these two cases occurs when we have complex values of x', x', ... , x",

the second for real x', x', ..., x".

8.2


87

The points with coordinates (1, 0, . . ., 0, 0), (0, 1, ... , 0, 0), ... , (0, 0, ..., 0, 1), (1, 1, . . ., 1, 1) are called first, second, ..., n-th and unit coordinate points, respectively; the first n of these points form the coordinate n-hedron. Reasoning as in §§ 5, 6, it is readily verified that we may select as coordinate points any n + 1 points of the space P", provided only all n of them are independent (k points are independent, if the k systems

of n numbers which are the coordinates of these points are linearly independent). A projective transformation of the space P. is defined by the equations

xi = paza,

(8.3)

where the determinant of the transformation d = I Pal

by (8.3), every point M(x1, x2,

0;

(8.4)

... , x") of the space P. corresponds to a

point M (z I, j2'. . ., X") of the same space (the coordinates of both points are referred to the same system of projective coordinates); as a result of (8.4) this relationship is single-valued and invertible. If we denote

by q; the reduced minor of the element pa in the determinant d, we can verify by the same reasoning as in 6.2 that

z° = qr xi;

(8.5)

the only difference from § 6 will be that the indices i, j, a will now assume the values 1, 2, ..., n instead of 1, 2, 3. The projective transformations of

space form a group with n2 -1 parameters (Exercise 1). The Proof of Theorem 5.1 is extended without difficulty to the case of

arbitrary n. Thus, we have Theorem 8.1: There exists one and only one projective transformation of the space P. by which n+ 1 arbitrary points with the property that every n of them are independent correspond to n+ I specified points every n of which must also be independent. Introducing now, in the same way as it was done above, the concept of the projective geometric object, we can define the problem of (n -1)dimensional projective geometry: (n - I)-dimensional projective geometry studies those properties of geometric objects defined in P. and the quantities related to them which are invariant with respect to projective trans-

formations of the space P" . Obviously, invariants of projective transformations will also be inva-


88

CHAP. I

riants of the transformations of projective coordinates given by (8.1), so that the establishment of a projective geometry is in complete agreement with the equivalence of all systems of projective coordinates, stated above. If the coordinates of the point x1, x2, ... , x" and the parameters of the projective transformation pQ are real, we have projective geometry in the real domain; if we admit that these numbers can also have complex values, we arrive at projective geometry in the complex domain. Many problems

of projective geometry can be dealt with in the same manner whether they refer to the real or the complex domain; in cases where the distinction is essential, we will often refrain from stating this fact, especially as this will always be sufficiently clear from the context. We will now acquaint ourselves with some of the simplest concepts of multi-dimensional projective geometry. The set of all points of the space P. whose coordinates x', x2, . . ., x" satisfy the equation

u, x' = 0,

(8.6)

is called a h y p e r p l a n e of the space P"; the numbers u1, u2 , ... , u" of the

coordinates of this hyperplane are obviously homogeneous. The projec-

tive transformation (8.3) changes (8.6) into the equation u, pa z° = 0 or 4.z° = 0, where

Z. =

f

pQ u,;

the projective transformation relates the hyperplane u, to the hyperplane u, whose coordinates are determined in terms of the coordinates u, by (8.7). Next, we will consider a manifold Pt consisting of all those points of the space P whose coordinates satisfy the equations of the n - k

hyperplanes (2 5 k 5 n -1): 1

u, x' = 0,

2

u, x` = 0,

"-k

... ; u, x' = 0;

(8.8)

the hyperplanes (8.8) will be assumed to be independent, i.e., the n-k 1 2 "-k 2 "-k systems of n numbers u1, . . ., u., u1, ... , u,, , u1, ... , u" are linearly independent. In accordance with known results of the theory of linear homogeneous

equations, k points of the set PI, will be independent: x', x', ... , x'; 1

2

It

8.2

89


the coordinates x` of every other point of the manifold Pk can be represented by

x' _ A'x'+A2x'+ 1

... +Akx'.

2

(8.9)

k

If we renormalize the coordinates of the points x, x, . . ., x or replace 1

2

them by k other independent points of the set, the numbers A', A2, ..., Ak become subject to a linear homogeneous transformation. Thus, each of the points of the manifold Pk is determined by specification of k numbers Al, 22, . . ., 2k which are given exactly apart from a common multiplier, and these numbers may be subjected to an arbitrary linear transformation which must be the same for all points of Pk. Consequently, Pt is a projective space of order k and the numbers A', 22, ..., Ak may be conceived

as projective coordinates of its points; therefore the manifold Pk is called a (k -1)-dimensional plane of the space P.. We note that we can now characterize in geometric terms the dependence or independence of points: k points of a projective space will be dependent if they lie on the same (k - 2)-dimensional plane, they will be independent if it is impossible

to draw such a plane through the k points. For k = n -1, there is only one equation (8.8), and as a consequence an (n - 2)-dimensional plane is a hyperplane. A two-dimensional plane is called simply a plane, so that for n = 4 the concepts of hyperplanes and

planes coincide. For k = 2, we obtain the one-dimensional projective space which is called ordinarily a straight line of the space P,,; if x, y are two of its points, the coordinates of any point z of this straight line are given by z' = .lx'+µy'; (8.10)

.reasoning as in 6.3, we can, starting from (8.10), define the cross-ratio of four points of the straight line and pairs of points which divide or do not divide each other. It is not difficult to see that in a projective space of order n the duality principle applies to points and hyperplanes. Thus, we may introduce the

concepts of first, second, ... , n-th and unit coordinate hyperplanes; one may use for this purpose any n + 1 hyperplanes every n of which are independent; the projective coordinate system is then completely determined. If u1, v1 are two different hyperplanes, all the hyperplanes wI = Aul+p.vi

(8.11)


90

CHAP. I

will pass through the (n - 3)-dimensional plane in which the hyperplanes u,, v, intersect; they form a pencil of hyperplanes. In connection with the pencil of hyperplanes, one can also introduce the concepts of cross-ratio of four hyperplanes and of pairs of hyperplanes which divide or do not divide each other (cf. 6.3). We will stop at this point; later on, as we take an interest in the geometric propositions of invariant theory, we will study multi-dimensional projective geometry in greater detail.

8.3 We proceed now to the consideration of multi-dimensional affine space. Let there be given a manifold A. the elements of which may be placed in a single-valued invertible relationship to a system of n numbers xx2, ..., x". We will call an element of A. a point and its components, the system of numbers x', x2, ..., x", the coordinates of the point M. In A,,, we may have different coordinate systems; we will select from them

a set of systems which are interrelated by equations of the form x' = pi. xt" + hi,

i = 1, 2, ... , n, ' i' = 1', 2', ... , n',

(8.12)

and we will call them systems of affine coordinates (cf. beginning of 8.2). We will call the manifold A. with the systems of affine coordinates given within it an affine space of dimension n (or of order n).

An acne transformation of the space A. is given by the equations

x' = paza+h`,

(8.13)

where it is assumed that the determinant of the transformation (8.14) d = IPaI # 0; the equations (8.13) relate every point (x', x2, ..., x") of the affine

space A. to another point of this space whose coordinates with respect to the same system of affine coordinates are (z', z2, ..., z"); by (8.14), this correspondence is single-valued and invertible. Solving the equations

(8.13) for the z', as has been done in 6.2, we find .za

= 4ix'+)r",

(8.15)

where q, is the reduced minor of the element pa in the determinant (8.14),

so that a

t = Sa

gIP6

a

and

ha = - qih`.

8.2-8.3


91

Affine transformations form a group with n2+n parameter (Exercise 1). The definition of an affine geometric object, given in 4.7, remains in force in a space of an arbitrary number of dimensions; the problem of affine n-dimensional geometry comprises the study of those properties of geometric objects, defined in A,,, and of quantities related to them which

are invariant with respect to any affine transformation. A distinction must be made between the affine geometries in the real and complex domains; in this respect, we could repeat everything said in 8.2 with respect to the two projective geometries. In affine geometry, the simplest and at the same time the basic geometric

object, apart from the point, is the contravariant vector. One can arrive at this concept in the following manner. From the affine transformations we select those for which pa = 5 , i.e., the so called parallel translations; changing the notation, we can give every one of these transformations

an equation of the form

yl = x'+a'.

(8.16)

A parallel translation is determined by the numbers al, a2, ..., an. We will now subject the space to the affine transformation (8.15); it follows then from (8.16) that

qjy'+hl = qjx'+hl+q7a', i.e.,

y` = x°4`, where

a° = qja',

8.17)

and consequently aI

(8.18) = thus, in the case of an affine transformation of a space, a parallel translation goes over into a parallel translation.

A geometric object which is determined by n non-homogeneous coordinates al, a2, ..., a' and for affine transformations changes in accordance with (8.17), (8.18) is called a contravariant vector; every contravariant vector determines a parallel translation. Obviously, a parallel translation will be specified if we state for one point of space x' the point y' which

corresponds to it as a consequence of the translation; therefore it may

CHAP. I


92

be said that a contravariant vector is determined by specification of two points and a statement of their order. The point x' is called the starting point of the vector, the point y' its end point; the components of the vector will be y'-x'. We note that from this point of view we may consider the coordinates of the point x' as the components of a vector which starts from the point (0, 0, . . ., 0) i.e., from the origin of coordinates, and ends

at the point x; this vector is called the radius vector of the point x. Two contravariant vectors are equal if their components are equal. Let each of a pair of equal vectors be determined by a pair of points; then there exists a parallel translation which shifts the starting point x of the first vector to the starting point z of the second vector and the end point y of the first vector to the end point t of the second vector. In fact, if

yl-x' = ti-ZI, then

z' -

x' = t' - y';

(8.19)

let the common values of the left- and right-hand sides of (8.19) be equal

to a'; then the parallel translation (8.16) will also have this property. The vector Aa' is the product of a number A, and the vector a'; vectors

which are derived from each other by multiplication by a number are called collinear. The vector a' + b' is called the sum of the vectors a' and b'. This operation on vectors obviously has the same property as the corres-

ponding operation in two- and three-dimensional spaces. In particular, in the case of addition of vectors, we have the known triangular rule. The invariance of the above definitions of operations on vectors with respect to affine transformations (8.18) can be established directly. The contravariant vectors e (1, 0, . . ., 0, 0), e (0, 1, . . ., 0, 0), ... 2

e(0, 0,

... , 0, 1) are called coordinate contravariant vectors. If a contra1

variant vector a has components a', a2,,..., a", then a = a'e+a2e+ ... +a"e. 1

2

w

Since the coordinates of a point are the components of its radius vector, an affine coordinate system is completely determined by specification of

the origin of coordinates and the coordinate contravariant vectors; it is readily seen that we can select as origin an arbitrary point of the space A. and as coordinate vectors n arbitrary independent contravariant


8.3

93

vectors of the same space (which have linearly independent components). A manifold consisting of all points of an affine space whose coordinates satisfy the linear equation

ci x+ h = 0

(8.20)

is a hyperplane of this space; the numbers cl, c2, ..., c,,, h are the homogeneous coordinates of this hyperplane. Reasoning as in 8.2, we can show that the affine transformation (8.13) transfers a hyperplane into a hyperplane and that the law of transformation of the coordinates of a hyperplane is

ea = pact,

h

= cih'+h.

(8.21)

Two hyperplanes one of which is defined by (8.20), the other by

cix'+h=0,

huh,

(8.22)

do not have a single common point and are therefore called parallel. It is clear from (8.21) that there exists a parallel translation which transfers the hyperplane (8.20) into the hyperplane (8.22), parallel to it; the vector a' of this parallel translation must satisfy only the condition ci a' = h - h [the parallel translation (8.16) is obtained from (8.13) by letting pa

hi = -a'].

(8.23)

= as

It follows from this that two pairs of planes, the first with coordinates ci, h and ci, k, the second with coordinates ei, h and ci, k can be transformed into each other by a parallel translation, if and only if

ci = pci,

h-k = p(h-k)

(where we have taken into consideration the homogeneity of the coordi-

nates of the plane), i.e., if ci

h-k

h-k

Let

h-k'

(8.24)

in this way we have defined a new geometric object with homogeneous coordinates u,; u2, ..., u,,; in the case of an affine transformation of the

94

CHAP. I


space, these coordinates change according to the law (8.25)

ua = par u! .

as is shown by (8.21). The geometric object defined in this manner is called a covariant vector or a doublet; it is given by a pair of parallel hyperplanes and a statement of their order. If two pairs of hyperplanes can be transformed into each other

by a parallel translation, they have been shown above to correspond to the same covariant vector, or they are said to be equal covariant vectors. Just as in 4.5, one can define multiplication of a covariant vector by a number or addition of covariant vectors for any n, and both these operations

can be given an geometric interpretation analogous to that of 4.5. We note also that by (8.21) the first n coordinates c1, c2, ..., c of a hyperplane determine a covariant vector exactly apart from a numerical factor. In conclusion, we will also introduce the concept of a k-dimensional plane of the space A,,; this is the name given to the manifold At of all those points of the space A. whose coordinates satisfy the equations of the n - k hyperplanes 1

a-k

2

2

1

uix'+h = 0, ufx'+h = 0, 1

a-k

...; ui + h = 0,

(8.26)

a-k

2

..., ui are assumed to be independent.

where the covariant vectors u1, U1,

As a consequence of this assumption, the system (8.26) has solutions; if x' is one of these solutions and a', a', ..., a, are k linearly independent 1

k

2

solutions of the system of equations obtained from (8.26) by setting all h

equal to zero, the general solution of the system (8.26) will be

x' = x'+A'a'+22a'+ 0

1

it is readily seen that the a', all 1

2

2

... +Aka1;

(8.27)

k

., at also represent vectors. k

We see that each point of the manifold A. is determined by the specification of k numbers Al, A2, . . ., )k; we conclude from this (cf. 8.2) that Ak is a k-dimensional affine space, and hence the terminology introduced above for this manifold is justified.

Obviously, an (n -1)-dimensional plane is a hyperplane; the twodimensional plane is called simply a plane. For k = 1, we obtain the

8.3-8.4

95


straight line

x` = x'+).a';

(8.28)

0

the points of the straight line (8.28), for which 0 5 A 0

(8.30)

and

for any system of n numbers a', a2, . . ., a" at least one of which is nonzero C). The law of transformation of the coordinates of this geometric ) Below (Theorem 29.6) we show that the relation (8.29) is automatically satisfied if the condition (8.30) is fulfilled.

96


CHAP. I

object for transition to a new system of affine coordinates is P; P; gl;

(8.31)

The geometric object g;; is called the metric tensor of an n-dimensional Euclidean space; by (8.31), it is defined in any system of affine coordinates. We will now study those affine transformations which leave the metric tensor unaltered; this means that the parameters pa of the transformation

satisfy the relation gap = pa P'p g,>;

(8.32)

the manifold of all these transformations forms a group (Exercise 3) which is called the group of motions and reflections of n-dimensional space.

Euclidean n-dimensional geometry studies those properties of affine geometric objects and the quantities related to them which remain invariant for any of the transformations of the group of motions and reflections. Since the group of motions and reflections is a subgroup of the affine group, invariance with respect to the affine group also implies invariance

with respect to the group of motions and reflections. Hence all the concepts of 8.3 are retained in Euclidean geometry. However, in addition, there will also occur special Euclidean invariants. Let a', b' be the components of two contravariant vectors*) a, b which in a certain transformation of the group of motions and reflections (with parameters p', h') correspond to the vectors aa, ba; it follows from (8.32) that gap aabp = Pa aap.J0bpg,J, i.e., [cf. (8.18)]

gap aabp = g;; a'b'.

(8.33)

Thus, gtja'bi is an invariant of the vectors a, b with respect to the group of motions and reflections (the numbers g;, here remain unaltered and must be regarded as numerical coefficients); this invariant is called the scalar product ab of the vectors a and b. The scalar product of the vector a by itself is called its scalar square and denoted by a2. By (8.30), one always

has a2 > 0; -I/a2 is the length of the vector a. The distance between two points. is equal to the length of the vector joining these points. *) As we will see below (§ 29, Exercise 14), the distinction between contravariant and covariant vectors disappears in Euclidean geometry and the word "contravariant" may be omitted.

97


8.4-8.5

All known properties of the scalar product are readily seen to hold true in n-dimensional space also. With the aid of the scalar product one can define also the concept of the angle between two vectors. Let a and b be two vectors with coordinates a', b'; let x, /3 denote their lengths, so that 91J b'bJ =

91J aia' = `x2,

a > 0,

/32,

/3 > 0.

By (8.30), one has for any A g;J(Aa'+b')(1.aJ+bJ) > 0, i.e.,

cc2A2+2abA+Q2 > 0;

hence the quadratic equation in I 0(

2A2+2abA+/32

=0

must have imaginary or equal roots and (ab)2-x2132 < 0, or

labs 5 a#.

The last inequality permits us to write ab = afl cos gyp,

where qp is called the angle between the vectors a and b. Thus, in ndimensional Euclidean space also, the scalar product of two vectors is equal to the product of the lengths of the vectors and the cosine of the angle between them. The concepts of length and angle form the foundation

of n-dimensional Euclidean geometry; a more detailed treatment of this work can be found in [191, Chapter IV. 8.5

The arguments pursued in 7.2, 7.3 obviously also remain in force

in the case of a space of an arbitrary number of dimensions. Thus, in n-dimensional space also, affine invariants can be reduced to projective invariants and Euclidean invariants to affine ones, and hence to projective invariants. This allows us later on to place the main emphasis on projective invariants. Exercises

1. Show that the manifold of all projective transformations and the manifold of all affine transformations (8.13) form groups.

98


CHAP. I

2. Explain the geometric meaning of the coordinates of a point x and of a hyperplane

u in projective space of order n. 3. Generalize the proposition of § 4, Exercise 2 to the case of an affine n-dimensional space.

4. Prove that all those affine transformations which satisfy the relation (8.32) form a group the number of parameters of which is n(n+ 1)/2. 3. Prove that the determinant of any transformation of the group of motions and reflections [cf. (8.32)) is equal to ± I. The transformations for which the determinant is + 1, are called motions in n-dimensional Euclidean space. Show that all motions by themselves form a group.

CHAPTER 11

THE FOUNDATIONS OF TENSOR ALGEBRA

Tensors and operations on tensors

§ 9.

In Chapter 1, the significance of the concept of the invariants of a group of transformations in contemporary geometry has been explained. In proceeding to a systematic study of the foundations of the theory of 9.1

algebraic invariants, we will now give its problems a purely algebraic formulation; certain geometric terms, which we will retain for this purpose in view of their general acceptance, will make it easier for us to step over to the geometric interpretation of the theory which we will not leave

completely out of sight in what follows. A contravariant vector is determined by the values of n numbers x', x2, ..., x" (its components); contravariant vectors are subject to linear transformations by which each of their components change in accordance with the formulae

i,a=1,2,...,n;

x'=p;za,

(9.1)

in this context it will always be assumed that the determinant of the transformation (9.1) A = IpQI * 0.

(9.2)

We will say that the equations (9.1) define a linear transformation of a space of order n. All linear transformations of a space are readily seen

to form a group (Exercise 1). Solving the equations (9.1) with respect to JCa, we find Xa

= q'i'x`,

i, a = 1, 2, . . ., n,

(9.3)

where qi is the reduced minor of the element pQ in the determinant (9.2),

so that p`B q1 = S;

(9.4)

FOUNDATIONS OF TENSOR ALGEBRA

100

CHAP. U

[cf. (6.7)]; it follows from (9.4) that IP'I

Iq°I = 1,

whence

Iq°1 = A-' # 0.

(9.5)

The term "contravariant vector" has been taken from affine geometry of n dimensions [8.3]; the transformation (9.1) can be regarded either as an affine transformation of a contravariant vector or as a transformation of its affine components. Obviously, from the algebraic point of view, the distinction is irrelevant. In projective geometry of order n, a contra-

variant vector is interpreted as a point, and the transformation (9.1) as a projective transformation or as a transformation of projective coordinates; in this context it must not be forgotten that the projective coordinates of a point are homogeneous, so that the contravariant vectors

r' and .fix' determine the same point for any ). # 0. Side by side with contravariant vectors other geometric objects are also

considered in relation to linear transformations of space. Every such geometric object is determined by a system of numbers, its coordinates, and by their law of change under the transformation (9.1), where one has

to fulfill the requirements formulated in 4.7 (replacing, of course, the words "affine group" by "group of linear transformations of space"). Apart from the contravariant vector the simplest geometric object is the covariant vector. A covariant vector is specified by its n components ul , ul , ... , u which transform under a linear transformation of space in accordance with the law

uQ=p,u;.

(9.6)

Here and later on, the absence of specific statements indicates that the

indices assume the values 1, 2 . . ., n. The geometric meaning of the covariant vector is as follows (§ 8): in affine geometry, it is a doublet, in projective geometry, a hyperplane. As has already been mentioned, the transformations (9.1) and (9.6) are called contragredient. We will solve equation (9.6) with respect to the u; by the method of 6.2.

It is readily seen that Pa q' = Sf ;

(9.7)

in fact, if Pa is the cofactor of pa in the determinant (9.2), then the sum

9.1-9.2

TENSORS AND OPERATIONS ON TENSORS

Y

101

i

Pa 1'a

a=1

is known to be equal to A, if i = j, and equal to zero, if i:0 j; dividing by A and recalling that

we obtain (9.7). From (9.6) we find a

q; ua

ai = = 4;Paul

u;

since the Kronecker Delta

h;=

l

0

i=j

for i

the right-hand side of the last equality is equal to u;. Replacing j by i, we find, finally, (9.8) u; = q°ua. For the sake of simplicity, in what follows we will often denote vectors

by symbols without indices, and contravariant vectors will be distinguished from covariant vectors by a tilde ( -), i.e., contravariant vectors will be denoted by a, $, . . .; however, to simplify writing, we will agree that the symbols x, y, z and t are to denote contravariant vectors even without tildes. The theory of invariants for geometric objects of a general form is only at its beginning; however, for a very large number of problems of geom-

etry and its applications, it is sufficient to deal with a narrower class of geometric objects which have been studied in greater detail, namely the

so-called tensors. We will now proceed to their investigation, confining ourselves for the time being to the algebraic aspects of the matter; a geometric interpretation of tensors will be given in §§ 10 and 11.

9.2 A covariant tensor of order r is specified by the numbers a;,;2 ..;, which transform under a linear transformation of space in accordance with the law aa,a2

...

= Pa; Pa; ... Pa

(9.9)

since each of the indices i,, i2, ..., i, assumes independently of the others the values 1, 2, ..., n, the number of components a;,;2,,.;, of a covariant


102

CHAP. 11

tensor of order r is n'. For r = 1, one obtains from (9.9) the equation (9.6); consequently, a covariant vector is a covariant tensor of order unity.

In order to simplify the writing in what follows, we will use r = 3; then (9.9) will assume the form (9.10)

Qapy = Pa 6 Py atjk;

proceeding as in the case of the covariant vector in 9.1, it is not difficult to solve the equation (9.10) for the a1 jk: y gf909i,aady = 9rPa99Pp9kPYQ;jk = SfVg8hatjk = af,h *),

or, changing the notation for the indices, 9.11

aijk = Qigj4kaapy

Let x, y, z be three covariant vectors; we find from (9.11), by use of (9.3), that t j k atlk x Y z = 4taxt9j BY!qk rza 6y = aavy ay zr

x,

i.e., the expression (p(x, y, z) = aljkx`Yjzk

(9.12)

is an invariant of the tensor atjk and the vectors x, y, z with respect to linear transformations of space; the function cp(x, y, z) is called a trilinear form in the vectors x, y, z. Conversely, if a trilinear form in contravariant vectors x, y, z is invariant with respect to linear transformations of space,

independently of the choice of the vectors x, y, z, the coefficients aIjk

of this form constitute a covariant tensor of order three; in fact, the equality at1kxtyjzk = aa0r xay#zY, or [cf. (9.1)]

j k aljkPaPiPYxxx =aQQyxxx, t

implies the equality (9.10) as a consequence of the arbitrariness of Ya

y ,z.

Obviously, the above reasoning is valid for any value of r; in that case one speaks of a multilinear form in r covariant vectors. ) Cf. (9.7).

9.2-9.3

103


9.3 A covariant tensor is said to be symmetric with respect to two indices, if the components obtained by interchange of these indices are identical; thus, the tensor aijk will be symmetric with respect to the first

and third indices, if aijk = akji.

The symmetry of a tensor is a property which is invariant with respect to linear transformations of space: aa9v -

Pi

a l$

pky

ai jk = pky pj 8 pa akji = ayaa .

A tensor is said to be symmetric with respect to several indices, if it is symmetric with respect to any pair of these indices. Finally, a tensor is called symmetric, if it is symmetric with respect to all its indices. Thus, the third order tensor aijk will be symmetric if

aijk = aikj - ajki = ajik = akij = akji In Chapter 1, we have encountered several times symmetric covariant tensors of order two [cf. (4.39), (5.21), (6.29), (7.19), (8.31)]. Obviously, the symmetry of a tensor changes the number of its effective

components, i.e., of those components which are not equal to each other by virtue of the conditions of symmetry. We will now evaluate the number of effective components of a symmetric covariant tensor of order

r; it is equal to the number of permutations of n elements 1, 2, ..., n among the r elements with repetitions (i.e., we admit among the combinations the presence of identical elements). We will associate every

permutation with repetitions it i2 ... i, (iI < i2 < 13 < the permutation without repetitions i1, i2+1, i3+2,

...

< i,) with

..., i,+r-1;

this correspondence will be invertible and single-valued. Thus, the number of permutations of n elements among the r elements with repetitions

is equal to the number of permutations of n+r-1 elements of the r elements without repetitions; the number of effective coordinates of a

symmetric tensor of order r is thus equal to n(n+1) ... 1

(n+r-1) - 2... r

- (n+r-1)! (n-1)!r!

(9.13)

There exists a connection between a symmetric covariant tensor of

104


CHAP. 11

order r: a1112..., and the r-th order form a1,12 ... 1, X11X12

... x`r

(9.14)

of a covariant vector x or, as is often said, of the n variables x', x2, ..., x"; for r = 2, 3, the form is called a quadratic and a cubic form, respectively. It is not difficult to determine the coefficients of the form (9.14). Consider

first the cubic form a,,, x1xjXk,

(9.15)

where a;;, is a symmetric tensor. The coefficient of (x')3 will be a,,, Terms involving (x')2x2 are encountered three times: for i = j = 1,

k=2, for i = k = 1, j = 2, for j = k = 1, i=2; by virtue of the symmetry of the tensor, the coefficients of these three terms are identical, so that the coefficient of (x')2x2 in (9.15) will be 3a112. Terms involving

x'x2x3 will occur six times, so that the coefficient of x'x2x3 in (9.15) will be 6a123. Analogous reasoning is applicable to the form (9.14) for any value of r. If the indices i,, i2, ..., i, are all different, the terms involving the product X1i X12... x" in (9.14) will be encountered r! times. If some of the indices among the i1, i2, ..., i, are the same, this quantity changes; for example, if the indices i1, i2, ., i,, are equal to each other, then the h! terms which would correspond to all possible orders of these indices, if all of them were different, would be replaced by one. Hence we arrive at the following conclusion: if among the permutations of i1, i2, ..., i, the indices have q distinct values, where the first of them occurs h1 times, the second h2

times, etc., so that h, + h2 +

.

.

. + hq = r,

(9.16)

then the corresponding product x"x'z . . . xi, will occur in the sum (9.14) r!/(h1!h2! ... hq!) times and the coefficient of this product in the sum (9.14) will be r!

(9.17)

h,!h2!...hq!

It is readily seen that the numerical multiplier which we have just derived is nothing else but the multinomial coefficient in the expansion of a multinomial form of degree r in n variables:

(x' +x'+ ... +X" )r.

105


9.3-9.4

By the result proved above [cf. (9.12)], the form (9.14) is an invariant of the tensor ai, j, ... i, and the vector x with respect to linear transformations of space. Conversely, if the form (9.14) involving the contravariant vector x (which is assumed to be arbitrary) is an invariant for these transformations and its coefficients are symmetric with respect to all indices,

the set of coefficients forms a symmetric covariant tensor; in order to prove the last statement, we can employ the theorem that identically equal polynomials have identical coefficients; the values of these coefficients have just been computed [cf. (9.17)]. 9.4 A covariant tensor is called skew-symmetric with respect to two indices, if on interchange of these indices two of its components differ only by their signs: the tensor aijk is skew-symmetric with respect to the first and third indices if

aijk = -akji, it follows immediately from this that for i = k one must have aijk = 0. A covariant tensor will be skew-symmetric with respect to several indices if it is skew-symmetric with respect to any pair of these indices. If skewsymmetry occurs with respect to all indices of a covariant tensor, it is said to be a skew-symmetric tensor or a polyvector. Polyvectors of orders 2, 3, 4, . . ., r are called bi-vectors, tri-vectors, quadri-vectors, ... , r-vectors, respectively. Thus, for a covariant bi-vector,

vij = -vji; for a covariant tri-vector Wijk = - Wik j = Wjki = - Wjik = Wkij = - Wkji

.

If any of the indices of the components of a polyvector happen to be the same, these components must, obviously, be zero. Hence the number of effective, i.e., of non-zero and distinct (essentially distinct, i.e., differing not only in sign) components of a covariant polyvector of order

r is given by

cr - n(n-1)... (n-r+1)

n!

(9.18)

for r > n, all components of a polyvector are equal to zero. We will consider further the case of the polyvector of order n : ei,i,... (the n-vector, where n is the order of the space) which is important for


106

CHAP. 11

what follows; the n-vector has only one effective component, namely e,2,.,,,. In the case of linear transformations of space, it changes in accordance with the law i,

I

12

12 ... n = PI Pi2 ... Pn ei,i2 ... i

(9.19)

Only those n! terms in the sum on the right-hand side of (9.19) are non-zero for which i1 i2 ... i is a permutation of 1, 2, ..., n; in that case

ei,12...i - ±e12...n, where the plus (+) sign occurs for even, the minus (-) sign for odd permutations i, i2 ... in- It follows from this that one can take e12 ... n outside the sum in (9.19) and that the sum becomes the determinant formed by the pa, i.e., the determinant of the transformation A. Thus,

e12...n - d ' e12...n; The only effective component e12 ... n of the covariant n-vector ei,12...1,, is a relative invariant of linear transformations of weight

Theorem 9.1:

+1. A detailed discussion of polyvectors will be given in Chapter VII of this book. 9.5

A contravariant vector of order r is specified by n' numbers ai,12 ... i.

which for linear transformations of space change in accordance with the law ala2 ... ar

whence it follows that ai,iz...ir

al 22 ar 1ii2 ... it 91,9/2... qi,a

PaQa,a2...ar

PaPa

(9.20)

(9.21)

For r = 1, one obtains from (9.21) the formula (9.1). Thus, a coiltravariant vector is a contravariant tensor of order 1. The results of 9.3 and 9.4 regarding symmetry and skew-symietry of covariant tensors also extend to contravariant tensors; this is an expression of the principle of the duality of space with respect to contravariant and covariant vectors. There exist connections between the contravariant

tensors a""-" and the multilinear form 2

1

r

ai,i2 ... Irui, ui2 ... ui,. , 1

2

(9.22)

r

involving the r covariant vectors u, u, ..., u and a symmetric contra-

9.4-9.6


variant tensor of order r, and the r-th order form a'"2...tru Ui2 .. U

107

(9.23)

involving the covariant vector u; both forms (9.22) and (9.23) are invariants with respect to linear transformations of space. In the case of the contravariant n-vector e"" '^, reasoning, analogous to that leading to the proof of Theorem 9.1, shows that its only effective component e12..." is multiplied for linear transformations of space of order n

by the determinant of the qj , which is known to be equal to the determinant of the transformation d to the pcwer - 1. Thus, one has Theorem 9.2: The only effective component e' Z - " of the contrais a relative invariant of linear transformations variant n-vector e""'

of weight -1. Next, we will define so-called mixed tensors. The set of numbers ct,k which change under linear transformations of space in accordance 9.6

with the law kCtm

z" = qtz gmP.P0P7 " t adr l;k,

(9.24)

specify a mixed tensor of covariance 3 and contravariance 2, i.e., of order 5. In an analogous manner, one can define mixed tensors with any orders of covariance and contravariance (where the covariance is equal to the

number of subscripts, the contravariance is equal to the number of superscripts, and the order is equal to the sum of covariance and contravariance); a mixed tensor of order r has n' components. Covariant tensors can be regarded as mixed tensors of zero contravariance, contravariant

tensors as mixed tensors with zero covariance. There exists a link between the mixed tensor cijk and the multilinear form cilk xIYizkU1 va,

(9.25)

which is invariant with respect to linear transformations and involves three contravariant vectors x, y, z and two contravariant vectors u, v. In the case of a mixed tensor, the concept of symmetry is only applicable

to subscripts or superscripts separately. Symmetry with respect to a pair of indices one of which is a subscript, the other a superscript cannot be considered, since it has no invariant significance. In fact, if

a,=a I


108

one has

t

j a'2

t

z

j

CHAP. It

i

at = gipta;; az = q;pzaj, in the first of these sums, the term containing al will be q; p2 ai , in the second gipl a', and therefore these sums are not identically equal, i.e., the symmetry vanishes as a result of a linear transformation. The above statements extend also in equal measure to skew-symmetry. In conclusion, we note also the case of the tensor of order zero; since no = 1, such a tensor must have one component a which remains unaltered for linear transformations of space; it is called a scalar. 9.7 Two tensors of the same orders of covariance and contravariance are said to be equal if in them all the components which correspond to the same values of the indices are equal; in particular, a tensor is equal to

zero, if all its components are equal to zero. Next, we will consider operations on tensors, i.e., operations which create new tensors. Let there be given a second order covariant tensor ci j which, generally

speaking, is not symmetric. Let (9.26)

aij = 3(Cij+cji),

assuming this relation to be unaffected by any linear transformation. It is readily verified that a, j is also a tensor: i 46 = papCij,

t

j

Cea = ppipacij

= ptjpiaCjt

(where in changing the notation for the summation indices we do not alter the sums). It follows from these two equations that aa9 = lCap+Cpa) _ Jpap%(Cij+Cji) =-- papQatj,

which proves the above assertion. Obviously, we find from (9.26) that aij = a ji, so that the tensor aij is symmetric. The above operation converts a non-symmetric tensor into a symmetric tensor; for this reason the process is called symmetrization. Obviously, symmetrization of a symmetric tensor leaves the tensor unaltered. In a similar manner, one can define symmetrization for covariant tensors of third order. In order to symmetrize the tensor cijk, one must define a new geometric object aijk which is invariant with respect to linear transformations: aijk = 6(CIJk+Cikj+Cjki+Cjik+Ckij+Ckji);

(9.27)

9.6-9.7


109

it is readily verified that the ai jk form a symmetric tensor. In order to symmetrize a covariant tensor of order r, one must subject its indices to all possible permutations, add the results and divide by r!; as a consequence, one arrives at a symmetric tensor of equal order. Obviously, symmetrization of a tensor with respect to some of its indices also gives new tensors; symmetrization will be indicated by placing brackets around

the symmetrized indices. Thus, the formulae (9.26) and (9.27) may be rewritten aij = C(fJ),

aljk = C((Jk).

If the brackets include indices which must not be symmetrized, they will

be provided with vertical lines b(imk)l = J(biJkl+bkj11),

h(iilkllm) = 6`hijklm+himklj+hjmkii+hjiklm+hmiklj+hmjkli).

Symmetrization of contravariant and mixed tensors can be defined in the same manner, where in the case of mixed tensors one may symmetrize either with respect to certain superscripts or with respect to certain subscripts; symmetrization with respect to indices comprising superscripts as well as subscripts has no significance, since it does not lead to a tensor. Symmetrization is encountered in connection with forms involving a

single vector. Let aijk be a non-symmetric covariant tensor of order three; then the cubic form i al Jk X X J Xk

(9.28)

in the contravariant vector x has been shown above to be invariant with respect to linear transformations of space. What are the coefficients in the form (9.28)? The coefficient of (x1)3 is a,,,; there are three terms involving (xl)2x2, so that the coefficient of (xl)2x2 will be a112+a121+a211 = 3a(112);

finally, the product XIx2X3 occurs six times and its coefficient in (9.28) will be a123+a132+a231+a213+a312+a321 = 6a(123)

Thus, aijk x'x'xk = a(iJk) x'x'xk

(9.29)

and the form (9.28) is not related to the tensor a,jk, but to the tensor

110


CHAP. IT

a(; jk). As a consequence, if it is known that (9.28) is an invariant of linear

transformations for an arbitrary contravariant vector, it does not follow at all that the numbers aijk form a tensor in all cases; this may only be asserted with respect to a(;jk) *). For an analogous reason it does not follow from the fact that the form (9.28) vanishes identically that ai jk = 0,

but only that c(;jk) = 0. The above results apply to tensors of any order r; in particular, one has

Theorem 9.3:

If the equality a;,;, ... ;, x

x'Z ... X'r =0

(9.30)

i s true f o r any values o f the variables x', x2, ..., x", then

aU16...4) = 0,

(9.31)

where the indices i1, i2, ..., i, can assume the values 1, 2, . . ., n independ-

ently of each other. However, it is impossible to assert under those circumstances that at,l, ...I, = 0 for all possible permutations of the subscripts. 9.8 In what follows, a second, very important operation on tensors will be alternation. We call alternation of a tensor c with respect to h of its superscripts or h of its subscripts the process of construction of a new tensor, each component of which is obtained from the corresponding components of the tensor c by combining all possible permutations of these h indices; the terms which correspond to even permutations are

given a plus (+) sign, those which correspond to odd permutations a minus (-) sign, and they are then added and divided by hi. The reader will readily verify that this relationship between c and a is invariant, i.e., it is retained under any linear transformation. Alternation will be indicated by placing square brackets around those indices to which it applies; indices which are excluded from the alternation will be separated by vertical lines. Thus CIIIilk1 = 1(Cijk-Ckji),

htijky = a(hijki-hikjl+hjkil-hjikl+hkij,-hkjit)I etc. Obviously, alternation of a tensor leads to a tensor which is skew*) The symmetrization operation can, of course, be applied; it retains its significance

for any system of numbers

9.7-9.9


III

symmetric with respect to the indices subjected to alternation. If more than n indices are included under the alternation sign, one obtains as a result a tensor which vanishes identically; for example, the tensor (9.32)

ati02...in+17i.+2... = U,

since all its components vanish. In fact, if a tensor is skew-symmetric with respect to several indices, then only those of its components for which all the stated indices are different can differ from zero; in the case of the tensor (9.32), this is obviously impossible, since the number of values which the indices i1, i2 , ... , 1 can assume is equal to n. 9.9 We will now consider three operations on tensors which we will require later on: addition, multiplication and contraction. Let a'jk and bA be two tensors both of which are contravariant of order unity and covariant of order two. We will define n' numbers cjk by the equalities i i t Cjk = ajk+bjk;

(9.33)

the relation (9.33) will be assumed to be invariant, i.e., to remain valid after any linear transformation of space. We will show that under this assumption the numbers ck form a tensor. A linear transformation changes the coordinates of the tensors a and b in accordance with the laws

j;y

aj i i = gipppvbar = gipap,bjk, k

a

a

k

whence follows a0 +b07 = g7P'apr(ajk+bjk), C01 = qi p, p1 c jk .

Thus, the cjk form a tensor which is also contravariant of order one and covariant of order two. It is called the sum of the tensors djk and b'jk. Addition of two or more tensors of any orders of contravariance and covariance may be defined in an analogous manner; it is seen from the proof above that equality of the covariance and contravariance of the tensor terms is necessary. Next, we will select two tensors aik and b', and determine the system of ns numbers c'k,,, which is invariant with respect to linear transformations: iI i cjkM = ajk,,,. b'

(9) .34

CHAP. II


112

After a linear transformation of space, we find aj i Aml a a ,A ax Sc

A

j

Sc

m it

cayu = aar bu = qi Pa Py ajk qI pp bm = qi q, Pa Py Pu ctkm;

thus, the Cjkm form a tensor which is called the product of the tensors ajk and b' . In contrast to the case of addition, we can form products of tensors of any covariance and contravariance; the number of tensors which can be multiplied is also arbitrary. We will now note certain particular cases of multiplications of tensors: let x' and y' be two contravariant vectors. By the above, the quantity x'yj and, consequently, also x1' j,jl and xl'yjl will be contravariant second order tensors: the second of these is symmetric, the third skew-symmetric. The bi-vector 2xt' yjl = x' yj - y' xj is called the alternating product of the vectors x, y and is denoted by [xy]. This statement extends to any number of contravariant and covariant vectors; thus, the alternating product of r 1

2

covariant vectors u, u, ... , u is the covariant polyvector of order r with the components 1

2

r

r! Ult. tli=... Ui.l 12

and is denoted by [uu

r

.

.

.

u]. *)

If ajk is a tensor and o a scalar, trajk is likewise a tensor of the same order as ajk. 9.10

The last operation on tensors which is to be discussed here is that of contraction of a tensor. The process of contraction of the tensor hkim with respect to the first subscript and the second superscript leads to the following new tensor, if it is found to be invariant with respect to linear transformations: i lj `lm = hjlm

(9.35)

The proof that k;m is a tensor is readily given; one has

a aa a a k I m ij kAu = hazy. = gigjpep.Ip hklm,

(9.36)

and, since by (9.7) S;, only those terms on the right-hand side of (9.36) will be non-zero for which j = k = 1, j = k = 2, ..., j = k. = n. *) In the case r = n, this notation has already been used above [(4.29), (5.13), (6.19)),

not for the n-vector itself, but for its single effective component; as is easily seen, this duplication in notation cannot lead to misunderstandings.

113


9.9-9.10

However, for all these terms gfpe has the value unity. Thus, is aµ

=

a

mh'j

l

qi P,i P,,

a

mk

I

it. - q i Pa P;,lm+

which proves the assertion. Obviously, contraction can only be applied to mixed tensors and it lowers the covariance and contravariance of the tensor each by unity, i.e., its overall order by two. Naturally, one can perform contractions with respect to two, three, etc., pairs of subscripts. Thus, one can obtain by contraction the covariant vectors h;; ,, hj;m, h'k;, etc., from the tensor hkim. If a tensor has equal numbers of superscripts and subscripts, it can be contracted with respect to all indices; as a result one obtains a zero order tensor, i.e., a scalar (an absolute invariant). In that case one speaks of total contraction. For example, if a',; is a tensor, a;; and a;1 will be invariants. It is recommended that the reader attempts to verify this result using the same reasoning as in the case of the tensor k,m above. Multiplication of tensors can be combined with contraction. From the tensors alik and by , we obtain after multiplication the tensor a! kb"" of total order 6; by contraction, we arrive at the lower order tensors Im

i

Im

C;k = a;k bi

it

Ij

i

Ctkp = a;k bP +

m

i

jm

ek =ajk bi

I

+

i

!j

R ' ajk bi

etc., and at the scalars ajkb ",

ajkb;'.

We can use this method to derive from a tensor aij and a contravarian vector xi the covariant vector aij xj. On the basis of this study, we can investigate directly the invariance of many of the expressions encountered

earlier; cf., for example, the formula (4.34), Exercise 8 of § 5, (8.33), (9.12), (9.25). Multiplication accompanied by contraction is often referred to briefly as contraction. With a view to what follows it is important to note Theorem 9.4: The result of total contraction of a tensor or of a product

of several tensors is an invariant of linear transformations of space. Exercises I. Show that for all possible px the transformations (9.1) forma group and that in the case of multiplication of two transformations of the form (9.1) their determinants

are also multiplied. Hence show that the set of those transformations of the form (9.1) whose determinants have the value unity, also form a group (the group of unimodular linear transformations).


114

CHAP. II

2. Prove that the transformations of any tensor [cf. (9.9), (9.20), (9.24)], for all possible linear transformations of space, form a group, homomorphic to the group of transformations (9.1).

3. For what values of c is the relationship a,lk = cakli possible for any values of i, j, k = 1, 2, ... , n (under the assumption that not all the n' numbers ailk are zero)? 4. Let the tensor R,Iki be skew-symmetric with respect to the first two and with

respect to the last two subscripts. In addition, let i,j, k, I = 1, 2, .. , n. Rilki = Rklif, What is the number of effective components of the tensor R,ltl? 5. A multilinear form q(x, y, z, . . .) of the contravariant vectors x, y, z, ... is said to be symmetric with respect to the vectors x, y, if g'(x, y, z, ...) = 97(y, X. z, ...). Show that the symmetry of the form entails the symmetry of the corresponding tensor, and vice versa. An analogous result applies in the case of skew-symmetry. From the assumptions above a new proof of the invariance of symmetry or of skew-symmetry follows.

6. Prove that the components of the tensor o,$k which is symmetric with respect to its first two subscripts and skew-symmetric with respect to the last two subscripts, are

equal to zero. 7. What are the components of the tensor a`l = S' after a linear transformation of space?

8. Solve the equations (9.24) for c,'-,t. 9. Prove the following assertion: If the form (9.25) in three contravariant vectors x, y, z and two covariant vectors u, v is invariant with respect to linear transformations of space for any vectors x, y, z, u, v, the quantities c,'- form a tensor of covariance 3

and contravariance 2. 10. If a tensor a;lk is symmetric with respect to its first two indices, then

auk = j(ai,k+alki+aitl), a[uikIPi] = 0; if a tensor cilt is skew-symmetric with respect to the first and the third indices, then C[ilk] - s(Cilk+Cfki+CtU)

Verify these statements.

11. Prove that every second order covariant tensor a,l can be represented in the form of a sum of a symmetric and a skew-symmetric tensor. 12. Show that

auk = a(uk)+a[irk)+}a[il]k+fat k1Ji+ s(il)k-}at(,l). 13. If for the vectors u, o the relationship

u(ivl) = 0

(9.37)

holds true, then one of these vectors is equal to zero. Give the proof of this statement. 14. Prove the following statement: If a tensor ii,tl satisfies the relationship rifkixix5Y7Y1 = 0

for any choice of vectors x and y, then i(,l)(ti) = 0. 15. Let x and y be two contravariant vectors; under what conditions will the tensor x'y' be symmetric? Under what conditions will it be skew-symmetric?

GEOMETRIC MEANING OF TENSORS

10.1

115

16. If for a symmetric tensor a,, one has the relation a,Jx'x1 = 0, provided only the contravariant vector x fulfills the condition 9.38) p, x' = 0 where p is a covariant vector, then a,, = p(igj), where q is another covariant vector. However, if for any vector, satisfying (9.38), one has the equality ai,xi = 0, then ail = ap,p,. Prove these statements and generalize them to the case of a symmetric

covariant tensor of any order. 17. If the equality CJkxiXkpi = 0

is true for any two vectors x and p, satisfying the relation (9.38), prove that

fit - bUgk), where q is a covariant vector. 18. Let au and b,, be two covariant tensors; let ;,# be obtained from all as the result of the linear transformation S, and bo from b,, as a result of a linear transformation which is contragredient to S. Show that

E

E aitbu+

where in the summation the indices assume the values 1, 2, ..., is independently of each other.

§ 10. 'The geometric meaning of tensors; the projective point of view In relation to the two possible interpretations of linear transformations of space (cf. 9.1), tensors also can be interpreted in two ways: in affine n-dimensional geometry and in projective geometry of order n. We will begin with a study of the projective aspects of tensors and limit ourselves, for the sake of simplicity, to their simplest forms. By virtue of the homogeneity of the coordinates of points, in projective geometry a geometric significance can be attached only to tensors which are specified apart from a numerical factor: two tensors which differ from each other only by a scalar multiplier correspond to the same geometric image. For the same reason, a tensor, all components of which are equal to zero, cannot have a projective geometric meaning; therefore all tensors in

this section will be assumed to differ from zero. The symmetric, second order, covariant tensor ajj. As we know (cf. 9.3), there exists a link between such a tensor and the quadratic form, 10.1

invariant with respect to linear transformations of space, a,, xixl

(10.1)

116


CHAP. It

which involves the contravariant vector x; conversely, the form (10.1) defines the symmetric tensor a.. In projective geometry, a point corresponds to the contravariant vector x'; by virtue of the homogeneity of the coordinates of a point, a geometric significance can only be given to the case when the invariant (10.1) is equal to zero. The geometric locus of points whose coordinates satisfy the condition

a;jx`x' = 0,

(10.2)

which is invariant with respect to linear transformations of space, is called (for n > 4) a second order hypersurface. The same hypersurface is also determined by the tensor aa1 (a 96 0). For n = 4, the hypersurface is the same as a surface; the cases n = 2 and n = 3 have been considered earlier. Thus, a symmetric covariant second order tensor a;;, specified apart from a scalar multiplier, determines (for n ? 4) a second order hypersurface; the numbers a;; can be called the homogeneous coordinates of the second order hypersurface. As we have seen, the tensor a.j is also linked in an invariant manner to the bilinear form a;fx'y' in two contravariant vectors x, y. Again taking

the homogeneity of the coordinates of a point into consideration, we must only seek a geometric interpretation for the equation a;; x'y' = 0.

(10.3)

As in 6.6, for this purpose one must solve the problem of the points of intersection of the straight line joining the points x, y with the hypersurface (10.2). If we write the coordinates of an arbitrary point of this straight line in the form (8.10), the further reasoning can proceed as in 6.6; the only difference, namely that the indices will now assume the values 1, 2, . . ., n instead of 1, 2, 3, does not affect the external form of the formulae, thanks to the Einstein convention., Thus we arrive at the following conclusions: Theorem 10.1: Any straight line intersects a second order hypersurface in two points (which may be distinct or coincident). Theorem 10.2: Two points x, y whose coordinates satisfy the relation (10.3) are conjugate with respect to the hypersurface (10.2), i.e., they divide the points of intersection of the hypersurface (10.2) and the straight line joining the points x, y harmonically.


10.1-10.4

117

In order that the above assumption will always be true, one must fall back on the point of view of projective geometry in the complex domain; in addition, if one of the points lies on the hypersurface, then Theorem 10.2

will only be true under an extended interpretation of the concept of a harmonic set of four points (§ 5, Exercise 6). The symmetric r-th order covariant tensor a;,;2...;,. Reasoning as in Example 1, we will verify that such a tensor, given apart from a scalar multiplier, determines a geometric locus of points x whose coordinates 10.2

satisfy the equation a

x" x'2

... x` = 0,

(10.4)

which is invariant with respect to linear transformations of space. This geometric locus of points is called (for n > 4) a hypersurface of order r. For n = 2, the relation (10.4) will be an equation of degree r in x2 : x' ; therefore it determines r points on a straight line (some of which may

coincide). For n = 3, the same equation leads to a curve of order r, and for n = 4 to a surface of order r. A hypersurface whose equation can be reduced to the form (10.4) with an arbitrary r, is said to be algebraic. Thus, the projective theory of algebraic hypersurfaces coincides to a significant degree with the theory of invariants of symmetric covariant tensors. 10.3

The symmetric covariant second order tensor a''. There exists a

link between this tensor and the invariant quadratic form a'' u; uj involving the covariant vector u. Since the coordinates u; of a hyperplane are

also homogeneous, only the equation a'' u; uJ = 0.

(10.5)

can be given a geometric interpretation. The manifold of hyperplanes whose coordinates satisfy the equation (10.5) determine a hypersurface of the second class (for n > 4; for n = 4,

one speaks of a "surface" rather than of a "hypersurface"). In 29.7, it will be shown that under given conditions a second class hypersurface

is a second order hypersurface. In an analogous manner, a hypersurface of the r-th class corresponds to

a symmetric covariant tensor of order r, specified apart. from a scalar multiplier. 10.4 The contravariant and covariant bi-vectors. As has been mentioned in 9.4, a bi-vector is a skew-symmetric second order tensor.

CHAP. It


Its

A contravariant bi-vector is said to be simple, if it is equal to the alternate product (9.9) of two contravariant vectors x, y: IX: x1 v'' = 2x[+y1] = x+y1- y'x1 = y y1i

or, more briefly,

u = [xy].

(10.6)

Let Ax+py, px+ay be two distinct points of the straight line joining the points x, y, so that Aa-pp # 0. The alternate product of the corresponding contravariant vectors of the points .lx+py and px+ay is the bi-vector with the components

I

px'+ay' pxl+ayl

1P

a

l'

1'

y'

Y'

it is seen that it differs from v11 only by a non-zero multiplier. Thus, if a

bi-vector is given apart from a scalar factor, it determines any pair of points of some straight line; in other words, one has Theorem 10.3: The simple contravariant bi-vector (10.7)

v'1 = 2x['yi}

given apart from a constant multiplier, determines a straight line through the points x and y. Employing the abbreviation of (10.6), we will often denote this straight

line by [xy]. For n = 2, the bi-vector 0 has only one effective component v' 2, and it is easily seen that it is always simple; two contravariant bi-vectors differ from each other only by a scalar multiplier. Since from the projective point of view tensors are of interest only apart from a multiplier,

one cannot derive any geometric interpretation of the bivector v' in projective one-dimensional geometry. Obviously, this result also applies to the case of the covariant bi-vector vi1. For n = 3 also, every contravariant bi-vector is simple. In fact, since, on the basis of the observations

at the beginning of this section, the bi-vector v'1 is assumed to have non-zero components, one can assume without restricting the generality that V12 0 0. Select two points with the coordinates

xI = I v12

if, vU

' y =

21

(10.8)

10.4


119

these points will be distinct even if the second coordinate of the point x

is unity and the same coordinate of the point y is zero. It is readily verified that v'f = 2x11yf 1, and this proves the assertion.

Thus, in two-dimensional projective space, every contravariant bivector corresponds to a straight line. Comparing (10.7) and (6.16), we see that the coordinates u1 of this straight line are linked to the three effective components v23, v31, v12 of the bi-vector v`' by the relations = V31 u3 = V12 u1 = V23, U2

which can be rewritten in the invariant form k

(10.9)

where egJk is an arbitrary, non-zero, covariant tri-vector. The truth of the last statement can be verified in the following manner: let e123 = p # 0.

Setting in (10.9) i = 1, only two terms corresponding to j = 2, k = 3 and j = 3, k = 2 will be non-zero on the right-hand side and we obtain u1 =

l1

3e123 V23

+e132 v32 = pV23

in an analogous manner, we find: u2 = pv31, u3 pv12. By virtue of the homogeneity of the coordinates, the presence of the superfluous factor p

is not important. The principle of duality shows immediately that for n = 3 a point with the coordinates x1 = leuikvJk ,

(10.10)

corresponds to every covariant bi-vector v1J, where e1Jk is an arbitrary contravariant (non-zero) tri-vector.

For n = 4, a contravariant bi-vector is not necessarily simple. This fact is most easily demonstrated in the following manner: a simple bi-

vector 0 is expressed in terms of two contravariant vectors x, y by the equality (10.7). Expanding the determinant x'

x2

x3

x4

Y1 x'

Y2 x2

Y3 x3

Y4 x4

YI

Y2

Y3

Y4

I

which is obviously equal to zero, by use of Laplace's method with respect

to the first two rows, we obtain V12V34 _ V13V24 + v14V23 = 0.

(10.11)

FOUNDATIONS OF TYNSOR ALGEBRA

120

CHAP. 11

For n = 4, a bi-vector is specified by its six effective components v12, v13, v'4, V23, v24, v34 which may have arbitrary values; if then (10.11) is not fulfilled, the bi-vector will not be simple.

Conversely, if (10.11) is satisfied for a given bi-vector, it is simple; in order to prove this statement, we will assume again (without reducing the generality) that v12 is non-zero and determine the contravariant vectors x and y from (10.8). Then it is found that (10.7) is true: for i = 1,

j = 2, 3, 4, for i = 2, j = 3, 4, it is fulfilled identically, and for i = 3, j = 4 it is true by virtue of (10.11). We note that the condition (10.11) can also be written in the form v[12v34)

(10.12)

= 0;

in fact, those among the terms on the left-hand side of (10.12) which differ from each other by the order of the indices in one or in both factors,

for example, v42v13, _ v24v1 3, v24v31, or by the order of the factors (i.e., v13v24 or -v 24 V 13), will be the same. Hence the 24 terms on the left-

hand side of (10.12) consist of three sets each of eight equal terms; the three different terms coincide with the terms on the left-hand side of (10.11). The condition (10.12) for a bi-vector to be simple reveals directly that this condition is invariant: the left-hand side of (10.12) is the single effective component of a contravariant 4-vector and therefore after a linear transformation of space (Theorem 9.2) the equality (10.12) assumes the form d-1u[12634] = 0, i.e., v[126341 _ 0.

Thus, in three-dimensional projective space, a straight line is determined by six homogeneous coordinates v12, v13, v14, v23, v,24, v34,

interrelated by (10.12) (v'' = -v'1); they were introduced first by Plucker [J. Plucker (1801-1868) ] and are called Plucker radial coordinates of the straight line. As an example of their application, we will introduce the condition for the intersection of two straight lines which are specified

by their Pucker coordinates v'' and w''. Let x, y be two distinct points on the first and z, t two points on the second straight line, so that vt"

= px['y'],

w'j = Qz(Itil,

pa#0.

The straight lines v'' and w'' will intersect if and only if the points x, y, z, t lie in one plane, i.e., if their corresponding contravariant vectors

are dependent; hence we must have

10.4


x1

x2

x3

x4

Y1

Y2

Y3

Y4

Z1

Z2

Z3

Z4

t3

t4

121

= 0.

t2 tL

Expanding the determinant by use of Laplace's method with respect to the first two rows, we find the required condition v12W34_v13w24+v14W23+v23W14_v24W18+v34w12

= 0,

or, in abbreviated form v[12W34]

= 0.

(10.13)

Vii - 2e7jk1 vk1,

(10.14)

For n = 4, the invariant equality

where etjkl is an arbitrary non-zero covariant 4-vector, relates every contravariant bi-vector v'j to the covariant bi-vector v.j. Let eL234=p# 0;

reasoning as above for n = 3, we find V12=PV

34

Pv42,

V13 = v24 = pv31,

V23 = pv14,

V14°Pv

23

v34 = pv12

s

(10.15)

If a bi-vector v'J is simple, the relation (10.11) applies; the equality shows that it can be rewritten in the form V34V12-v42V31+V23V14 = 0,

V[12 V341 = 0.

(10.16)

Let again v12 # 0; then also v34 # 0. We will specify two covariant vectors p, q by the equations

pi = .

1

V310

q1 = V41;

(10.17)

V34

these vectors are not collinear (since p4 = 1, q4 = 0). The relations (10.8) and (10.17) give, by (10.15),

p,x1=AY`=q.x'=qiy'=0, so that the planes corresponding to both of the covariant vectors p, q pass through the points x, y, i.e., through the straight line v''. By (10.17)

122


CHAP. n

and (10.16), we now obtain v1j = 2pujgjl.

(10.18)

The relation (10.18) determines the straight line as the intersection of two planes; the numbers v12, v13, v14, v23, v24, v34, given apart from a common multiplier and interrelated by (10.16) (vij = - v j,), are called Plucker axial coordinates of the straight line (Plucker called a straight line, given as the geometric locus of points, a ray, and a straight line, given as

the axis of a pencil of planes passing through it, an axis). The relations (10.15) show that the axial coordinates differ from the radial coordinates only by their order; the link between the radial and axial coordinates is given in invariant form by the equation (10.14). As has already been noted, the projective geometric meaning of the simple contravariant bi-vector 0 is given for n = 4 by Theorem 10.3; in this case, the numbers v'1 are called Plucker contravariant coordinates of the straight line. By the principle of duality, we now have Theorem 10.4: The simple covariant bi-vector vi j = 2p 1 q11,

(10.19)

given apart from a scalar multiplier, determines a (n-3)-dimensional plane in which the hyperplanes p and q intersect. Theorem 10.4 is readily proved directly, without application of the duality principle, by reasoning which is completely analogous to that leading to Theorem 10.3. The numbers yr j are called Plucker covariant coordinates of the (n-3)-dimensional plane. Next, we will give a projective geometric interpretation of non-simple

covariant bi-vectors vi j (for n Z 4) or, what is the same thing, of the equation v1j x' y1 = 0,

(10.20)

related in an invariant manner to such bi-vectors, where x, y are contra-

variant vectors. The equality (10.20) can be rewritten in the form vp x1y' = 0,

i.e.,

-vijyY = 0;

adding this equation to (10.20), we obtain vijx"y11 = 0.

(10.21)

If the points x, y coincide, then (10.21) is satisfied identically; however,

10.4-10.5


123

if they are distinct, then xtiyJ1 represent the Plucker coordinates of the straight line joining them. Thus, a manifold of straight lines whose Pucker coordinates satisfy a single linear relation corresponds to the bi-vector vi j: such a manifold is called a linear complex of straight lines. We will now consider all those straight lines of a linear complex which

pass through a given point y; if we let viJyJ = ui,

(10.22)

the relation (10.20) assumes the form

u, x` = 0; we see from this equality that all points of the stated straight lines lie on a single hyperplane u.

Theorem 10.5: All the straight lines of a linear complex passing through a given point of a projective space lie in one hyperplane. We show in an analogous manner that a contravariant bi-vector determines a manifold of (n - 3)-dimensional planes whose PlUcker coordinates satisfy a single linear equation. Such a manifold is called a linear complex of (n-3)-dimensional planes; this is the object which is dual to the linear

complex of straight lines. The reader will readily formulate the dual theorem to Theorem 10.5. The covariant second order tensor cil which, generally speaking, is neither symmetric nor skew-symmetric. If we contract the tensor c1, 10.5

together with an arbitrary contravariant vector x, we obtain (cf. 9.10) the covariant vector ui = c.1 xJ;

(10.23)

the relation (10.23) is invariant with respect to linear transformations of space. Thus, with the aid of the tensor c,J, every point of the projective space is made to correspond to a hyperplane u of the same space. Thus, let the point x correspond to the hyperplane u and the point y to the hyper-

plane v; then the points Ax+µy correspond to the hyperplane ,lu+pv, i.e., a pencil of hyperplanes corresponds to the points of the straight line. This correspondence, which refers every point of projective space to a hyperplane and has the above property, is called a correlation. Thus, the tensor c,J (given apart from a constant multiplier) determines a correlation of projective space.

In particular, consider the case when the tensor cij is symmetric or

124


CHAP. 11

skew-symmetric. In the case of a symmetric tensor aij, the correspondence

ui = aijxi,

ail = aji,

(10.24)

is directly related to the second order surface (10.2): The point y lies in

the hyperplane u, if and only if i.e., (cf. Theorem 10.2) if the points x and y are conjugate with respect to the hypersurface (10.2). Consequently, the hyperplane u, defined by (10.24), is the geometric locus of points which are conjugate to the point x

with respect to the hypersurface (10.2). The hyperplane u is called the polar hyperplane of the point x with respect to the second order hypersurface (10.2), and in this case the correlation (10.24) is said to be a polarity with respect to the hypersurface (10.2). From the fact that conjugacy of two points is a reciprocal property there follows

Theorem 10.6: If in any polarity a hyperplane passing through the point y corresponds to the point x, the point y corresponds to a hyperplane containing x. It is readily shown that the possession of this property with respect to all points of space characterizes a polarity in the manifold of all eorrela-

tions (Exercise 7). In the case of a bi-vector cij, the correlation

uj = vitx-,

vij = -vji,

(10.25)

is called a null-system. Comparing (10.25) with (10.22), we establish a simple link between null-systems and linear complexes of straight lines: Theorem 10.7: In the null-system defined by the bi-vector vi j, the hyperplane u, which corresponds to the point x, is that in which the straight lines through x of the linear complex corresponding to the same bi-vector lie. From this follows a property characterizing null-systems (IExercise 8):

Theorem 10.8: In a null-system, the hyperplane u corresponding to an arbitrary point x of the space, passes through this point. 10.6 The mixed second order tensor A'. Writing

Z' = Afxj,

(10.26)

we can with the aid of this tensor relate every point x of projective space

10.5-10.6


125

in an invariant manner to a point z of the same space. If the point x corresponds in this manner to a point z, and the point y to the point t, the point , x+py will correspond to the point 7.z+µt; in other words, the points of a straight line correspond likewise to points of a straight line. Such correspondence of points is said to be a collineation; thus, a collineation of projective space corresponds to a tensor A!, given apart from a scalar multiplier.

It is readily seen that in a collineation the points of a k-dimensional

plane (k = 2, 3, 4, ...) become points which lie on a k-dimensional 96 0, the collineation is simply a projective plane. If the determinant transformation. If we also add to the types of tensors considered in the examples 1, 3, 4, 5, 6 above, the non-symmetric tensor c`1, all possible forms of second

order tensors, i.e., of the most studied category of tensors, will have been exhausted. We will return to a more detailed study of them in Chapter VI. Exercises

1. Explain the geometric meaning of the invariant

1-

(ar,xtyf)'

a symmetric second order tensor and x, y are two different points neither of which lies on the hypersurface (10.2) corresponding to the tensor a,,. 2. Formulate the propositions dual to Theorems 10.1 and 10.2. 3. Write down the necessary and sufficient condition for the point z' to he on the

straight line v" in invariant form. 4. Prove that a simple contravariant 3-vector given apart from a scalar multiplier defines a plane in projective space. What is the projective geometric meaning of a simple covariant 3-vector? 5. What is the significance of the linear complex of straight lines determined by the

equation (10.21) for n = 3? 6. Letting n = 4, explain the significance of a linear complex of straight lines corres-

ponding to a simple bi-vector v,,. 7. If a correlation c,, has, the property of Theorem 10.5 with respect to all points x of projective space, it is a polarity. Give the proof. 8. If a correlation c has the property of Theorem 10.7, it is a null-system. Provethis statement. 9. The simple covariant bi-vector v is equal to the alternating product of covariant bi-vectors p, q. Explain the significance of the null-system corresponding to the bivector v,,.

126


CHAP. 11

§ 11. The geometric meaning; the alhne point of view In this study of the geometric significance of tensors in affine n-dimen-

sional geometry we will confine ourselves again to the simplest cases. 11.1

The symmetric covariant second order tensor ai,. For its geometric

interpretation, we link the tensor aij to the invariant equation

a,,xY = 1,

(11.1)

where x` is a contravariant vector. We will select from the entire manifold

of vectors whose components satisfy the equation (11.1) those which start from the origin of coordinates 0(0, 0, . . ., 0); then x` will be the coordinates of the end points of these vectors. The geometric locus of points whose coordinates x' satisfy equation (11.1) is called a central hypersurface of second order; we will denote it by Q0. The centre of this hypersurface will be the origin of coordinates (since, if the point x' lies on (11.1), the point -x' also lies on it). Those vectors whose components satisfy (11.1) and which begin at a point i;', are obtained from those just considered by parallel translations Tz, determined by the vectors '; consequently, the ends of these vectors are arranged on the central second order hypersurface Qg which is obtained from Q0 by the parallel translation TT; the centre of the hypersurface

QQ is at the point '. Thus, the equation (11.1) determines a system of hypersurfaces of Second order which are obtained from each other by parallel translations or, using a term introduced in 7.3, a T-system of such, hypersurfaces. Thus, one has Theorem 11.1: A symmetric covariant second order tensor determines in affine n-dimensional geometry a T-system of central second order hypersurfaces. 11.2 The symmetric covariant r-th order tensor afi12,.. ,. Reasoning

as in 11.1, we can verify that there exists a link between this tensor and a T-system of r-th order hypersurfaces one of which is determined by the equation ar,i2 ... it x r'x

6

.. X'r = 1,

( 11.2 )

where x' are the coordinates of the points of the surface. It is readily seen that the hypersurface (11.2) is not a general r-th 'order hypersurface: it has those properties which belong not only to the point x' but also to the point ex', where & is any r-th root of unity.

11.1-11.3

11.3


127

The contravariant simple bi-vector v''. We will begin with the case

n = 2. In two-dimensional affine space, all contravariant bi-vectors are simple and differ from each other only by a constant multiplier (cf. 10.4). We select one of these bi-vectors v = [xy]

(11.3)

0

(where x, y are non-collinear, contravariant vectors) as base; then, for every other contravariant bi-vector, we will have is = coil,

(11.4)

0

where co is a certain scalar. The number co is called the ratio of the bivectorv to the bi-vector D. For w = 1, the bi-vectors will be equal. 0

A simple geometric interpretation may be given to the equality of bivectors. Let one of the equal bi-vectors v" be given by (11.3); let the other be 0

V = [Zy],

where z is a contravariant vector. Since the space is two-dimensional, we have z = Ax +µy,

5 = [zy] = A[xy]+p[yy] = A[xy].

By assumption, v` = 15; consequently, one has A = 1 and the parallelo0

gram determined by the vectors z, y (or, simply, the parallelogram z, y)

Fig. I I

is obtained from the parallelogram x, y by an operation called shear (i.e., x is replaced by x+µy; cf. Fig. 11). If

v = [zt] = v = [xy], 0

(11.5)


128

CHAP. 11

where

z = Ax+µy,

t = A'x+p'y,

several shears may be required. Let us select a vector p" such that

P = z+at = x+Py,

(11.6)

whence

x+Py and

A+aA' = 1,

p+aµ' = P,

where a and P will only be defined if A' :A 0 (if A'= 0, we can make it non-

zero, for example, by a preliminary shear in which t is replaced by z+t). It follows from (11.6) that

[xy] = [ft],

[z1] = [Pt],

and consequently, by (11.5),

[Py] = [Pt]. We see now what shears must be applied: the first of these shears converts the parallelogram x, y into P, y, the second p, y into p, t, the third P, t into z, t. Thus, parallelograms which correspond to equal bivectors can be converted into each other by several shears; therefore the same area may be ascribed to them. Let us select the area of the parallelogram corresponding to the bivector 15 as the unit of the measure of area. Then to every parallelogram 0

z, t (where the order of the vectors is essential) will correspond a number w which is equal to the ratio of the bi-vector [ztl to the bi-vector v; the 0

number w is usually called the orientated area of the parallelogram, Introducing the dimension of area for parallelograms in this manner, we can extend it in the usual manner to figures bounded by any convenient curves.

We note that in this context the sign of an area turns out to be linked to the direction of the circuit of the perimeter of a figure; for brevity we will confine ourselves to the case of the triangle with vertices 1, 2, 3 (Fig. 12). Let z, t be the vectors which start from the point 1 and end at the points 2 and 3. The direction of a circuit corresponding to the order of vertices 1, 2, 3 will be assumed to be positive, if the area of the paralle-

11.3

129


logram z, t is positive, and negative in the opposite case. Since [tz] = - [zt], the circuits 1, 3, 2 and 1, 2, 3 will have opposite signs. In this definition, the vertex I plays a special role; it is readily shown that in actual fact all vertices are of equal importance. The vectors which

start from the point 2 and end at the points 3,

1

are t-z and -z,

respectively; then the bi-vector

[(t-z)(-z)]

= -[tz]+[zz] = [zt].

The same results are obtained for the vertex 3. Summarizing these results, we can state that the specification of a bivector in two-dimensional affine space establishes in this space the dimensions of an area and orientates it, i.e., it makes it possible to distinguish in it between positive and negative circuits of triangles (and any closed curves).

t

Fig. 12

Now let there be given in an affine space of an arbitrary number of dimensions a simple contravariant bi-vector d which is equal to the 0

alternating product of two contravariant vectors x, y: v" = [xY] 0

The vectors x, y determine a manifold of parallel planes the coordinates

of whose points are given by

'+Axl+py'

(11.7)

0

[cf. 8.3, (8.27)1. Vectors lying in one of these planes (i.e., determined by two points in such a plane) will have components of the form .lx'+µy'.


130

CHAP. 11

The alternating product of two such vectors is given by [(Ax + µY)(A'x +,U'Y)] = w[x y] = coo,

w = Ay' -.µ.

0

Next, we can repeat word by word everything that has been stated with reference to the case n = 2. In this way we arrive at

Theorem 11.2:

A simple contravariant bi-vector determines in n-

dimensional affine space a manifold of parallel planes and establishes in each

of them the orientation and dimension of area. If a bi-vector is given apart from a positive scalar multiplier, it defines

an orientated plane (or two-dimensional) direction. A system of parallel straight lines is determined by a contravariant vector x; it establishes on each of these straight lines a positive direction

and dimension of length (as the ratio of any vector to the vector x). We see that there is a profound analogy between contravariant vectors and contravariant bi-vectors. 11.4 The mixed second order tensor A;; there exists a link between

such tensors and the invariant equation

z' = A'x1

(11.8)

which relates a contravariant vector z of affine space to every contravariant vector x of the same space. In this context, if the vectors x, y are related to the vectors z, t, then the vector 2z+pt will correspond to the vector ,lx+py. As a consequence, one says that by (11.8) z is defined as a linear vector function of the vector x. The argument x and the function z

are here contravariant vectors; in such a case we have a linear vector function of the first kind. Thus, the mixed, second order tensor defines a linear vector function of the first kind in acne space. If the determinant JA,I $ 0, then (11.8) gives an affine transformation of the contravariant vectors of the space; thus, the theory of affine transformations of vectors constitutes a part of the theory of linear vector functions of the first kind. By virtue of this link with affine transformations

the mixed tensor AJ is often called an affinor. 11.5

The covariant,* second order tensor ci;. On the basis of the inva-

riant equation

u1 = cjjx',

(11.9)

this tensor relates a covariant vector u of affine space to every contra-

11.3-11.6


131

variant vector x of the same space, where, if the vectors x, y correspond to. the Sectors u, v, the vector i!x+py corresponds to the vector .Lu+uv; we have again a linear vector function in which the argument is a contravariant, and the function a covariant vector. This is a vector function of the second kind. The covariant second order tensor c;l defines a linear vector function of the second kind in affine space. The mixed tensor C k of covariance 2 and contravariance 1, which is skew-symmetric with respect to its subscripts. This tensor plays a very important role in the theory of continuous Lie groups. In order to demon11.6

strate this role, we will confine consideration to the case of the group consisting of matrices of the form A = IIAaII,

a,# = 1, 2, ..., M.

(11.10)

This group of matrices is said to be of order n, if the elements of its matrices are functions of n parameters. By Lie's basic theorem, the study of the local structure of a matrix (i.e., of its structure in a certain neighbourhood of the unit matrix) reduces to a study of the properties of a certain Lie matrix algebra of order n. The manifold of matrices of the form (11.10) is called a Lie matrix algebra, if it contains together with matrices A and B also the matrices AA+pB (where A, p are arbitrary scalars) as well as the commutator

[AB] = AB-BA

(11.11)

of the matrices A and B. We call the order of a Lie matrix algebra the largest number of linearly independent matrices in this algebra. In a Lie matrix algebra s/ let the matrices A1, A2, . . ., A. be linearly independent; we will select these matrices as a base of the algebra sat, i.e., we will express any matrix A of the algebra .sad in terms of the matrices A1, A2, ..., A,

A = A1A1+A2A2+ ... +2"A" = A'A;,

(11.12)

as a consequence of which every one of the matrices of 0 will be determined by n numbers ,l', 12, ... , A". By definition of a Lie matrix algebra, the commutators [AIAk] must all belong to sad, whence follows the relation [AI Ad = c';k A; ;

(11.13)

the numbers c'Ik determine the entire structure of the Lie algebra.


132

CHAP. 11

As base of the algebra d one may take any other n of its matrices Aa = pQ A1,

i, a = 1, 2, ... , n,

(11.14)

provided only that they are linearly independent, i.e., provided that the determinant (11.15) 0; IP=l

for any matrix A of s1, we will have = A*P'A1, whence it follows, taking into consideration (11.12), that A = A'A2

Al = P i

(11.16)

Thus, each of the matrices of the manifold s1 will be determined by specification of n numbers A', A2, ..., A° which may be exposed to the

linear transformations (11.16). Consequently, the Lie algebra a may be represented by an affine space in which the matrices A play the role of vectors [since the transformation (11.16) is linear and homogeneous]. Consider now the changes in the numbers ck for a linear transformation of the coordinates in the space .sad, i.e., for transformation of the base in accordance with (11.14); it follows from these formulae that

Al = q;Aa,

(11.17)

where q, is the reduced minor of the element pQ in the determinant (11.5).

Further, we will have [cf. (11.13), (11.17)] [AI Ar] = p '0 py[A1 Ak] = P 'B py c;k Al = pB pr cik qt A,

[Aa AY] = ceY As

where esy = q;p'a py c;k .

The relation (11.18) shows that in that affine space which represents the algebra .sad the numbers c;k form a tensor of covariance 2 and contra-

variance 1 which is called the structural tensor of the Lie algebra d. Obviously,

[BA] = -[AB]; therefore, by (11.13), i ; ckl = -cik.

( 11.19 )

11.6-12.1

RELATIVE TENSORS AND THEIR OPERATIONS

133

It can be shown (Examples 2 and 3) that the tensor c;k also satisfies the relation c[ik c1'

= 0.

(11.20)

In the general theory of finite Lie groups*), one has a tensor with the same properties which likewise determines the structure of a group in the

neighbourhood of its unit. Thus, the theory of both finite and continuous Lie groups coincides to a large extent with the theory of the tensor cjk, satisfying the conditions (11.19) and (11.20). Exercises 1. Show that in three-dimensional affine space parallelepipeds corresponding to equal simple contravariant 3-vectors [xvzJ and fx'y*z'J may be transformed into each other with the aid of displacements (where a displacement is said to be the replace-

ment of the vector z by z-axfly, etc., in 2. Prove that for any three matrices A, B, C there applies the relation [A[BC]]-} [B[CA]]+ [C[ABJ] = 0.

(11.21)

3. Prove that for the structural tensor cik of any Lie matrix algebra one has the relation (11.20). 4. Let the components of a tensor cj4l satisfy, in addition to (11.19) and (11.20), the following condition: The equality C,kAk = 0

only applies when h' = A _ ... = J.^ = 0. Show that matrices whose elements are defined by

A'_c.)k ,tt

for all possible R', At, ..., 3.^ form a matrix Lie algebra of order n for which the tensor

cIk is the structural tensor.

§ 12.

Relative tensors and their operations

With a view to later work, we must still introduce into consideration one other type of geometric object: relative tensors and their partic12.1

ular case -- relative scalars. The name relative scalar of weight g will be given to a geometric object with one component a which changes under linear transformations

of the space in accordance with the rule Or =

*) Cf., for example, Chebotarof, The theory of Lie groups, 1940.

(12.1)


134

CHAP. 11

where d is the determinant of the transformation (9.1). The scalar mentioned above (in 9.6) corresponds to the case g = 0 and, in contrast to a relative scalar, is called an absolute scalar. Obviously, a relative invariant is a relative scalar. By Theorems 9.1 and 9.2, the only essential component e12 ..." of the covariant n-vector e;,12 ... is a relative scalar of weight + 1,

and the only essential component e' 2 " of the contravariant n-vector e"" -- -'- is a relative scalar of weight -1. The law of transformation of relative tensors differs from that of the absolute tensors considered in § 9 by the occurrence of the additional multiplier d9. As an example we will define the relative tensor of covariance 3, contravariance 2 (and, consequently, of general variance 5) and weight g; such a tensor is determined by specification of n5 numbers Cam, where its components change under linear transformations of space in accordance with the law cyxµ = d°9tgiPyPAPNcklm,

(12.2)

A being the determinant of the transformation. An absolute tensor is a relative tensor of weight g = 0. A relative scalar is itself a relative tensor of zero variance and an absolute scalar is a relative tensor of zero variance and zero weight. All the results of 9.2 to 9.6 extend without change to relative tensors: one can speak of covariant and contravariant relative tensors, of symme-

try and skew-symmetry of relative tensors, etc. Also the link between tensors and algebraic forms remains in force, except that relative tensors correspond to relative forms which are themselves relative scalars of the same weight as the tensors. Thus, a symmetric covariant relative tensor a;jk of weight g is related to the cubic relative form atjk xix1xk

in the contravariant vector x, which is a joint relative invariant of weight

g of the tensor a and the vector x. 12.2

The definitions of the operations of symmetrization, alternation,

addition, multiplication and contraction presented in 9.7 to 9.10 also extend to relative tensors. In this context, it is not difficult to see that symmetrization, alternation and contraction do not change the weight of a tensor; one can only add tensors of the same weight g and obtain as a result a tensor of the same weight; one can multiply relative tensors of any

weights, where on multiplication the weights of the tensors are added.

12.1-12.3


135

For example, let ak and Cjk be relative tensors of covariance 2, contra-

variance 1 and weight 4, and bm a mixed relative tensor of variance 2

and weight - 1; then d jk/= ajk+C,k

is also a relative tensor of covariance 2, contravariance I and weight 4; ejkm = C jk b '

is a tensor of covariance 3, contravariance 2 and weight 3; im = a jk bk

is a tensor of covariance 2, contravariance 1 and weight 3;

a=b is a relative scalar of weight -1; finally, gkm - aj(k Cm)i

vkm = aj(k Cml;

are two covariant tensors of variance 2 and weight 8; the first of these is symmetric, the second skew-symmetric; in other words, vkm is a covariant bi-vector of weight 8. Note also the following proposition which is important for the later work (cf. Theorem 9.4): Theorem 12.1: The result of total contraction of a relative tensor of weight g is an invariant of weight g. 12.3

The proof of Theorem 9.1 is also valid for the case of a relative n-vector; consequently, the only essential component e, 2 . , .,, of the relative covariant n-vector of weight g : e,11, .., is a relative scalar of weight (g + 1). If one takes g = -1, then the component e12 ... R will remain unaltered for linear transformations of space; we set it equal to 1. Thus,

one has Theorem 12.2: The relative covariant n-vector ; of weight - 1, for which the component 812 ...,, = 1, retains the values of all its components after any linear transformation of space; in what follows we will call it the unit covariant n-vector. For an analogous reason (cf. Theorem 9.2), one has Theorem 12.3: The relative contravariant n-vector E'''= of weight + 1, with component 812 .." = 1 which is called the unit contravariant


136

CHAP. 11

n-vector retains the values of all its components after any linear transformation of space. Later on, the covariant and contravariant unit n-vectors will be denoted by e ji2 in and Ei,;z...; We will note certain properties of unit n-vectors. First of all, we will

show that eili2...in

e"r=...i" =

n!

.

(12.3)

In fact, in the sum on the left-hand side of (12.3), only those terms are

non-zero for which it i2 ... i are rearranged in the order 1, 2, ..., n; there will be n! such terms and each of them will be equal to + 1. Next, consider the tensor which is obtained as a result of contraction of the unit contravariant n-vector with any tensor, for example,

i.;:... "

(12.4)

By the results of 9.10 and 12.2, the weight of the tensor (12.4) exceeds that of the tensor a by unity and its variance is smaller by n units. On the

other hand, the sum on the right-hand side of (12.4) involves only n! non-zero terms, in fact, those for which i, i2 ... i can be rearranged in the order 1, 2, . ., n; if this permutation is even, then the factor si't'e = 1, if it is odd, the multiplier is equal to -1. Hence it is clear that the right-hand side of (12.4) is equal to .

n!a[12..n1in* ...;,

We arrive at an analogous result when we contract e" '^ with any tensor with respect to any n subscripts. Thus, contraction of a tensor (of covariance > n) with 0'2with respect to n of its subscripts is the same as replacing these indices by 1, 2, ... , n, alternating with respect to them and multiplying the result by n!. This operation is referred to as total alterna-

tion. We arrive at the following Theorem 12.4: The total alternation of a tensor of covariance r > n and weight g n-ith respect to any n of its subscripts leads to a tensor with the some contravariance, covariance r - n and weight g + 1. Similar reasoning shows that contraction with e;,;,.. ,,, is equivalent to total alternation with respect to is superscripts; hence one has Theorem 12.5: The total alternation of a tensor with contravariance s > n and weight g with respect to an)' n of its subscripts gives a tensor with the same covariance, contravariance s -n and weight g -1.

12.3


137

We have seen above (Theorem 9.4) that the operation of total contraction provides a method of obtaining invariants; total alternation is another procedure leading to the same goal. In fact, let there be given a covariant

tensor of weight g and variance r = nq, where q is an integer. Then, dividing its subscripts in an arbitrary manner into q groups with n indices

each and performing a total alternation with respect to each of these groups, we obtain by virtue of Theorem 12.4 a tensor of weight g+q and variance r-nq = 0; thus, we have Theorem 12.6: If one performs q total alternations on the subscripts of a covariant tensor of weight g and variance nq, where q is an integer, one

obtains a relative invariant of linear transformations of space of weight g + q.

The reader will readily formulate the theorem which is dual to Theorem

12.6 and relates to contravariant tensors. Unit n-vectors can be applied to rewrite in a more convenient form certain formulae which we have encountered earlier. Thus, for n = 2, we can rewrite (5.13) in the form

[xy] = F,ix'y';

(12.5)

the relationship between the components of a contravariant and a covariant vector which in projective one-dimensional space correspond to one and the same point (cf. end of 5.2) can be expressed in the form tr; = Fii x',

(12.6)

x' = F"u;.

(12.7)

For n = 3, the formulae (6.19) and (6.21) can be rewritten with the aid

of unit n-vectors in the form [xyz] = F;jk X Y'zk,

[uvw] = F"kttl vj Wk

(12.8)

(cf. footnote p. 112) and the formulae (6.16) and (6.18) in the form Ui = EiJk x'yk,

x' = F''ku; vk

(12.9)

[cf. (10.9), (10.10)] or in the form 2x 1yk) = f."ktli ,

2u1i Uk]

- F;jk Xi.

(12.10)

Note that, as a consequence of the statements (12.5) and (12.8). Theorems 5.2, 6.2 and 6.3 are immediately obvious. By the first of the


138

CHAP. II

formulae (12.9), the quantity u; is defined as a relative, and not as an absolute covariant vector (cf. § 6, Exercise 2); in view of the homogeneity

of the coordinates of the straight line, this difference is obviously not essential. An analogous remark may be made in relation to (12.9) and (12.6).

Exercises 1. Verify that the transformations of a relative scalar (cf. (12.1)] for all possible linear

transformations of space form a group which is homomorphic to the group of transformations (9.1). Derive the same result for transformations of a relative tensor, for example, for the transformations (12.1). 2. Prove that i02... in ain] E. (12.11) = it biz is in 3. Prove that

fills ...ik akrl

k!(n-k)! ill b'!J2 ... t5it] is

an

(12.12)

4.') Let a, b be two covariant and x, y two contravariant vectors. Also let c = [xy]. Prove that label = (ax)(by)--(bx)(ay).

5.') Prove the following identities (where a, b. c, d are covariant, x, y, contravariant vectors): ([abI[xy1)

(12.13)

(ax) (by) - (ay) (bx),

[[abjx] (ax)b - - (bx)a, [(abI [cd]] = lacd]b .- [bcd]a = [abc]d- [abd]c,

(12.14)

[lca)(ab]] = [abc)a,

(12.16)

([bc][cal[ab]]

(12.15) (12.17)

[abc12,

and write down the identities which are dual to the identities (12.14)-(12.17). 6.') Let a, b, c, a', b', c' he covariant vectors and [bc] = x, [b'r'] = x',

[ca] le'a') =

y. y',

[ab] = z. ja'b'] = z'.

Prove that

[[xx'][yy'][Z1']] _ label[a'b'c'][laa'][bb'][cc']] (12.18) It is readily seen that (12.18) contains the well known theorem of Desargues on perspective triangles as well as its inverse. ') In 4-6, it has been assumed that n = 3; also [xyzj and [urw] do not there denote not a hi-vector, but a relative covariant tensor, defined by the first of the formulae (12.9); similarly, a and bin [ab] are covariant vectors and (ax) is an abbreviation for aix'. 3-vectors, but relative scalars defined by (12.8), and

CHAPTER III INVARIANTS AND CONCOMITANTS OF TENSORS AND THEIR SIMPLEST PROPERTIES § 13.

Invariants and concomitants of tensors

13.1 We have already employed above the concept of an invariant of linear transformations of space; we will now give a rigorous definition of this concept for the case of one or several tensors. Let a be any absolute or relative tensor with N essential components; we will number these components in any definite order and denote them by a,, a2, ..., aN. As a result of a linear transformation of space [cf.

(9.1)], these components written in the same order will assume the values a*, az,

..., aN. A function cp(a,, a2, ..., aN) of the components

of the tensor for which (p(ai, a*, ..., aN) = AA' cp(a,, a2,

(13.1)

where d is the determinant of the transformation (9.1), will be called an invariant of the tensor a. The equality (13.1) must be satisfied as a result of the rule by which the components of the tensor a vary for linear transformations of space, for any such transformation and for any values of the components a,, a2, ..., aN. In other words, if in (13.1) we replace the components a,,, , az i ... , aN by their expressions in terms of a,, a2, aN and the coefficients pa of the linear transformation of space in

accordance with the law of transformation of the components of the tensor a, then the equality (13.1) must become an identity with respect to

a, , a2, ... , aN as well as with respect to all coefficients pa (provided A = IpaI : 0). The number g on the right-hand side of (13.1) is called the weight of the invariant; for g = 0, the invariant is said to be absolute, for g :A 0, relative. Note that in this terminology one always understands by an invariant of a tensor an invariant with respect to linear transformations of space.

140

INVARIANTS AND CONCOMITANTS OF TENSORS

CHAP. III

In the same manner one may define joint invariants of several tensors

(which may comprise absolute as well as relative tensors): Such an invariant is a function of the components of these tensors with properties analogous to those expressed by (13.1).

It may be said that the equality (13.1) indicates that the function

(p(a,, a2, ..., aN) of the components of the tensor a is itself a relative scalar of weight g. In this way we arrive at the more general concept of a concomitant of a tensor. Let there be given several functions

b, _ P,(a, , a2, ... , aN), h2 = 92(a 1 , a2, ... ,

aN),

...,

(13.2)

bM = (pM(a1, a2, ..., aN)

of the components of a tensor a with the following properties: If one replaces the numbers a,, a2, ..., aN by the values a1, a2, ..., aN which these numbers assume after a linear transformation of space, then the numbers b1, b2, ..., b,,, change as if they were the components of a certain absolute or relative tensor. In that case one says that b1, b2, ..., bM define together a concomitant of the tensor a. In this context, it is under-

stood that the equalities, expressing the fact that after replacement of a1, a2, aN by a*, a2, . . , aN, the quantities b,, b2, ..., b,,, transform like the components of a tensor, must become identities with respect to a,, a2, . , an and the coefficients pa of the linear transformation (for d 96 0), if one replaces a,, a2, . . ., aN by their expressions in terms of a,, a2, ..., aN and the pQ which correspond to the law of transformation of the components of the tensor a. In an analogous manner, one may define a joint concomitant of several tensors. In particular, it follows from this definition that after performing one or several operations of the types defined in 9.7-9.10 on the tensors

a, b, c, ... we will obtain joint concomitants of these tensors. If in (13.2) the components a,, a 2 ,. .., ay are functions of the components of a third tensor c which varies in accordance with the tensor law, i.e., if a is a concomitant of the tensor c, then obviously also b will be a concomitant of the tensor c; an analogous result applies to the case of concomitants of several tensors. Thus, one has Theorem 13.1: A concomitant of concomitants of several tensors is likewise a concomitant of these tensors. As has already been stated above, an invariant is a particular case of a concomitant (namely that in which the concomitant is a relative scalar). It is easy to proceed from any concomitant to an invariant: Let k be a


13.1-13.2

141

concomitant of the tensors a, b, c, ... which is itself a tensor of covariance

r and contravariance s; contracting it with r contravariant vectors x, 1

2

s

1

X, ..., x and s covariant vectors u, u, ..., u, we obtain a multilinear form 2

r

which is an invariant of the tensors a, b, c, ... and the vectors x, x, ... , x, 1

1

2

u, u,

2

r

S

.

. ., u. Conversely, given a multilinear form of contravariant and

covariant vectors which is an (absolute or relative) invariant of these vectors and certain tensors a, b, c, . ., then the coefficients of this form together constitute a concomitant of the tensors a, b, c, ... (cf. 9.2 and .

§ 9, Exercise 9). On the basis of these results, the theory of concomitants

reduces to that of invariants. These definitions of an invariant and a concomitant will be elucidated by a number of examples in § 14. The terminology used in classical texts on the theory of invariants and in a majority of contemporary books differs somewhat from that used here, since we aim to approximate the terminology of tensor algebra. The classical theory rests on the basis of the study of multilinear and higher order forms, and not of the tensors formed by the manifold of their coefficients. An invariant of a form is what we have called an invariant of the tensor connected with this form. In the classical theory, special attention (is given to forms of the type 13.2

cv(x', x2, ..

>

x") = anti...) x X

... x"

(13.3)

involving one covariant vector or, as one says, one set of variables x', x2, ..., x" [the coefficients a;lt,...i, in (13.3) are symmetric with respect to all their subscripts]. The number r is called the order of the form; for n = 2, a form is called binary, for n = 3, ternary, for n = 4, quaternary and, for

arbitrary n, n-nary. In the present treatment, symmetric covariant tensors correspond to such forms (cf. 9.3): the symmetric covariant tensor

a,,,,... jr of variance r corresponds to the form (13.3). Let there be given a concomitant of the system of tensors a, b, . . . to which there correspond the forms 9, I/', . . .; by contracting it with a )suitable number of contravariant and covariant vectors, we obtain a form w. In the classical theory, this form w is called a concomitant of the forms cp, 0, . . . (whereas in the present text a concomitant is the system of the coefficients of the form w). Concomitants can be subdivided into several types.

CHAP. III


142

If a concomitant of the forms cp, 0, ... is a form in a number of contravariant vectors, it is called a covariant of the forms cp, L', ...; the set of coefficients of the form co constitute then a covariant tensor. Very frequently one is concerned with a covariant which is a form in one contravariant vector, i.e., with a covariant of the type (13.4)

b1 2...;,x''x':... x''

(where b.1L2... i, is symmetric with respect to all subscripts); in the present treatment, the quantity corresponding to this concomitant is a symmetric

covariant tensor. The number s is called the order of the covariant; if 611;2...j. are integral rational functions of the coefficients of the forms cp, 41, . . ., then the highest degree of these functions with respect to the coefficients of one of these forms, for example, cp, is called the degree of the covariant with respect to the form co. If a concomitant is a form in a number of covariant vectors, it is called a contravariant (and its coefficients, the components of a contravariant tensor). Finally, if a concomitant is a form in covariant and also contravariant vectors, it is called a mixed concomitant (to which there corre-

sponds a mixed tensor). Invariants may also be regarded as particular cases of concomitants, and, in fact, as concomitants of zero order. The classical theory has advanced especially far in the region of the study of binary forms of a single contravariant vector, to which we will devote Chapter V. We will introduce for this case certain notational abbreviations: the components of a contravariant vector will be denoted by x, y instead of by x', x2. Since the coefficients a111,...r, of the form (13.3) are symmetric with respect to all their subscripts, they will differ from each other only by the number of ones or twos among the subscripts; therefore we will write 13.3

ao = alll ... 1, a1 = all ... 12,

ak = all ... 122...2,

..., a,=a22...2.

r-k

k

(13.5)

As has been shown in § 9 [cf. (9.17)], the coefficient of x'-kyk in the form

(13.3) (for n = 2) will be k!(r- k)!

ak = Ckr ak .

(13.6)

13.2-14

EXAMPLES OF INVARIANTS AND CONCOMITANTS

143

Thus, a quadratic binary form may be written

f = aox2+2a,xy+a2y2;

(13.7)

cp = aox3+3a,x2y+3a2xy2+a3y3;

(13.8)

a cubic binary form

a fourth order binary form

0 = aox4+4a,x3y+6a2x2y2+4a3xy3+a4y4

(13.9)

an r-th order binary form

as xr+raI xr- Iy+ +

r(r-1) a2 xr-2 Y2 1.2

r(r-1)__(r-k+1) akxr-'yk+ ... +a, yr.

1.2...k

(13.10)

For n = 2, the distinction between contravariant and covariant vectors

is not essential in view of the simple relationships (12.6) and (12.7) which hold between them; correspondingly, in the theory of binary forms, a distinction is made between only two types of concomitants: invariants and covariants. In this context, one usually considers covariants which are

forms in one contravariant vector; such a covariant of several binary forms ip, 0.... is a form in the components x' = x, x2 = y of a contravariant vector which is invariant with respect to linear transformations of space. The coefficients of this form are functions of the coefficients of

the forms 9, Iy, ... . In what follows, we will employ alongside the terminology of tensor algebra, given in 13.1, the classical terminology of 13.2 and 13.3; in view of the simple and close link which exists between them there is no danger that this will lead to confusion. § 14. Examples of invariants and concomitants We will give now several examples of invariants of tensors and also of

their concomitants which are additional to those encountered earlier. Only a few of these invariants and concomitants will be given geometric interpretations; for the remaining ones this will be done below after a more detailed investigation of the corresponding tensors.


144

CHAP. III

1. A contravariant vector x and covariant vector u have the joint invariant (their scalar product)

(ux) = (xu) = uix'.

(14.1)

By virtue of the homogeneity of the coordinates x' of a point and of the coordinates ui of a hyperplane, a projective geometric meaning can only be attached to the case when the invariant (14.1) is equal to zero:

the point x then lies on the hyperplane u. The affine geometric meaning of the invariant (ux) can be established in the same manner as was done in §§ 4, 5 for the case n = 2; for this purpose, one has to start out from the fact that the covariant vector ui determines a pair of hyperplanes, the first of which is given by

u.S'+h = 0,

(14.2)

u;5'+h = 1

(14.3)

the second by

(where c' are the coordinates of the points of the hyperplane, cf. 8.3) We will place the contravariant vector x so that its starting point ' lies on the hyperplane (14.2). Then the straight line 0 S' =

S' + Ax" 0

on which the vector x is located, intersects the hyperplane (14.3), as can be seen from a simple computation, at the point + t

-X.

(ux)

o

Consequently, the invariant (ux) is equal to the ratio of the collinear vectors x' and 1

0

X

1

0

However, if (ux) = 0, the same reasoning shows that for this choice of

its origin the vector x will lie completely in the hyperplane (14.2).

2. Construct from the components of n contravariant vectors x, x, ... , x the determinant ' 2

x

145


14

i1 X1X2...Xni

111111

1

1

1

X1X2...x" 2

xIx2 n

2

2

= n!X[1x2 ... x"] = CXX ... X],

... X"

2

12

n

(14.4)

n

n

n

which is called the determinant of the n contravariant vectors x, x, ..., X. I

2

n

The second part of (14.4) is nothing else but the expanded expression for

the determinant employing the alternation symbol; it shows that the determinant (14.4) is equal to the result of the total alternation of the tensor x" x" ... x`". Using a theorem dual to Theorem 12.6, we thus 1

2

obtain

Theorem 14.1: The determinant of n contravariant vectors is a relative invariant of weight --1 (cf. Theorems 4.2, 5.2, 6.2). In connection with the notation in the last part of (14.4), which is very important for what follows, we must refer again to the remark in the footnote on p. 112. For the same reason as in Example 1, and also by virtue of the relative nature of the invariant (14.4), a projective geometric meaning can only be attached to the case in which the invariant is equal to zero: The points

x, x, ..., x are then dependent, i.e., they lie in one hyperplane. The 1

2

n

affine geometric meaning of (14.4) will be considered below (cf. § 32, Exercise 20). If the vectors x, x, 1

..., x are linearly dependent then

2

n

it1x+X12x+ 2

1

...

+A"x = 0, n

...,

A" are scalars at least one of which is non-zero; this where A', A2, dependence is readily seen to be preserved after any linear transformation of space. Hence we see that the number of linearly independent vectors among the vectors x, x, . . ., x cannot be increased by a linear transfor1

2

n

mation; however, by virtue of the invertibility of the transformation, this number also cannot decrease, and consequently must remain unaltered. We have here an exampe of a so-called arithmetic invariant. The rigorous definition of this concept is as follows: a natural number which can be

CHAP. III


146

evaluated in terms of the components of one or several tensors is called .an arithmetic invariant of these tensors (or of the corresponding forms), if it remains unaltered when we replace the components of the tensors by the values which they assume after a linear tranformation of space.

The reasoning which has just been presented obviously applies for any number of vectors; the number of linearly independent vectors among vectors x, x, . ., x is known to be equal to the rank p of the .

1

2

q

matrix I Ix` I I ( = 1, 2, ..., q), formed from the components of these A

vectors, and it is called the rank of the system of vectors x, x, ..., x. Thus,

we can write down Theorem 14.2:

q

2

1

The rank p of a system of q vectors is an arithmetic

invariant.

Geometric interpretations of the rank p of a system of vectors in projective as well as in affine geometry are readily given. In projective space, those p points of the system x, x, ... , x which are linearly independent 1

q

2

define a (p -1)-dimensional plane (cf. 8.2); the points x, x, ... , x, 1

2

q

being dependent on these p points, are all located in a single (p-1)dimensional plane completely determined by them. In a similar manner, we can verify (cf. 8.3) that the contravariant vectors x, x,. . ., x of a system of rank p, if they are represented as starting 1

2

q

from one point of affine space, all lie in one definite p-dimensional plane.

3. The determinant

1

2

R

1 2

R

= n !u11 u2 ... URA = 1U U ... U] n

R

(14.5)

n

u1u2...Un 1

2

is called the determinant of then covariant vectors u, u, ..., u; from Theo-

rem 12.6 follows

Theorem 14.3: The determinant of n covariant vectors is a relative invariant of weight + 1 (cf. § 4, Exercise 8, Theorem 6.3). The projective geometric meaning of the vanishing of the invariant

147


14

1

2

(14.5) is as follows: all the hyperplanes u, u, ..., u pass through one point. The concept of the rank of a system of q vectors together with Theorem 14.2 carries over to the case of covariant vectors. If the rank of the system 1

2

1

q

2

q

u, u, ..., u is equal to p, then (cf. 8.2) all the hyperplanes u, u, ..., u pass through one (n - p -1)-dimensional plane completely determined by them.

4. Let there be given a quadratic form a;; x'x',

(14.6)

where x' is a covariant vector and the a. form a symmetric second order tensor *). The determinant

all a12 ... a,, a21 a22 ... a., = nl al1111 a2121 ...

la;;l = I

a.2 ... a,,

(14.7)

I

is called the discriminant of the quadratic form (14.6) or the discriminant of the symmetric covariant second order tensor a1 , corresponding to this form.

By a well known property of determinants, the last part of (14.7) is also skew-symmetric with respect to all those subscripts to which the alternation does not extend; therefore it does not change its value if we

include all these subscripts under a second alternation sign; then we derive from Theorem 12.6 Theorem 14.4: The discriminant of a quadratic form is a relative invariant of weight +2 [cf. (5.23), (6.30)]. We will elucidate the projective geometric significance of the vanishing of the discriminant of a quadratic form. If l a;tl = 0, there exists, by a

well known result of the theory of linear homogeneous equations, at least one point xo for which a;t x0 = 0

(14.8)

from which follow directly a11xo4 = 0,

a.1y'xo = 0,

*) The tensor is understood to be absolute; in the case of relative tensors, we will always state their weight.

CHAP. III


148

where y' is any vector, i.e., by Theorem 10.2, the point x0 is conjugate to any point y of space with respect to the second order hypersurface (14.9)

and lies on this hypersurface. Such a point of a second order hypersurface is called a singular point. Thus, if the discriminant of the quadratic form (14.6) vanishes, the second order hypersurface (14.9) has one or several singular points.

The rank of the matrix 1)a;;II formed from the coefficients of (14.6) (the components of the tensor) is said to be the rank of the quadratic form (14.6) or the rank of the symmetric tensor ail. We will now prove Theorem 14.5:

The rank p of the quadratic form (14.6) is an affine

invariant.

In fact, by the very definition of the rank of a matrix, the quantity p is equal to the number of linearly independent vectors among 2

1

ui = ai2 = aij e', ... , ui = a,, = aiJ ei,

ui = ail = a1J e',

(14.10)

"

2

1

where e, e, ..., e are the contravariant coordinate vectors (cf. 8.3). 1

2

"

Consider the manifold of all covariant vectors of the form

aijx',

(14.11)

where x is an arbitrary contravariant vector. The vectors (14.10) belong to this manifold, and therefore the number of vectors (14.11) which are linearly independent cannot be less than p; on the other hand, it cannot be larger, since any one of the vectors (14.11) is a linear combination of the vectors (14.10): 1

z

a,1xf = xIui+x2ui+

...

"

+x"ui.

Thus, the quantity p is the number of vectors of the form (14.11) which are linearly independent, and from this follows the invariance of p on the basis of reasoning of the type which led to the proof of Theorem 14.2. The projective geometric meaning of the rank p of the quadratic form (14.6) is also linked to the equations (14.8). If p = n-1, the equation (14.8) has only one linearly independent solution and the hypersurface (14.9) has only one singular point; if p = n-2, the solutions of (14.8) will have the form Alx'+A2xi, where x and x are independent points; 1

2

1

2

14


149

the hypersurface (14.9) now has a straight line of singular points. Similarly, it may be verified that for arbitrary values of p z 1, the hypersurface (14.9) will have an (n - p -1)-dimensional plane of singular points (p < n);

for p = n, there will be no singular points. 5. On the basis of Theorem 12.6, one can readily construct joint invariants of the symmetric covariant tensor.a;! [or, what is the same thing, of the quadratic form (14.6)] and of the covariant vector u;. It is easily seen (cf. § 9, Exercise 10) that, if both, subscripts of the tensor all occur within the same alternation symbol, we obtain the result zero: to every

combination involving these subscripts there will correspond another combination which differs from it only by the order of the indices; the two terms of the alternation corresponding to two such combinations will differ only in sign and will cancel. For a similar reason, the same thing will occur if one has inside an alternation sign two indices belonging to the product ulul. Consequently, in a total alternation, one can include one index belonging to the vector u and (n -1) indices belonging to each of n- I factors of the type ail. Thus, we arrive at a joint relative invariant of weight + 2 of the tensors a;j and ul (n l)2u[1 u[1 a22 a33

. . .

Ci 02i2 ... rnFl2i2h ...

uil at:12 a,213

.. ar.i.; (14.12)

in the first part of (14.12), one alternation sign extends over all first subscripts of the symbol a, the other alternation sign embraces the second subscripts. Incidentally, one observes here the inconvenience connected with the notation employing square brackets. This inconvenience does not occur in the equivalent presentation of the second part of (14.12); but then the second way of writing the formula is more complicated. In the classical terminology (cf. 13.2), the quantity (14.12) is a covariant of the

quadratic form (146) of weight + 2, of order 2 and of degree (n-1). In order to elucidate the geometric meaning of u[1 U(1 a22 a33 ...

0

(14.13)

(under the assumption that the rank of the tensor at! is n), we select the plane u to be the first coordinate plane (cf. 8.2); then u1 = 1, u2 = u3 =

... = u = 0 and (14.13) assumes the form a12121 a3131 ... an] = 0

(14.14)

(the need for the second alternation sign disappears, since the left-hand

CHAP. III


150

side of (14.14) is itself skew-symmetric with respect to all those subscripts

to which the alternation sign shown does not extend). The hyperplane u, whose equation will now have the form x' = 0, is itself a projective space P"_, of order (n-1) the points of which are determined by the (n-1) homogeneous coordinates x2, x3, . . ., x"; those points of the hypersurface (14.9) which lie in this space form in it a

hypersurface of second order with the equation a; J x'x' = 0,

i, j = 2, 3, ... , n.

(14.15)

The left-hand side of (14.14) is the discriminant of the quadratic form corresponding to this hypersurface; therefore (cf. Example 4) the equality (14.14) indicates that the hypersurface (14.15) in the space F _ I , representing the manifold of points common to the hypersurface (14.9) and the hyperplane u, has at least one singular point. In such a case it is said that the hyperplane u touches the hypersurface (14.9). Thus, the equality (14.13) is the condition of contiguity of the hyperplane u and the second order surface (14.9); it is called the tangential equation of the hypersurface (14.9) [cf. (6.38)].

6. Following the model of the invariant (14.7), one can construct joint relative invariants for two, three, etc., symmetric tensors a,J, b,J, c,,, ... (i.e., two, three, etc., quadratic forms) with weight + 2 (Theorem 12.6): (n1)2b[1[1 a22 a33 ... a"1"] ,

(14.16)

(n 1)26[,[1 b22 a33 a44 ... a",,,,

........................

(14.17)

(n1)2b[It, C22 a33 a44 ...

(14.18)

......................

in all these expressions, the first alternation sign extends to the first subscripts of the symbols a, b, c, . . ., the second over the second subscripts.

For n = 2, the quantity (14.16) becomes (5.25). 7. Let u, be a covariant, x' a contravariant vector, aj j, b,, symmetric covariant second order tensors which correspond to the quadratic forms

(p=a,,x'x'

and

i/i=b,Jx'x.

By Theorem 12.6, the expression (n 1)2u[I alk,[, b22 b33 ... b"l"] xk = 6 '''2 ...1.eil z". J'u11 akj, b

b

b

xk

(14.19)


14

151

is a joint relative invariant of weight +2 of the tensors ui, x5, ail, b51; it is a linear form in the contravariant vector x' and the covariant vector ui. In the classical terminology [cf. 13.2], the quantity (14.19) is a joint mixed concomitant of the forms (p and 41. The coefficients of this bilinear form Ct = E1112 ... InEiii2 ... J,

bi212 ... bi.,1. ak11

k

(14.20)

are (§ 12) the components of a mixed relative second order tensor of weight +2; in the terminology of tensor algebra (cf. 13.1), this tensor ck will be called a concomitant of the tensors ail and b;1. 8. Let there be given an n-nary form of r-th order

t

..1à{112...rxx'2

...x

(14.21)

where a,,i2 ... j, is a symmetric covariant tensor of order r and xt is a

contravariant vector. Since 8x'

,

Ox'

- bk ,

we find for the derivative off with respect to xk

of '3x

... xtr+ax,12 ... it x''S'2xi3 = a I,i2 ... it 611X12 k

... x''

+ ... +ai,i2...t,x''

... xI_'8k.

(14.22)

However, 0 for 1

Ik, i1 = k;

therefore only those terms for which ii = k remain in the sum on the right-hand side of (14.22) and 6 is replaced by unity in all these terms.

Thus, the first sum is equal to akr2 ...

:, x2

... x''.

(14.23)

A similar reasoning applies to the remaining sums on the right-hand side of (14.22); by virtue of the symmetry of the tensor a, all of them turn out to be equal to the expression (14.23). Thus, one finds

of axk

rakr2

... ,r x

;2 . .

. x,,.

CHAP. III


152

Introducing the notation 1 of = r axi

A

a2f

1

J, fij = r(r-1)oxox i

_1_

(14.24)

a'f

fiik - r(r - l xr - 2)

Ox'axjaxk ,

etc., we can rewrite our result in the form

fi = aU2i,...i,x X ... X ii

f,

i,

(14.25)

.X

14.261 (14.26)

In an analogous manner, we find

fi; = ai;i,i.... t, fijk = a11114,5 ... i,

z,,x"

.

.

x x ... X ,

(14.27)

etc. The formulae (14.25), (14.26), (14.27) and similar ones show directly

that the fi(i = 1, 2,

..., n) are the components of a covariant vector,

that the fig form a covariant, symmetric second order tensor, that thefi;k form a covariant, symmetric third order tensor, etc. This situation makes it possible to derive from an already known invariant of tensors of this type covariants of one or several forms of the type (14.21). I

2

For example, let f, f, . . ., f be n-nary forms of orders rl , r2i ..., rn, 2

1

respectively. Then fi, f1,.. ., fi will be covariant vectors and, consequently, (Example 3) I

1

2

I

I

n

nlf[If2 ... fnl = A f2

fn A

f2...f is an invariant under linear transformations of weight + 1. Since the 1

2

n

invariant (14.28) depends on the coefficients of the forms f, f, ..., f and is itself a form in the contravariant vector x, it is a joint covariant of the 1

2

forms f, f, I,

2

(f, f,

n

..., f which is called their Jacobian; it is usually denoted by ,f): n

153


14

The Jacobian

Theorem 14.6:

1

1

of

ax' OX2

ax"

2 1

2

n

1

-

(f,f,.--,f)=

-

r1 r2 ...rn

2

2

2

Of of

of

ax' ax2

ax"

n Of

1

1

Of of

n

(14.29)

n

of

of

OX'ox2

ax"

n

of the n forms f, f, ..., f of orders rl , r2, ..., r", respectively, in a contravariant vector x is a joint covariant of weight + 1. For a single form f of order r, one can obtain a covariant starting from the fact that the f,; form a symmetric second order tensor; consequently (Example 4), f11f12 nif[1I 1Jf2121

..

fl.

fln

f21f22 f2n ifn1fn2

1

r"(r -1)"

fnn a2f a2f (ax')2 ax'ax2

a2f

ax'ax"

02f a2f Wax2 (ax2)2

ax2ax"

a2f

02f a2f a2f ax"ax' ax"ax2 ::: (ax")2

(14.30)

I

is a covariant of the form f of weight + 2; it is called the Hessian of the form f. Theorem 14.7: The Hessian of the form f of order r in a contravariant vector x, defined by (14.30), is a covariant of weight +2 of the form f. 9. On the basis of the results obtained so far (Example 8) one can ob-

tain the simplest concomitants of a cubic binary form ( [cf. (13.8)]. By Theorem 14.7, the Hessian of this form

INVARIAN'S AND CONCOMITANTS OF TENSORS

154

o2q)

H = 3 Ia

CHAP. III

2

i axz axay a2(p

as(p

axay aye

_ (aoa2-ai)x2+(aoa3-a1 a2)xy+(a1 a3-a2)Y2

(14.31)

is a covariant of weight +2. Consequently,

h11 = aoa2-a2,. h12 = h21 = #(aoa3-a, a2), h22 = a1 a3-a2 (14.32) are the components of a covariant symmetric tensor of weight + 2, where the relations (14.32) are preserved after any linear transformation of space. The tensor h;1hk, will have weight 4; therefore, by Theorem 12.6, the discriminant of the tensor h,1 or, what is the same thing, the discriminant

of the Hessian H of the form T 111,

h12 1

h21

h22

= 2ht111jh212 = 2ht1[1 h2121

will be an invariant of the form q1 of weight 6. Four times the value of this invariant is called the discriminant of the cubic binary form qp; we will

denote it by the symbol D:

D = 4(aoa2-a2Xala3-a2)-(aoa3-ala2)2 = 3a2a2+6aoala2a3-4aoa2-4axa3-aoa3.

(14.33)

Theorem 14.8: The discriminant D of the cubic binary form q' [cf. (13.8), (14.33)] is a relative invariant of weight 6. Since the form H has weight 2, one finds in an analogous manner Theorem 14.9: The Jacobian Q of a cubic binary form V and of twice its Hessian H [cf. (13.8), (14.31)] is a covariant of the form cp of weight 3.

The expanded form of the expression for the covariant Q is

Q = (aoa3-3a0a1a2+2a3)x3+3(a(,ala3+aia2-2a0a2)x2y +3(-aoa2a3-aIa2+2aia3)xy2+(-aoa3+3ala2a32a2)y3. (14.34) The concomitants H; D and Q play an important role in the theory of cubic binary forms (§ 23). 10. In tensor notation, the binary fourth order form (13.9) may be rewritten 0 = a1/k,xlxlxkxl, (14.35)

155


14

where the a,jkl form a symmetric tensor and x1 is a contravariant tensor. On the basis of Theorem 12.6, we see that (14.36)

i = 8a[,[1[1[1 a2]2]2]21

is an invariant of the form (13.9) of weight 4. For the purpose of expanding the expression for the invariant i, we reason as follows: in expanding the right-hand side of (14.36), the term a1 111 a2222 occurs only once, namely,

when in all alternating pairs of subscripts I occurs on the left and 2 on the right; its sign is obviously positive (+). For the same reason, the term a2222 all I, occurs only once; since in that case in all four alternating pairs 2 will be on the right and I on the left, its sign will also be positive (+). We obtain the term a,112a2221, if in three alternating pairs we place

1 on the left, and in the fourth on the right, and consequently its sign will be negative (- ). Since the choice of those pairs which have 1 on the right is arbitrary, there will be C4 = 4 such terms. Similar reasoning can be applied to the term a2221 a1112 Finally, the term a1122 a2211 will .have a positive (+) sign, since two alternating pairs will have I on the left, two will have 1 on the right; it will occur C4 = 6 times, because the choice of those two pairs among four pairs among which 1 is on the left is arbitrary. Further, one must still take into consideration that each alternation gives a multiplier 1/2 in front. Thus, i = 1[2a1111a2222-8a1112a2221+6(a1122)2]

or, in the notation of 13.3 [cf. (13.5)],

i = aoa4-4a1a3+3a2.

(14.37)

11. Using multiplication and contraction (cf. 9.10), one can construct from a given mixed second order tensor (affinor) Al other affinors 2

3

A; = A; Aa,

A = Am i AP Aa,

... ;

(14.38)

a

2

obviously, the matrix I IA4I I is the square of the matrix I IA(I I and the matrix 3

2

3

I1Aill the cube of the same matrix, etc. The affinors Ai , 4,... are absolute concomitants of the affinor A; . Total contraction (Theorem 9.4) leads to the following absolute invariants of the affinor A : S1 = A:,

S2 = AaA;,

S3 = AQA;Ad,...

(14.39)


156

CHAP. III

The same procedure can be applied to several affinors; for example, one can construct from the two affinors Al and B; a third affinor

C = A; Ba ;

(14.40)

C is a joint absolute concomitant of the affinors A and B{ and the matrix IIC II is the product of the matrices II Ai II and I IB)Il.

12. The concept of rank, given in Example 4 above for a covariant symmetric second order tensor, can be generalized to the case of such a tensor of any order. The number of linearly independent vectors among all

the covariant vectors of the form (14.41)

a;;k x1yk,

where x, y are any contravariant vectors in space, is called the rank of the symmetric covariant third order tensor a.1k. As in the case of the tensor

a, j, it is readily verified that the rank of the symmetric tensor a;1k is equal to the rank of the matrix lla;lkll, where the subscript i determines the

number of the row and the pair jk that of the column of the matrix, so that the matrix in question has n rows and n(n + 1)/2 columns. In a similar

manner, one can define the rank p of a symmetric covariant tensor a;,;, , _ ,;, of any order r: p is the number of vectors which are linearly independent among all covariant vectors of the form a;;,t,

.

, . t,

x xh 2

. .

3

. X tr

(14.42))

r

where x, x, ... , x are (r- 1) arbitrary contravariant vectors in space. 2

3

r

The same procedure is also applicable to skew-symmetric covariant tensors. The rank of the bi-vector v;1 is the rank of the matrix IIvt1Il or, what is the same thing, the number of vectors which are linearly independent among all covariant vectors of the form v;1x ,

(14.43)

where x is any contravariant vector of space. The rank of the tri-vector Ir;1k is equal to the number of vectors which are linearly independent among all covariant vectors of the form w;1k x1 yk

(14.44)

(where x, y are two arbitrary contravariant vectors); in other words, the rank of )v,jk is the rank of the matrix IlwrlkII of n rows and n(n-1)/2


14

157

columns,.a row being determined by the subscript i, a column by a pair

of distinct subscripts jk. Analogously, one may define the rank of a covariant r-vector for any r. It is readily shown how one can determine the rank of a symmetric or of a skew-symmetric contravariant tensor. In the case of non-symmetric and mixed tensors, one has to introduce the concept of the rank of a tensor with respect to one of its indices. This concept will be explained by means of a particular case: the rank of the tensor c, k with respect to the first superscript is the number of vectors which are linearly independent among all contravariant vectors of the form (14.45)

c;kx'uk,

where x is any contravariant and u any covariant vector of space. It is readily seen how this rank may be defined as the rank of some matrix. An exception is the case of a second order tensor for which, in accordance with a well known property of the rank of a matrix, the ranks with

respect to each index always coincide; therefore one can speak of the rank of the non-symmetric second order covariant tensor c, or of the rank

of the affinor A; without referring to a specific index. From these results follows Theorem 14.10:

The rank of a tensor (if necessary, with respect to

one of its indices) is always an arithmetic invariant. From this theorem follows the obvious, but important Theorem 14.11: The rank of any concomitant of a tensor a is an arithmetic invariant of the tensor a. Exercises

1. Prove that if the hypersurface (14.9) is a pair of hyperplanes, the rank of the 'quadratic form (14.6) is 2 or I. 2. Prove that if the rank of the quadratic form (14.6) is equal to 1, the hypersurface (14.9) is a pair of coincident hyperplanes. 3. Show that the determinant Ia"i, formed from the components of the contravariant, symmetric, second order tensor a" (the discriminant of the tensor a') is a

relative invariant of this tensor. Determine its weight. 4. If we introduce the notation a ti

_

(n-I)!

E

191 3

i"

a,:Jta4

(14.46)


158

the equality (14.13) can be rewritten in the form a'1u'u1 = 0;

CHAP. III

(14.47)

show that a'J is a symmetric, contravariant tensor and determine its weight. Further, prove that (14.48)

a'Ja,, = D611l.

where D = ja;1] is the discriminant of the tensor a11, and that the discriminant of the tensor all (cf. 3) is equal to D°-'. It follows from (14.48) that a'1 is the cofactor of the element all in the determinant (14.7). 5. If u is a polar hyperplane of the points x (i.e., the geometric locus of points conjugate to x), the point x is called the pole of the hyperplane u. Find the formula expressing the coordinates of the pole x with respect to the hypersurface (14.9) of the hyperplane u in terms of the coordinates of u. 6. Show that the invariant (14.12) can be represented in the form of the bordered determinant `

all a12 ... alw ul all ass ... as.. us (14.49)

ulus...

U. 0

whence there follows a corresponding way of writing the tangential equation (14.13) of the hypersurface (14.9).

7. Letting n = 3, write down the condition that the straight line u intersects the curves a;1x'xJ = 0 and blfx'xl = 0 in two pairs of points which divide each other harmonically.

8. Show that the invariant (14.16) can be written in the form of a sum of n determinants. 9. Show that for n = 3 the vanishing of the invariant (14.16) has the following pro-

jective geometric significance (under the assumption that the discriminant of the tensor a,1 is non-zero); the polar with respect to the curve a11x'x1 = 0 of atny point lying on the curve b1Jx'xJ = 0 intersects the two curves in two pairs of points which divide each other harmonically. 10. What happens to the invariant (14.19) when bit = a11?

11. Explain the projective geometric meaning of the vanishing of the invariant (14.19), assuming the rank of the tensor bl, to be equal to n. 12. If a binary form is a power of a linear form, its Hessian vanishes. Verify this statement. 13. Following the model of the invariant (14.36), one can construct an invariant for a binary form of any order r; show that for odd r this invariant is equal to zero and write down its expanded expression for even r. 14. Given three quadratic binary forms

f = aox'-'r2alxy+a:y2, 9, = bex'+2b1xy+b,y', , = coxz+2caxy+cly2, show that the determinant ao

at bl

as

co

Cl

C2

I

E-

bo

b2

is a joint relative invariant of weight 3 of the forms f, 97 and +p.

(14.50)

SIMPLEST PROPERTIES OF INVARIANTS

15.1

15. Express the invariants 1, =

1, = AsIaAPAYI

159

(14.51)

of the of nor A; in terms of the invariants S1, S,, S, [cf. (14.39)]. What happens to the invariant 1, for n = 3? 16. Show that for n = 2 one has for the affinor A' the relation o1AQAP1 = 0; represent it in the form

A,Aa--SA!+I,&. =0

(14.52)

(cf. (14.501.

§ 15.

The simplest properties of invariants

As has been stated above (cf. 13.1), a concomitant of a system of tensors can be reduced to an invariant of these tensors and several 15.1.

(contravariant and covariant) vectors. For this reason, we will study now the properties of invariants. In the definition of an invariant [cf. (13.1)], the function ip has been assumed to be arbitrary; however, a completely developed theory has only been constructed for algebraic invariants, i.e., for those which can be expressed by an algebraic function cp. Further, in this theory, it has been assumed that the weight of every tensor for which invariants are to be constructed is an integer. Therefore we will consider below only such invariants and without special mention will assume the weights of all tensors to be integers. The first task which we will pose will be to prove that algebraic invariants can be reduced to integral rational invariants, i.e., to integral rational functions of the components of tensors. In order to attain this goal, one must first consider a certain problem. In accordance with (13.1) (or, in the case of several tensors, with an ana-

logous equality), under linear transformations of space a relative invariant is multiplied by some power of the determinant d of the transformation; it is natural to attempt to generalize the concept of a relative invariant by demanding that the multiplier which arises after a linear transformation should be some function co of the coefficients of the transformation. It turns out that one does not arrive at an actual generalization in this manner, since w must necessarily be a power of A. We will prove this result first for the case of an integral rational function of the

components of the tensors. Theorem 15.1: Let the integral rational function cp of the components of the tensors a, b, ..., not identically equal to zero, have the following

160

CHAP. III


properties: its value after a linear transformation of space differs from its former value by a multiplier which is a function of the coefficients pa of the linear transformation only, and this property holds true for any values of the essential components of the tensors a, b, . . . and the coefficients p.'. Then this multiplier is an integral power of the determinant of the transfor-

mation, and hence p is a relative invariant of the tensors a, b, .. . Let the essential components of the tensors a, b, ... be a1, a2, ..., aN b] , b2, . . ., bM, ...; by the condition of the theorem

* ,a2i...,aN,b],b2,...,bM,...) w(Pa)cp(a1, a2, ... , aN; b] , b2, ... , bM; ...) (15.1)

., aN. b ] b2, ., bM, ., pl, ' pl2 . p", " where the components of the tensors with asterisks denote that they are the result of a linear transformation S with coefficients p'.1 and w(pa) is an abbreviation for w(pi , Pi , , pn) By the remarks at the beginning of 15.1, the weight of the tensor a is an integer; taking into consideration the transformation rule for tensors

for an y values a le a Zs

[cf. (9.9), (9.20), (9.24), (12.1), (12.2), etc. ], we see that ak (k =1, 2, .. ., N)

is a polynomial in p,,, qi (i, a = 1, 2, ..., n) and the a1, a2, ..., aN, possibly divided by some integral power of the determinant d of the transformation S. However, each of the q; , being the reduced minors of d = tp'l, themselves divided by d, is a polynomial in the coefficients pa. As a consequence, ak is equal to the quotient of an integral rational function of p, and the components of the tensor a divided by d', where I is an integer >0; the same reasoning applies also to the new components of the remaining tensors. If we now select in (15.1) the components a] , a2, ..., an, b1, b2, ..., bM, ... so that the function (p does not vanish and replace at , ..: , aN , b; , . . ., b * ,... by their expressions in terms of pa,

we see that w(Pa) =

wI(p,)

(15.2)

d s'

where w1(p') is a polynomial in the coefficients p' of the transformation S and s1 is a non-negative integer. We will now apply to (15.1) the linear transformation S', the inverse of the transformation S; the coefficients

of S' will be q;, and consequently (p(a], ...,aN; b1, ..., bM;...) =

aN; bi ,

... ,

b,*a; ...).

(15.3)

161


15.1

It follows from (15.3) and (15.1) that

(p(a,, ... , aN; bl

, ... , bM; ...) ` (qa) ' w(pa) ' p(a 1 ,

. .

, aN; b,

, ... ,

bM; .

.

.)

i.e.,

w(qa) ' w(pa) = 1.

(15.4)

Since the determinant lqaI is equal to A', we have, by (15.2), w(qa) = w1(ga) ' A" =

w Pa)

( 15.5)

where w2(pi) is another polynomial in pa and s2 is an integer or zero. From (15.4), (15.2) and (15.5) we obtain w1(P.) ' w2(Pa) = A*" +S2

(15.6)

The determinant d is known to be an irreducible polynomial of its elements pa **); therefore we conclude from (15.6) that w1(pa) = c ' ds-',

where c is a constant and s3 an integer >_ 0. Hence, by (15.2),

w(p,) = c ' d°' where g = s3-s1, i.e., g is an integer. Further, applying the identity transformation E to (15.1) we see that w(6i) = 1,

and hence c = 1. Thus, Theorem 15.1 has been proved.

Theorem 15.1 at times greatly facilitates the establishment of the invariance of some expression. Thus, in § 14, Exercise 14, it is readily shown that for linear transformations D is multiplied by some function of the coefficients p; and Theorem 15.1 relieves us in this case of the addi-

tional burden of proving that the function in question is a power of the determinant of the transformation. Note: It follows from the preceding study that the equality (15.6)

is true for all those values of pa for which d = IpQI # 0, whence it follows that it is an identity. In fact, if the polynomial

f(XI,x2,...,xN)=0 ) Cf. Bocher, M., Introduction to Higher Algebra, Macmillan, New York, 1907.

) Cf. Note at the end of 15.1 below.


1 62

CHAP. III

for all values of the variables x 1, X2, ... , xN for which another, not identically zero, polynomial (p(X I

, X2, ... , XN) : 0,

then the product of these polynomials will be zero for any values of x1i x2, ..., xN. However, if the product of two polynomials vanishes identically, one of them must vanish identically, and consequently 1(x1, x2 , ... , XN) = 0 identically.

15.2 We proceed now to the case when in (15.1) the function qp is a fraction whose numerator and denominator are polynomials in the components of the tensors a, b, . . .; for this purpose, for the sake of brevity,

we will write Qp(a, b, ...) instead of cp(a1 i a2, ..., aN; b1, b2, ..., bM ; ..), etc. Thus, let cp(a, b, ...) = ,(a, a

b', ...)

(15.7)

where 0 and X are integral rational functions of the components of the tensors a, b, ... with no common factors. It follows then from (15.1) that

Ip(a*, b*, ...)X(a, b, ...) = w(p.)q/(a, b, . . .)X(a*, b*, ...).

(15.8)

After substituting for a*, ..., a*, b*, ..., bk, ... in (15.8) their expressions in terms of a1, ..., aN, b1, ..., bM, ... this equality must become

an identity with respect to a1, ..., aN, bI, ..., bM, ... for any values P. for which A = lpaI 0. By virtue of (15.8), its left-hand side must be divisible by Ifi(a, b, . . .). However, > t and X have no common factors; consequently, 0(a*, b*, . . .), considered as a polynomial in a1, . . ., aN, b,, ..., bM, ..., must be divisible by It/(a, b, ...). Since the degrees of these polynomials are obviously the same, one has 0(a, b, ...) = coI(p')Ifi(a, b, ...), and [cf. (15.8)] X(a, h, ...) = w2(Pa)X(a, b,

...

By Theorem 15.1, one now has

wl(P') = d°i,

w2(Ps) = d°2,

(15.9)

15.1-15.3

163


where g 1 and 92 are integers and 0, x are relative invariants of the tensors

a, b, .. Theorem 15.2: Every rational (relative or absolute) invariant of one or several tensors is a quotient of rational invariants of those tensors.

Further, we have from (15.9) co( () =

w1(Pz) = 491 W2(P2)

i.e.,

Theorem 15.3:

The weight of a rational invariant is always an integer.

15.3 Finally, we will assume that in (15.1) the function qp is an algebraic function of the components of the tensors a, b, . . ., i.e., a function satisfying the equation

[q(a, b, ...)]k+I,(a, b, ...)[(P(a, b, +12(a, b.... )[cp(a, b,

.

.

...)]k-2+

... +1,(a, b.... ) = 0,

(15.10)

where II, I2i ..., Ik are rational functions of al , ..., aN; bl, ..., bm; ... Without reducing generality, we can assume that in (15.10) the exponent k is the smallest possible one; then the coefficients in (15.10) are determined uniquely (otherwise the difference of two such equations would give an equation for cp of a lower degree). By (15.10),

b*, ...)]k+ll(a*, b*, ...)[qp(a*, b*, ..

+ ... +Ik(a*, b*, ...) = 0, or, by (15.1),

[to(a, b, ...)]k+ Ii(a*, b*, ...) [rata, b, u'(pa)

+ ... + Ik(a*, b*, ...)

= 0. (15.11)

[w(pQ)]k

Since the coefficients in (15.10) are determined uniquely, it follows from

(15.10) and (15.11) that

II(a*, b*, ...) = c)(pa)1I(a, b, ...}, 12(a*, b*, ...) _ [w(Pa)]212(a, b, ... Ik(a*, b*, ...) _ [o4pa)]kIk(a, b,

...

(15.12)


164

CHAP. III

taking into consideration the results of 15.2, we find Theorem 15.4:

Every algebraic invariant of one or several tensors is

a solution of an algebraic equation whose coefficients are rational invariants of those tensors. At the same time, we have generalized Theorem 15.1 to the case when

cp is an arbitrary algebraic function of the components of the tensors

a,b,...

Let the first of the invariants It , I2 ,

... , Ik which is non-zero be 1.; then,

oy Theorem 15.3, [w(pa)]k = d°,

(15.13)

w(pa) = d°iti;

(15.14)

where g is an integer and Theorem 15.5: -ational number.

The weight of an algebraic invariant is always a

15.4 An integral rational invariant of the tensors a, b, . . . is said to be homogeneous, if it is a polynomial which is homogeneous with respect to each of the tensors a, b, ... We will prove now Theorem 15.6: Any integral rational invariant of the tensors a, b, .. . is a sum of homogeneous integral rational invariants of those same tensors. Since the transformation formulae for the components of any tensor are always linear and homogeneous in these components, every term of an invariant I of the tensors a, b, . . . of weight g becomes after a linear transformation a sum of terms of the same degree in the components of each of the tensors a, b, . . . We will take in the invariant I all the terms which have the same degree in the components of each of the tensors a, b.... and denote their sum by I,. Then, in the equality expressing the invariance of I, these terms on the left-hand side will be equal to the same terms on the right-hand side multiplied by d°, i.e., It is an invariant. Since

this reasoning may be applied to any group of terms of equal degree, Theorem 15.6 has been proved. 15.5

Another remarkable property of rational invariants is their

isobarism. In order to establish this property'), one has to introduce the concept of the weight of a term of an invariant. ') For the sake of simplicity, we restrict ourselves here to the case of absolute

invariants of tensors (cf. Exercise 1).

15.3-15.5


165

A term of an integral rational invariant of the tensors a, b, ... is a product of a numerical factor and certain components of these tensors; for each of these components, the super- and subscripts have definite values. The difference between the number of the subscripts with the value 1 and the number of superscripts with the same property is called the weight of a component of the absolute tensor with respect to the component x' of the contravariant vector x; in a similar manner, one can define the weight with respect to .x2, with respect to x3, etc. Thus, the

component a31? of the tensor a k' has weight - I with respect to x, weight + 1 with respect to x2, and weight 0 with respect to x3, x4, xs, .... x". The sum of the weights of all its factors is called the weight of a term of an integral rational invariant, so that, for example, the weight of a term with respect to x' is equal to the difference between the numbers of all its

subscripts and superscripts with value unity. Let an integral rational invariant cp of weight g of the absolute tensors

a, b, ... consist of the terms Q1, Q2,

...,

Q,:

(P = Q1+Q2+ ... +Q,;

(15.15)

then it is natural to assume that all similar terms have been summed. The weight of the term Q, (i = 1, 2, . . ., s) with respect to x' will be denoted by gi. We now apply the linear transformation SA with coefficients

Pi=A#0, pz=p3=...=pn=l,

pi

=0for i#a;

(15.16)

then

d=2, qi=

qi=0fori0a.

It is readily seen that after the linear transformation SA every component of the tensor will be multiplied by A`, where t is the weight of this component with respect to x'. Therefore the term Qi of the invariant is multiplied by AB', and the equality which expresses the invariance of (P with respect to SA assumes the form A9iQ1+A92Q2+

... +A"'Q3 = A°(QI+Q2+ ... +Q,);

(15.17)

the equality (15.17) must be an identity with respect to A and the compo-

nents of the tensors a, b.... Therefore

91 =92=...=9,=9,


166

CHAP. III

A similar reasoning can be applied to the weight with respect to x2, x3, etc. Thus, one arrives at

Theorem 15.7:

All terms of an integral rational invariant of absolute

tensors have the same weight with respect to x', where i is one of the numbers 1, 2, . . ., n, this weight being equal to the weight of the invariant.

The property described in Theorem 15.7 is called the isobarism of the invariant. It is not difficult to verify its truth for the invariants discussed in

§ 14: the weight of every term with respect to x' or with respect to x2,

etc., of the invariant (14.4) is - 1, of the invariants (14.7), (14.12), (14.16), (14.17), (14.18), (14.19) it is 2 and of the invariants (14.39) it is zero.

The property of isobarism assumes a particularly clear form for invariants and covariants of binary forms, if one uses the notation for their coefficients given by (13.5); then the index of the coefficient gives the number of times the index two appears in the tensor notation, i.e., the weight of this coefficients with respect to x2. Therefore in an invariant of binary forms with coefficients ao, a,, ..., a,; bo, b1, ..., b,; ...,

the following term of the invariant: caa a1

...

a;''b0 b;`

...

b", .. .

will have (with respect to x2 = y) the weight

+rA,+0-µo +I, µ1+2µ2+

... +sµ,+ ...

(15.18)

The isobaric property is expressed by the fact that the sum (15.18) is the same for all terms of the invariant. The weight of a term of a covar-

iant of the form (13.10), equal to CaZOaA1 V 1

.

aA, P 9 r

Y

will be (with respect to x2 = y)

0-.10+1-A1+2.12+

... +rA,-q,

(15.19)

and similar results apply for covariants of several forms. A single glance at the formulae (14.31), (14.34), (14.33), (14.37) is sufficient to verify that (with respect to x2) all terms of the covariant H have weight 2, of the covariant Q weight 3, of the invariants D and i weights 6 and 4, respectively. As regards the invariant D of § 14, Exercise 14, we see straight away that its weight is 3.

167


15.5

We note that from Theorem 15.7 there follows Theorem 15.8:

The weight of an integral rational invariant of covariant

absolute tensors is always equal to a positive number (provided only the invariant is not a constant), so that absolute invariants of such tensors are always fractions. From Theorem 1 5.7, we easily derive a relation linking the weight of an

invariant of several tensors to their co- and contravariances. Let there 1

2

k

be given k tensors a, a, ... , a with covariances rl, r2, ... , rk and contra-

variances sl,s2, ..., sk and an integral rational invariant cp of these tensors with weight g. Denote by d; (i = 1, 2, ..., k) the degree of any term Q of the invariant qp in the components of the tensor a and write down the n equalities expressing the fact that the weights of a term with respect to x1, with respect to x2, . . ., with respect to x" are all equal to g. Adding these equalities, we obtain on the left-hand side the difference between the number of all subscripts and the number of all superscripts in the term Q, i.e., I r; d; - I.s; d i , and on the right-hand side, the product ng, i.e., k

(r;-s;)d; = ng.

(15.20)

1=1

In the case of a single (absolute) tensor, the relation (15.20) assumes the form

(r-s)d = ng; for g #0, we conclude from this that the degree d is the same for all terms of an invariant, i.e., we have

Theorem 15.9: A non-absolute integral rational invariant of a single absolute tensor is necessarily homogeneous. In the case of a homogeneous invariant of one or several tensors, the degrees d, are the same for all terms; therefore the formula (15.20) leads to Theorem 15.10: Let 9 be a homogeneous integral rational invariant of 1

2

k

weight g of the absolute tensors a, a, ... , a; then the relation (15.20) i

applies where r; ands, are the co- and contravariances of the tensors a and d; is the degree of the invariant cp in the components of the tensor a (i = 1, 2,

...,k).

CHAP. Ili

INVARIANTS AND CONCOMITANTS Of TENSORS

168

It is readily verified that (15.20) is true for the examples of § 14. For the invariant (14.7), one has r = 2, s = 0, d = n, g = 2, and both sides of (15.20) are equal to 2n. For the invariant (14.33), we find r = 3, s = 0, cl = 4, g = 6, n = 2, and both sides of (15.20) are equal to 12. The covariant Q of a cubic binary form [cf. (14.34)] is an invariant of a symmetric covariant third order tensor and a contravariant vector,

where r1=3, s1=0, r2=0, s2=1, d1=d2=3, n=2, g=3; in this case, both sides of (14.20) are equal to 6.

The results above reduce the study of the structure of all algebraic invariants to that of the structure of homogeneous integral rational invariants, which will be undertaken in Chapter IV. For this purpose, we will assume there and in the following chapters that the weights of all tensors are integers, and unless stated otherwise, that all. invariants are 15.6

homogeneous integral rational invariants which are not equal to constants;

the same remark applies to the equations in the following exercises (except for Exercise 1, where the invariant may or may not be homogeneous). Exercises

1. The difference between the numbers of subscripts and superscripts which are equal to 1, added to the weight of a tensor is called the weight of a component of a relative tensor with respect to xI; as before, the weight of a term of the invariant is defined

as the sum of the weights of the factors (and similarly for the weights with respect to x2, x3, . . .). Prove Theorem 15.7 for the invariants of a system of relative as well as of

absolute tensors. What form is assumed by the relation (15.20) in this case? Does 7 heorem 15.9 retain its validity in this case?

2. Prove that an it-nary form cannot have non-zero covariants of the type (13.4) with negative weights. 3. Show thatall invariants, and likewise all covariants of the form (13.4) of positive

weight, of an n-nary form of order r, vanish in the case when the form in question is equal to a linear form raised to the r-th power (cf. § 17, Exercise 6). 4. Prove that in each invariant of an n-nary form the sum of the numerical coefficients of all its terms is always equal to zero. 5. Let K be a covariant of weight g of the n-nary form q' of the type (13.4) ; the coef lcients are polynomials in the coefficients bills.. f, of the form 97. Prove that for g> 0 in each of these polynomials the sum of numerical coefficients for all terms is equal to zero (cf. § 17, Exercise 6). 6. Show that a binary form of odd order cannot have an invariant of odd degree.

7. Given the (r+ 1) binary forms of order r

4't = aox'+ra,x'-Iy-r ... +a,f, 4'2 = box'+rb,x'-Iy-f- ... +brY', q), *I = tox'-rrllx' .'Y+ . . . +1,Y',

-

15.5-15.6

169


prove that the determinant

D= is a joint invariant of the forms q' 99a,

(15.21)

... q',+ what is its weight? 2

1

3

N

8. Let a be a tensor with N essential components and a, a, ..., a tensors of the 1

same type as a (i.e., with the same covariance r and contravariance s and with the same type of symmetry or skew-symmetry). Denote by D the determinant whose i-th row I

contains all N essential components of the tensor a and a column which is formed 1

2

N

from those components of the tensors a, a, ..., a for which all super- and subscripts 1

2

N

have the same values. Show that D is an invariant of the tensors a, a, ..., a, and find its weight.

CHAPTER IV THE FUNDAMENTAL THEOREM OF THE THEORY OF INVARIANTS AND ITS CONSEQUENCES

§ 16. Tensors with constant components As has been stated in 15.6, the immediate problem under consideration consists of the establishment of the structure of the homogeneous, integral, rational invariants of one or several tensors. In the examples of 16.1

§ 14, we have encountered invariants of this type; each of them was constructed from a given tensor with the aid of the operations presented in §§ 9, 12. In this context, exceptionally important roles were played by

the operations of total contraction and total alternation (Theorems 9.4, 12.1 and 12.6) as a result of which invariants were obtained. The fundamental theorem of invariant theory confirms that one can obtain by such means any homogeneous integral rational invariant, and thus solves

the problem posed above. In order to prove the fundamental theorem which will be presented in § 17, we must first of all find the general form of so-called tensors with constant components (all of which are integers, cf. Exercise 3).

16.2 A tensor is said to be a tensor with constant components, if the values of all its components remain unchanged after any linear transformation of space. Obviously, the tensor which is equal to zero has this property; the unit n-vectors (Theorems 12.2 and 12.3) are examples of non-zero tensors whose components are constant. As a third example, one can quote the unit affinor with components A;`= b,'; in fact, one has AA

= Pa Rx bi = pa q = [cf. (9.4)]

Sa .

On the basis of these three examples, one can construct any number of

non-zero tensors with constant components; for example, one has CiUx(Sy+ C2 i

3Xj, i

6JEi j tiz...>

CX,X2 ... X,. CVIY2 ... Y

16.1-16.3

TENSORS WITH CONSTANT COMPONENTS

171

etc. We will show that one can derive in this manner any non-zero tensor with constant components. First of all, the relationship between the covariance r, the contravariance s and the weight g of a non-zero tensor with constant components is readily established. After a linear transformation with the coefficients Pa = A5

,

A 0 0,

(16.1)

for which

d=A", qi=tb7, A

each of the components of a tensor is multiplied by .l'-S+"°; on the other hand, each must remain unaltered for any A. Consequently, if at least one of the components of the tensor is non-zero, one has

r-s+ng = 0.

(16.2)

Note that for n = 1 every linear transformation has the form (16.1), and therefore every tensor for which (16.2) is true will have constant components. Rejecting this trivial case, we will always assume in what follows that n > I (Exercise 4). 16.3 We begin with the study of absolute tensors with constant compo-

nents; for such tensors, the condition (16.2) assumes the form r = s, i.e., if the tensors are not equal to zero, their covariance and contravariance are always the same. We will take first the simplest cases of r = s = 1, 2, 3, assuming, as stated above, that n > 2; this permits us to elucidate the main difficulties which arise in the general case. The condition that a tensor Ci` with unit co- and contravariances has constant components can be written in the form C; = Pa qx Ci

(16.3)

Contracting (16.3) with respect to pz, we find Pi Ca = Pa Px q Ci = [cf. (9.7)] = pa bz Ci = Pa Ci

or, changing notation,

x

PACE = P7Ca

(16.4)

The left- and right-hand sides of (16.4) are linear forms in the n2 p', . ., p which are identically equal to each other [the fact that the pa satisfy the restriction (9.2) has no significance; cf. the variables p; ,

.

172

THE FUNDAMENTAL THEOREM OF INVARIANT THEORY

CHAP. IV

remark at the end of 15.11. In order to visualize more easily the coefficients of these forms, rewrite (16.4) pA Sa Ci = px S, Ca.

(16.5)

In these identities, the coefficients of any of the variables pA on the right- and left-hand sides must be equal, i.e.,

Sac; = S;Ca.

(16.6)

Now in the equalities (16.6) set the indices x and a equal to unity and introduce the notation C; = c; we find then

Ci = c6

(16.7)

.

Every absolute tensor with constant components and unit co- and contravariances differs from the unit affinor only by a numerical multiplier.

Now let r = s = 2; reasoning as before when deriving (16.4), we arrive at the conclusion that the condition for constancy of the compo-

nents of a tensor c; is x ,y CAp

PA µ

ij =

Pat

$Cxy

Pj a6,

or, what is the same thing, PA pu ba Sa C;1 = p pµ S; S; C

(16.8)

The equalities (16.8) are identities with respect to the n2 variables P1 Pi , . ., p., their right- and left-hand sides being quadratic forms in these variables. If the variables px and 4. are different, then (cf. 9.7, derivation of Theorem 9.3) the coefficient of the product pxpp in the forms on the left-hand side will be

bx c j +a$ ba c , however, if the variables pA and pµ are the same, the coefficient will be given by the same formula with the multiplier 1/2 in front of it. The coefficients of the forms on the right-hand side will be similar. Thus, the fact that (16.8) is an identity implies the relation SaS0Cj; +SPSQCi; =

6.16

jCae+b,SiCA;

(16.9)

setting here x = a = 1, y = /3 = 2 (which one is entitled to do since n ? 2) and introducing the notation 12

C12 = C1,

12

C21 = C2,

16.3


173

we obtain, finally,

C"= CIb;b;+c268`6j;

(16.10)

this is the general form of an absolute tensor with constant components of co- and contravariances 2.

If a tensor ck for which r = s = 3 has constant components, then [cf. (16.8)] PaPBPybabpbyC

jkV

= PAPNPybibjakC Br

(16.11)

we have on the left- and right-hand sides of (16.11) cubic forms in the n2 variables pi , P2 2, .,p:. The coefficient of the product p x pµ p; in any of the forms on the left-hand side in the case when all the variables p'A, pp PV are different will be a sum of six terms (cf. 9.7). In order to write this sum in a condensed form, we will number all permutations of the three elements a, fi, y in any definite order and denote by {afly}p the P-th of

these permutations. We will number the permutations of A, µ, v in a corresponding order and apply the same notation to them so that, if {afly} 3 = rya, then {A v} 3 = pv 1. Then the coefficient in question can

be written in the form 6 biabpbY)pCilk`)p

(16.12)

P=1

If two of the variables p.a, pµ, p*,' are equal and the third is different from the other two, one must multiply the expression (16.12) for the coefficient 'by the factor 1/2, while, if all variables are the same, one must multiply

by the factor 1/6. As a consequence, the identity (16.11) implies the relations 6

P=1

6

bta ba bY)PC

iik")p

= P=1

bizbi bk)PC(,r)p .

(16.13)

If n 3, one can proceed as above: Setting x = a = 1, y = fi = 2, 3 z = y = 3, C{123 )p = Cp, we find the following general form for absolute

tensors with constant components when r = s = 3: 6

C j k " = % CpbUAbJbk)°

(16.14)

P=1

For n = 2, this choice of the values of the indices x, y, z, or, P, y is now impossible and no other choice will give the same effect. Therefore we will proceed differently: we will regard (16.13) as a system of n6 = 64


174

CHAP. IV

equations in the six unknown tensors

Dkv=C;;k°)P,

P= 1,2,...,6;

(16.15)

P

each equation of the system is obtained by allotting the indices x, y, z, a, fi, y definite values. The tensors (16.15) have the same components, but differ from each other by their numbering; for example, if {Apv}1=Ayv and {a.ily}3=µv2., then D', i 1 = C1 i 1; however, for the tensor D, the component Di i i has 3

1

3

the same value. In this context, one must have the following important circumstance in mind: We must choose the constants cp(P = 1, 2,..., 6) in such a way that the equations (16.14) apply for any choice of the values of the indices A, p, v, i, j, k. Therefore it is impossible in these equations to fix the indices in any manner: For example, if one sets .? = µ = 1, v = 2, the factors cp are the same in pairs on the right-hand side of (16.14) and the search for all six cp is impossible. As a consequence, we must assume that the un-

knowns in the equations (16.13) are not the components of the tensors (16.15), but these tensors themselves, considering them to be a manifold of coordinates, numbered in a definite manner. Not more than 5 of the 64 equations of the system (16.13) have linearly independent left-hand sides, since in each of them the coefficients of the

unknowns are linked by the same linear relation

6'byb;] = 0

(16.16)

[alternation over three indices, not more than two of which are different, obviously gives zero, so that the relationship (16.16) is true for any values of the indices a, fi, y, x, y, z]. Therefore, for n = 2, the equations (16.13) cannot be solved for the tensors (16.15); one has still to add one more equation. We will select from the system (16.13) five equations with linearly independent left-hand sides, setting successively

1) x=a=y=/3=1,z=y=2, 2) x=a=z=y= 3) x=a= 1,y=/3=z=y=2, 4) x=a=z=/3= 1,y=y=2,

5)y=a=z=/i=1, x=y=2, which give

16.3-16.4

TENSORS WITH CONSTANT COMPONENTS 6

175

1

P, Cp 61 2 6 j k

C,jk + C1 Jk" _ P=1

6 2

C,jk" + C kA _

Cp S; _1

Ski P,

P=1 6

CA AV

3

Cpb m6V1 P,

(16.17)

P=1 6 4

C j,"+C,k =

Cp416j8k}P,

P=1 6 5

C fk + Ci jk =

Cp (S 16A v}P. P=1

Further, for n = 2, we have for any choice of the indices i, j, k, A, µ, v

C k"' = 0

(16.18)

which establishes a linear relationship between the tensors (16.15). After multiplying the second and the third of equations (16.17) by 2, the fourth and the fifth by -1, we add these and use also the first of equations (16.17) and the equality (16.18); letting 1

2

3

4

5

Cp = 6(cp+2Cp+2Cp-Cp - cp),

we again obtain (16.14). Thus, for every n the equality (16.14) gives the general form of the

absolute tensor with constant components for r = s = 3. For n z 3, the representation of the tensor Ck" in the form (16.14) is seen to be unique on the basis of this proof. For n = 2, such a representation is not unique: One can add to the right-hand side of (16.14) the expression cb!16 6:1,

where c is arbitrary; this expression is equal to zero and retains its form, while the coefficients change their values.

Before we proceed to the case r = s, we must prove Lemma 16.1: Let there be given the system of m linear homogeneous equations with n unknowns 16.4

ak,x'=0, i=1,2,...n, k = 1, 2,. .. M,

(16.19)


176

CHAP. IV

of rank *) (n-s), where s is a positive integer and all the coefficients aki are

real **). Further, let

x'=x`, a=1,2,...s, i=1,2,...n

(16.20)

a

be s linearly independent real solutions of (16.19). Then, by adding to the equations (16.19) the s equations n

a=1,2,...s,

y- XIX'=0,

i=1 a

(16.21)

one obtains a system (A) of (m + s) equations of rank n.

For the proof, it is sufficient to show that the system (A) has only a zero solution. Let x' = x' (i = 1, 2, . . ., n) be a solution of (A); since all the coefficients of this system are real, we may assume that all the xo are likewise real. The solution x' = xo satisfies all the equations (16.19),

and as a consequence it must be a linear combination of the solutions (16.20): 7

xo =

A. x1, i = 1, 2, a=1

... n.

(16.22)

a

Further, the xo also satisfy the equations (16.21), and hence also the equations n

s

A. E x'x' = 0,

i=1 a

a=1

which, by (16.22), may be rewritten in the form n Y-xox0.

i=1

Thus, = 0,

and, since all the xo are real,

xo=xo=...=xo=0. This proves Lemma 16.1. *) The rank of a system of linear equations will be called the rank of the matrix formed

from the coefficients of the unknowns in these equations. **) Cf. Exercise 1.


16.4-16.5

177

16.5 Now consider the case when the co- and contravariances of an absolute tensor with constant components are equal to some arbitrary

number r. Reasoning as in 16.3 for the cases r = 1, 2, 3, it may be shown that for the absolute tensor C, 12 ::: i; with constant components one has the relation 1

rl

P=1

r!

i;1P = E S(l'b j2... 8 r aa2 ... aa,)p Ci i1 2 ...

QY}p

P=1

(16.23)

[cf. (16.6), (16.9), (16.13)]. All r! permutations of the numbers 1, 2, ... r in (16.23) are assumed to be numbered in some definite order; {a1 a2 ... a,}p denotes that permutation of a1 a2 ... ar in which the subscripts form

the P-th permutation of 12 ... r, and a similar convention applies to {X1.1 A2

... A,)P.

If r < n, the further steps are very simple; we set in (16.23) x1 = al =1,

x2 = a2 = 2, ... x, = a, = r and introduce the notation C(12...r

_

P = 1, 2.... H;

12 ... r) p - CPt

then (16.23) assumes the form x,a2... Ir Cit 12..1, =

r! {1,

P=I

x2

Arlp

(16.24)

CPS 1t a12...a1r

The case r > n requires more involved arguments. We will regard (16.23) as a system of n2r equations in the r! unknown tensors D1,x2... Ar 1112 ... 1r

P

....Ir}p = l.. f02C!A,).2 ...ir

P = 1, 2,

... , r!;

(16.25)

each equation of the system corresponds to some definite choice of values of the indices x1, x2, ..., x,, a1, a2, ..., at, For r > n, the rank of this system without fail is smaller than r!, since there exist between the coefficients of the unknowns in each of the equations (16.23) linear relationships which are the same for all equations. An example of such a

relation is 8(a, aa2

... Van aan: >> san+ 2 ... 6kX = 0,

which applies for any values of the indices x1 x2, ..., x a1 , a2, a, ..., a,: We have under the alternation sign (n+ 1) indices so that some of them

will certainly be the same. Denote the rank of the system (16.23) by (r! - s), where s > 0.

178


CHAP. IV

Now consider the system of homogeneous equations r!

P=1 S{QI 6X2. ..Sa,)p2P = 0

(16.26)

in the r! unknowns A1, A21 ... , )pl.. All the equations of the systems (16.23) and (16.26) have the same coefficients for the unknowns and,

consequently, the rank of the system (16.26) is also equal to (r!-s); the coefficients of the equations (16.26) are obviously real (as they are all equal to either zero or unity). As a consequence of all this, the system (16.26) has s linearly independent solutions

4 =2

Q=1,2,...,s,

,

(16.27)

where all the numbers 1p are real. By Lemma 16.1, adding to the system (16.26) the s equations rl

(16.28)

AP AP = 0, P=1

we obtain a system of (n2r+s) equations in the AP the rank of which is r!

To each of the solutions (16.27) corresponds a relation of the form rl P=1

^P5(a15Qi...6a,)P = 0.

(16.29)

Let pX,X2._.Xr be an arbitrary tensor of covariance r; contracting it with respect to all indices on the left-hand side of (16.29), we obtain rI

) PPIs1492...Rr)p

P=1

=

0,

(16.30)

since, obviously, SXr SXl 5X2 Sl R2... Qr PXI X2 ... Xr

_

- PQIX2 ...Rr,

etc. The components of the tensor pXIX2 ... X, can be selected quite arbitrarily; as a consequence, (16.30) can only apply if the coefficients of each of the n' components p21a2 on its left-hand side are equal to zero. Hence it is clear that (16.30) remains in force if the indices of p are superscripts, as a consequence of which r!

E

2PC(ill(i2... r)P =

0,

a = 1, 2, ... s.

(16.31)

P=1

The equality (16.31) gives s independent linear relations between the

16.5


179

tensors (16.25). If we add (16.31) to the equations (16.23), we obtain a system (A) of n2'+s equations in the tensors (16.25); the matrix of the coefficients of the unknowns in the system (A) coincides with the matrix

of those in the system obtained by combining (16.26) and (16.28). Consequently, the rank of the system (A) is equal to r!, and one may select from it r! equations in such a way that the determinant formed by their system (B) is non-zero; the system (B) involves all the s equations (16.31) and (r!-s) equations of the system (16.23) obtained for certain definite values of the indices xI, x2 , ..., x at , a2 , ..., a,. The right-hand

sides of the last equations are linear combinations of the tensors biz` 8 22...5 }P. Solving the system (B) by use of Kramer's rule, we express the C;; 22::: i, in the form of lineat combinations of the right-hand sides of the system (B), i.e., again in the form (16.24). It is obvious that the right-hand side of (16.24) is always an absolute tensor with constant coefficients. Thus, one has Theorem 16.2: The formulae (16.24) for arbitrary values of the coefficients c ,,(P = 1, 2, ..., r!) determine all possible absolute tensors with constant components with co- and contravarianee r. If the covariance of an absolute tensor with constant components is not equal to its contravariance, it is necessarily equal to zero [cf. (16.2)]. The significance of the symbol {A1 A2 ... A,}P has been explained above in connection with the equality (16.23). Rearranging the multipliers in the terms on the right-hand side of (16.24), one can readily see that the

formula (16.24) can be reduced to the form CAx:... A,

rl

P=1

c, gxl{i,gx2 c P i2 ... WP

(16.32)

where cP are the same constants as the Cp, but numbered in a different order.

Note that for r > n the representation of an absolute tensor with constant components in the form (16.24) or, what is the same thing, in the form (16.32) is not unique, since one may add on the right-hand side of (16.32) any linear combination of the left-hand sides of (16.29); for r S n, this representation, on the basis of its derivation, is clearly unique.

The proof of Theorem 16.2 for the case r< n, given by Cramlet*) in *) Cramlet, A determination of all invariant tensors, Tohdku Mathem. Journal,

Vol. 28.

CHAP. IV


180

1927, has been reproduced here. For any r, Thomas *) gave a proof in 1926 which turned out to be wrong. In 1929, two proofs of Theorem 16.1 were given by Knebelman **), but both of them were unconvincing. In the first of them, Knebelman established only that every'component of a tensor Ci; x= . ; * 11 (for given values of At, i.2 , ..., 2,) may be represented as a linear combination of the corresponding components of the tensors Sri' S i ... 5;; however, this representation will be different for different combinations of A,, A2, ... , A,, so that it does follow from this that the tensor C,11,12 ::: can be represented in the form (16.24), where c,, c2 , ..., c,l are the same for all choices of the values of the indices A, , 22 , ... , A,. By the way, this can be seen from the fact that the representation of each component in the form given above is unique, whereas the representation

(16.24) for r > n is not unique. In the second proof, he performed an induction with respect to r for arbitrary n; in this context, an essential role is played by the assumption that certain of the coefficients cµ and eµ do not depend on n. However, in actual fact, they depend on n, and this proof looses its significance. The fact that both proofs due to Knebelman are untrue is revealed immediately by executing them for the simplest cases: first, for r = 3 and n = 2, second, for r = 2. The proof of Theorem

16.2 for the case r > n, presented above, was found by the Author. 16.6 Now it is not difficult to establish the general form of a relative tensor with constant components (with integral weight). First, we assume that the non-zero tensor with constant components C;,;2:::% has a positive weight g; then [cf. (16.2)]

s = r+ng.

(16.33)

By virtue of Theorem 12.2, the tensor CX,x2 ... xs

i1i2...i, Ejr+1 ...Jr+n...Eis-n+1 ... is

(16.34)

will be absolute and its components constant [in (16.34), the number of unit n-vectors by which the tensor C is multiplied is equal to the weight g; this follows from (16.33)]. By Theorem 16.2 [cf. (16.24)],

C...1,,F' ir+l... 1112

+n .

.

. Eis-n+ I...is - `Cp

a,

(16.35)

P=1

) Thomas, T. Y., Tensors whose components ..., Annals of Math., Vol. 27. ) Knebelman, Tensors with invariant components, Annals of Math., Vol. 30.

16.5-16.6


181

[where the notation is the same as in (16.23) with r replaced by s]. Contract both sides of (16.35) with g contravariant n-vectors with respect to the indices i,+ 1 , ...9i,+"9 , is- n+ 1 , ... , is; taking into consideration

(12.3) and introducing the notation CP -

c* P

(n1)B

we obtain S!

Cxtx2 ... x,

itiz...1, -

P=1

CP

tt{xtaxz t2... t,

Ei,.+ t

... Ir+n

...

t ... is

or, in equivalent form, Cxlxz

x, xr+ 1 ... Xn -X' -- z CPait{Xt ail ... hi,.E . X2

iti2...1,

.

.E

X.)

(16.36)

P=1

Theorem 16.3: T h e f o r m u l a (16.36), where the cp(P = 1 , 2, ..., s!) are arbitrary constants, gives the general form of a relative tensor with constant components of integral, positive weight g, covariance r and contravariance s; if the numbers r, s and g satisfy the relation (16.33); however, if this condition is not fulfilled, the relative tensor with constant components its equal to zero. In a similar manner, we can prove Theorem 16.4: The formula x, C-,.-.'::: 1 12I.

31

-

xt

P=1

x2

CP IS(it ai2...W. ai, Et,+1... 1,+n ... Ei.-n+a... t.)P

(16.37)

where cp (P = 1, 2, . . ., r!) are arbitrary constants gives the general form of a relative tensor with constant components of integral, negative weight g,

covariance r and contravariance s, if the numbers r, s and g satisfy the relation

r = s+nlgl;

(16.38)

however, if the relation (16.38) is not fuflled, a relative tensor with constant components of weight g < 0 is equal to zero.

The proof of Theorem 16.4 may be started from the representation of the absolute tensor with constant components in the form (16.32) and from Theorem 12.3. These proofs of Theorems 16.3 and 16.4 were given

by Knebelman in the work cited above (cf. 16.5). Theorems 16.2-16.4 solve completely the problem posed at the end of 16.1.


182

CHAP. IV

Exercises

1. Show that Lemma 16.1 remains true also in the case when the coefficients ak, and the solution (16.20) are complex numbers, provided the equations (16.21) are replaced by n

a= 1,2,...,s,

Ex'x'=0, i=1 a a

(16.39)

where x' denotes the conjugate of x'. a

a

2 For a tensor with constant components of weight g, those components are nonzero for which the difference between the number of superscripts equal to 1 and the number of such subscripts is equal to g, and similarly for the indices equal to 2, 3, etc. Prove this statement without falling back on to the formulae (16.24), (16.32), (16.36) and, (16.37).

3. Show that there do not exist non-zero tensors with constant components whose weight is not equal to an integer.

4. Verify the truth of Theorems 16.2-16.4 for the case n = 1. 5. Show that Theorems 16.2-16.4 are also true in the case when the components of the tensors and the coefficients pa of the linear transformations belong to any field with zero characteristics. 6. Let there be given in a two-dimensional vector space of the field Z, (consisting of c,,, = 0, two elements 0 and 1, where I + 1 = 0) a symmetric tensor c,,k, where c11, = c,,, = 1. Show that crrk is a tensor with constant components.

§ 17. Proof of the fundamental theorem

The results of § 16 ease the task of proving the fundamental

17.1

theorem. Let cp be a homogeneous integral rational invariant of weight g 1

2

4

of the system of tensors a, a, ... , a of degree di in the components of the i

tensor a (i = 1, 2, ... , q); in this context, it is assumed (cf. 13.1) that for each of the tensors the relations of symmetry and skew-symmetry among its components are given. We will assume hereafter that the essential components of the tensors are quite arbitrary. Each term of the invariant is equal to a product of the di essential components of the tensor i

a (i = 1, 2, ..., q), multiplied by a numerical coefficient. If we denote by Bi,i2 ...1,

XJX2...X,

the tensor obtained by multiplying d1+d2+ 1

(17.1)

... d,, tensors the first d, 2

of which are equal to the tensor a, the following d2 to the tensor a, etc., 4

the last d, to the tensor a, then it is clear that each term of the invariant v is the product of a numerical factor and one of the components of the

17.1

PROOF OF THE FUNDAMENTAL THEOREM

183

tensor (17.1); one can assume that for those components of this tensor which do not figure in the invariant the corresponding numerical multipliers are zero. Denoting these factors by V"12' : i;`, 11

(17.2)

.

we obtain for the invariant rp the form tP = Ki1',2 .:: i,'BXIX2 :: `X,

(17.3)

In this context, one must have in mind that the same product of the 1

q

2

essential components of the tensors a, a, . . ., a may occur several times in (17.3) for different combinations of the indices il, i21 ..., i x1, x2, ... , x,; for example, this may happen as the result of an interchange of two factors representing the components of the same tensor, or of the presence of some kind of symmetry or skew-symmetry in one of the tensors. In such a case, for all these terms, the coefficients must be assumed to be the same or, if necessary, to differ in sign only; if a term is encountered m times, the corresponding coefficient of (17.2) will, obviously, be

equal to the present coefficient of this term in the invariant multiplied by ± 1/m [cf. (9.17)]. It is not difficult to see that as a consequence every symmetry or skew-symmetry possessed by the tensor (17.1) influences the corresponding property of the coefficients of (17.2): For example, if the tensor B is skew-symmetric with respect to the indices x1, x2, the coefficients will also be skew-symmetric in x1, X2, etc. The representation of the invariant in the form (17.3) will be elucidated by two simple examples. 1. The discriminant

D=

alla22-(a12)2

(17.4)

of the binary quadratic form (5.22) is a relative invariant of weight 2 (cf. 5.5 or § 14, Exercise 4). Every term on the right-hand side of (17.4) is one of the components of the tensor Bjjpq = a1j aaq;

(17.5)

by virtue of the symmetry of the tensor aj j, this tensor will be symmetric in i,j and p, q; in addition, it does not alter on interchange of ::Ac pairs

of indices i, j and p, q Biipq = Bvtj


184

CHAP. IV

In order to find the coefficients of (17.2) in this case, we may reason as follows: the term a,, a22 of the invariant is obtained from (17.5) for two

combinations of the values of the indices i, j, p, q, i.e., for i = j = 1,

p = q = 2 and for i = j = 2,p=q= 1; therefore we let K'122 ` K2211 =

The term (a,2)2 is obtained four times: for i = p = 1, j = q = 2, for

j=p=1, i=q=2, for i=q=1, j=p=2 and for j=q= 1, i = p = 2; therefore we must write K1212 = K2112 = K1221 = K2121

_ -#;

the remaining 10 coefficients K'1Pq are zero. Now we can write (17.4) in

the form D = KIJP9aiJa..

(17.6)

The coefficients K'1" have all the symmetry properties of the tensor (17.5):

K'jm = Kj"",

K11P9 = KiJ9P,

KIJP9 = KP"IJ.

2. The quadratic ternary form = ¢1jXIXJ =

a,,(X1)2+a22(X2)2+a33(X3)2+2a12X1x2

+2a13 XIx3+2a23 X2X3

(17.7)

is a joint absolute invariant of, the symmetric covariant second order tensor a,1 and the contravariant vector x'. Each term of the last part of (17.7) is one of the components of the tensor BP4'j = aP4 x'xJ,

(17.8)

which is symmetric in both subscripts p, q and both superscripts i, j. The term a,1(x')2 of the invariant qp is obtained from (17.8) only once for

i = j = p = q = 1; therefore we must take

K;;=1; for a similar reason, one has

K22 = K33 = 1.

The term a, 2x1 x2 is obtained four times from (17.8): for p = 1,

q = 2, i = 1, j = 2, for p = 2, q = 1,i= 1, j = 2, for p = 1,q=2,

185


17.1-17.2

i=2,j=landforp=2,q=1,i=2,j=1, so that we have IZ

21

12

K12 = K21 = K12 =

-

K21 21 -- 42

In the same manner, we find 23

23

31

31 13 13 K13 = "31 = K13 - K31 = K23

32

32

K32 = K23 = K32 =

The remaining 66 coefficients KKP9 are zero and the invariant rp can be

written in the form

P=

(17.9)

Ku7ap9X'Xi.

The coefficients K9, like those of the tensor (17.8) are symmetric with respect to both superscripts and with respect to both subscripts. 17.2 We will now write down the condition that the expression (17.3)

is an invariant of weight g with respect to linear transformations S of space with the coefficients pa and determinant d : d/Kx,xs ... x.Bl,i2 ... !,

Kx,x2 ... x.Ba,a2 ... 2, a,a2...a,

l,i2.i, ..

x,x2...x.,

(17.10)

where the asterisk above B denotes the values of the components of the tensor after the linear transformation S, so that ,a2

Bx, l:.

-r = As

Bo X,

X2

d P2, px2

. .

x, a, a2 a, !,!2 ... i, . P,1, qt, q12 ... ill, Bx,x: .. X.

where go is the weight of the tensor B; as a consequence, (17.10) assumes

the form x, a, a2 a. ,i2...l, -/lia2...l, x, X2 ... PA,gt,gi2 d/oKa,az...a,Px,Pz2 ...Ql.gx,xz...x,

= K i2

2.

. i,.

X1X2 ... X2

(17.11)

Now introduce into consideration a tensor of covariance r, contravariance s and weight g' = g-go with components K;;,?::, - ;`'; after application of the linear transformation S-', its components become Vx,x2... X, = d/o-/ a, i,i2 . . i,

22

qt, q12

a. x,

X2

gi,PA, px2

x, K2,A2... A, P2, a,92 ...9,

(for the transformation S - ', the coefficients p and q interchange roles and

the determinant of S-' is J '). Thus, the formula (17.11) may be rewrit-

ten in the form (K ? .. i,. xs-K ',,

.

i;')BX;xz :. . s.

= 0.

(17.12)

The left-hand side of (17.12) is a polynomial in the essential components

186

THE FUNDAMENTAL THEOREM OF INVARIANT THEORY 1

2

CHAP. IV

q

of the tensors a, a, ... , a; since a linear transformation does not affect any of the relations of symmetry and skew-symmetry of the tensor K, the coefficients of this polynomial differ from Kx1x2...x,

1,12... it -

x,x2...x, ... I,

Ki112

only by integral non-zero multipliers (cf. 17.1). By (17.12), this polynomial vanishes for any values of the essential components of the tensors 1

2

q

a, a, ..., a and, consequently, all its coefficients are zero, i.e., x, il12 ... i,. x,x2

xix2 ... x,

= Ki1i2 ... 1,

(17.13)

As well as S and S-1 we include the entire group of linear transformations of n-th order space; therefore (17.13) denotes that Kids?;;i;` is a

tensor with constant components of weight g' = g-go. The weights 1

2

q

g1, g2, ..., qq of the tensors a, a, ..., a are integers (cf. 15.6); hence also

go = g1 dl+g2d2+ ... +ggdq

(17.14)

is an integer; g has the same property (Theorem 15.3), and consequently

so also has g' = g -go, and we can apply Theorems 16.2-16.4 to the tensor K. 2 q First, we will assume that all the tensors a, a, .. , a are absolute (which 1

is usually the case in practice); then go = 0, and the weight of the tensor K is g. We will distinguish three cases which correspond to the three Theorems 16.2 to 16.4. If the invariant qp is absolute, g = 0, s = r and the tensor K is given by (16.24); since x ..i... .I.

SiB...x... = B...i...,

one obtains from (17.3) tit

- P=1

CP B Ili2... 2 irr}P

(17.15)

Recalling that the tensor B has been constructed from the tensors 2 q a, a, ... , a with the aid of multiplications of tensors, we arrive at 1

Theorem 17.1:

Every absolute invariant *) of a system of absolute

) In accordance with the statements of 15.6, we understand here and later on by the word "invariant" a homogeneous integral rational invariant which is not equal to a constant.


17.2 2

1

187

q

tensors a, a, ... , a is a linear combination of terms each of which is obtained 1

2

q

from a, a, ..., a by means of the operations of multiplication of tensors and of their total contraction.

Taking into consideration that contraction with unit n-vectors is equivalent to a total alternation (cf. 12.3), we conclude from (17.3), (16.36) and (16.37) that for g > 0 Si

P=1

CPB( ii2...iI2...ni...[12...n])p

(17.16)

where s = r+ng, the number of alternation signs is g and the numbers cP differ from cp only by the order of numbering and factorial multipliers,

while for g < 0

y1 - cpB(17.17) r!

lili2. .i,,[12..."I...[12...n]1F

P=I

where r = s+nIgj and the number of alternation signs is Igj, i.e., one has Theorem 17.2: Every invariant of positive weight g of a system of 1

2

absolute tensors a, a,

q

..., a is a linear combination of terms each of which 2

1

has been constructed front a, a,

q

... , a by means of the following operations:

Multiplication of tensors and g total alternations with respect to the subscripts followed by total contraction. Theorem 17.3: Every invariant of negative weight g of a system of 1

q

2

absolute tensors a, a,

. .

., a is a linear combination of terms each of which 1

2

q

has been constructed from a, a, . . ., a by means of the following operations:

Multiplication of tensors and Jgj total alternations with respect to the superscripts followed by a total contraction. In the case of relative tensors, the formulae (17.15)-(17.17) are readily

seen to remain valid. The formula (17.15) will apply, if. the weight g' of the tensor K is zero, i.e., for g = go [cf. (17.14)], the formula (17.16), for g' > 0, i.e., for g > go and the formula (17.17) for g' < 0, i.e., for g < go; the number of alternation signs in the last two formulae will be

Ig-gol. Unifying all these results, we can state Theorem 17.4: Every invariant of a system of arbitrary tensors *) *) The weights of which are integers (cf. 15.6).

188


CHAP. IV

is a linear combination of terms each of which is obtained from these tensors with the aid of multiplication of tensors, total alternation and subsequent total contraction.

The fact that the operations of Theorem 17.4 will always lead to invariants, has been established above (Theorems 12.6 and its dual, 12.1, 12.4, 12.5 and 9.4). Theorem 17.4 is the fundamental theorem of the theory of invariants. The proof presented above was given by Cramlet *) in 1928; it retains its validity also in the case of any field with zero characteristic (§ 16, Exercise 5). The terms whose linear combinations are invariants are, by Theorem 17.4, obtained by contraction of (17.1) with tensors (17.2) of the forms xi6x2 z, hjh2...h. SSX, X2 Sr. k,k2...k i,

13

82

Ir

X,Eh,h

;, S;2 ... ()ir 2 ... h ... Ekik2 ... kn the components of all such tensors have only the values ± 1 or 0. Taking 6i11 169X2

into consideration that the coefficients of an invariant differ from the components of the tensor (17.2) only by integral multipliers, we arrive at Theorem 17.5. Every invariant is a linear combination of invariants all coefficients of which are integers. A consequence of Theorem 17.5 will be encountered in § 21. Exercises 1. Prove that Theorem 17.4 remains true for integral, rational invariants of systems of tensors whose weight is an arbitrary number. 2. Show that Theorems 15.7 and 15.10 are simple consequences of the fundamental theorem of the theory of invariants (cf. Theorem 17.4). 3. On the basis of the values of the components of the tensors K'"" and K," found in

Examples I and 2 of 17.1, represent these tensors in the forms (16.36) and (16.24), respectively.

4. Write the following invariants in the form (17.3), finding their coefficients (17.2)

in the manner in which this was done in Examples 1 and 2 of 17.1; then represent these tensors (17.2) in one of the forms (16.24), (16.36) and (16.37) and verify the correspondence between the properties of symmetry and skew-symmetry of the tensors (17.1) and (17.2):

a) the invariant A, of the affinor A' for any n; b) the invariant v12v34 v1*v42+v2'vE3 of the contravariant bi-vector v'° for n = 4 (§ 10, Example 4); c) the invariant a;,kx'x'xk of the symmetric covariant third order tensor a,,k and the contravariant vector x' for n = 3; d) the invariant (14.37) of a fourth order, binary form. *) Cramlet, The derivation of algebraic invariants ..., Bull. of Amer. Math. Soc., Vol. 34.

SYMBOLIC METHOD IN INVARIANT THEORY

17.2-18.2

189

5. For n = 2, the expression A',Ai-AA$ is an invariant of the affinor At,, and the

expression (c12)2+ (cE1)2-2c=IC22 an invariant of the non-symmetric covariant second

order tensor c,;. Prove this statement by the following approach: Represent these expressions in the form (17.3) and establish that the tensor (17.2) has constant components. In addition, verify the correspondence between the symmetry of the tensors (17.1) and (17.2) in these cases.

6. Prove that every absolute covariant of the form (13.4) of an n-nary form qp is equal to c q*, where c is a constant and k a natural number.

§ 18. 18.1

The symbolic method in invariant theory

By virtue of the fundamental Theorem, 17.4, every invariant

can be written in compact form, provided we have a convenient notation for contraction and alternation. The notation for alternation employing

square brackets at times encounters unsurmountable difficulties; its disadvantages have already been seen when writing down the simplest invariants in § 14. These difficulties grow with the use of unit n-vectors, when the writing down of invariants encounters extraordinary complications. In certain especially important practical cases this inconvenience may be overcome by the use of Aronhold's symbolic method. 18.2

Aronholds' method is most convenient for invariant forms

involving one contravariant vector (symmetric tensors) of the type which are at the centre of attention of the classical theory. The basic idea of the

method involves a special way of writing down the components of a symmetric tensor. The components a;,;2...;, of a covariant symmetric r-th order tensor can be written in the form ai, a12 .

.

. a;,,

(18.1)

where the ai are the components of an imaginary covariant vector a. The symbol a, by itself has no significance; only the product (18.1) of r such symbols has a meaning in that it denotes the component ai,;2 ... i, of the tensor. An analogous notation may be applied to contravariant tensors; we will always mark the imaginary contravariant vectors with tildes above the symbol. In order to distinguish the vectors encountered previously from the imaginary vectors, we will speak of them as real vectors; contravariant real vectors will likewise be provided with tildes above the symbols. However, we will agree, as has already been mentioned in 9.1, that the symbols x, y, z, t, will always denote real contravariant vectors.

190


CHAP. IV

Consider now several examples of invariants written down on the basis of the Aronhold method: 1. The n-nary cubic form (p = a.jk x'x3Yk

(18.2)

is an invariant of the symmetric tensor a,jk and the contravariant vectorx.

Introducing the imaginary vector a, we have (p = aiajakxixiXk = (a,x')3. Using the notation (14.1), viz.,

a; x' = (ax) = (xa),

(18.3)

we can write (18.2) in the form (p = (ax)3.

(18.4)

The expression (ax) is said to be a symbolic factor of the first kind. Analogous reasoning can be applied to the n-nary form f = a112 ...,, x ` x12

... x

(18.5)

for arbitrary r; by representing its corresponding symmetric tensor a,,,,, ...i. in the form (18.1) with the aid of the imaginary vector a, we find

f = (ax)'.

(18.6)

Further, noting that a(ax)

5(a1x'+a2x2+ ... +a"x")

ax,

az',

a,,

we obtain from (18.6)

f,=-15] -=(ax)r -I a;. r ax'

(18.7)

2. For n = 2, two symmetric second order tensors a,j and a,j have the joint invariant 6

= all x22+a22a11-2a12x12

(18.8)

(cf. 15.5). Letting

a;j=a;aj,

xij=aiaj,

where a and a are imaginary vectors, the invariant (18.8) may be written

191


18.2 in the form

6=

(a1a2-a2or 1)2.

(al)2(a2)2+(a2)2(a1)2-2a1a2a1a2 =

Next, introducing the earlier notation

[aa] = tall a21 =

a1

a2

a1

a2 1

(18.9)

we obtain, finally,

6 = [aa]'.

(18.10)

The quantity [ax], a symbolic factor of the second kind, is seen to be equal to the determinant formed by the components of the imaginary covariant vectors. 3. The two quadratic binary forms

f = a;j x'x' = (ax)2

(18.11)

(p = 0(;j x'x' = (ax)2

(18.12)

and

correspond to the symmetric tensor of Example 2 [cf. (18.6)]. We will write the Jacobian of the forms f and T using the Aronhold symbols. By (18.7), f; = (ax)a;, oi = (ax); , whence 9 =

A f2 (P1

(P2

(ax)a1 (ax)a1

(ax)a2 (ax)a2

= (ax)(ax)

a1

a1

a2 a2

i.e., [cf. (18.9)]

S = (f, (p) = [aa](ax)(ax).

(18.13)

In (18.13), we now change over from the symbolic to the real notation:

g = (a1a2-a2a1XaIx1+a2x2)(a1x'+a2x2)i the coefficient of (x1)2 is seen to be

(a,a2-a2a1)ata1 = (a1)2a1a2-a, a2(a1)2 = all a12-a12a11, and the coefficients of x'x2 and (x2)2 can be found in a similar manner.

Proceeding next to the classical notation (13.5) and letting x1 = x, x2 = y, we find, finally, $ - (a0al-alao)x2+(a0a2-a2ao)xy+(a10f2-a2a1)y2. (18.14)

CHAP. IV

THE FUNDAMENTAL. THEOREM OF INVARIANT THEORY

192

We will obtain the same result, if we compute the Jacobian 9 by starting

from its definition and employing differentiation. 4. In quite the same manner, we obtain the symbolic representation 2

1

2

1

it

n

of the Jacobian (f, f, ... , f) of the system of n-nary forms f, f, ... , f of orders r1, r2, .

.

., r, respectively. On introducing an imaginary vector

A

a (A = 1, 2, ..., n) for each of these forms, we obtain [cf. (18.6), (18.7)] A, i = 1, 2, ... , n.

fi = (ax)""--' ai,

f = (ax)'A,

From the definition of the Jacobian (cf. § 14, Example 8), one has 1

1

1

A 12 ... 2

1 2

(f,f,

2

it

. . .

2!

,f)= It

it

I

I

(ax)

'

2

_ (ax} r2

r-I a2 ... (ax)r1 a a I (ax) ' 1

1

1

I

2

2

2

2

n

n

n

n

2

a rt

a3 ...

(ax)e"-1 a,

(ax)" -'(Qx)'2- 1

(ax)'--1

a

it

2

1

I

a,(ax)'2- 1 a2...(ax}n2

n

it

fn

11f2

P-1

.

(ax)'"- 1,

or, applying the notation (14.5), 1

2

12

it

it

(f,f, ... f) = [aa ...

1

it

2

a](aX)r"-1(ax)'2-1

(18.15)

(ax)'^-1

1

2

it

where again the symbolic factor of the second kind [a a ... a] denotes the determinant formed from the components of then imaginary covariant vectors.

For the symbolic representation of the invariants of contravariant tensors, one requires, as well as these factors of the first and second kinds, also the symbolic factor of the third kind [d d ... d] which denotes the 12

n

determinant formed from the components of the n imaginary contravariant vectors a, a, . . ., a [cf. (14.4)]. 1

2

n

18.2-18.3


193

18.3 In Examples 1-4 of 18.2, all invariants were of the first degree in the components of the tensors. In the case of invariants of higher degree, Aronhold's notation becomes somewhat more complex. For example, let it be required to write down the symbolic representation of the product a;; akI for the symmetric tensor a,1. Introducing the imaginary vector a

by the equality

a,, =ala,, we obtain for the above product a;aj aka,,

which may denote a;kal, and as well as a;,akl. In order to remove this multi-valuedness, we introduce a second imaginary vector b for which

also a1j = b,bf. Thus, the imaginary vectors a and b are different, but the products a,aj and bib; are the same; two such vectors a and b are called parallel imaginary vectors or parallel symbols. We now obtain a representation which excludes all possibility of misunderstanding: ,7,j ak, = a,ofbkb,

or

aijak, = b;bj at a,;

both these representations have, of course, the same validity. In the case of invariants of third degree in the components of a tensor, one must introduce three parallel symbols, and so on. We will consider examples where this complicated representation can be employed. 5. The discriminant D [cf. (17.4)] of the binary quadratic form (18.11) is an invariant of degree 2 in the coefficients of the form. Let

a,j =a;al=b,b1,

(18.16)

where a, b are parallel imaginary vectors; by (18.6), we obtain for the same form the alternative representations

f = (ax)2 = (bx)2.

(18.17)

The discriminant (17.4) can be written in the form D = all a22-(a12)2 = (aI)2,(b2)2-a, a2 b1 b2 = a, b2(a1 b2-a2 b1);

interchanging the symbols a and b, we obtain D = (b1)2(a2)2-b1 b2 a1 a2 = b1 a2(b1 a2 - b2 a1).


194

CHAP. IV

Taking half the sum of these two expressions for D, we find

D = J[aj bz(a1 hz-az b1)-6! az(a1 b2-a2 b1)] _ J(a1 b2-azb I)2 i.e.,

1) = 1[ab]2.

(18.18)

Conversely, expanding the brackets in /(18.1/8), we have D = 3(a1 b2-a2 h1)2 =

%[(a1)2(b2)2+(a2)2(b1)2-2a1 a2 b1 b2]

_ j(a11a22-a22a11-2aI2a12) = a11a22-(a12)2.

(18.19)

6. The expression w = a;j ak, x'x'yky1

(18.20)

is an absolute invariant of the symmetric tensor a; j and the two contravariant tensors x, y of second degree in the components of the tensor a,j. Again introducing the notation (18.16), we obtain w = a; a j bk b, x'xjyky' = (ax)2(by)2.

(18.21)

In the same manner, we find for the invariant cut = a,,aklx`yjxky'

(18.22)

co, = (ax)(ayXbx)(by).

(18.23)

the expression

Without the introduction of parallel symbols we would have obtained for both invariants the representations (ax)2(ay)2 and could not have distinguished between them. The notations (18.21) and (18.23) are applicable for arbitrary n. 18.4

The examples of 18.2 and 18.3 are sufficient to explain the essence

of Aronhold's notation. However, in this case, as in any other case of the employment of symbolic methods, there arise doubts regarding the legality of the application of the rules of ordinary algebra to the Aronhold symbols. The best way for the removal of these doubts is to adopt the following point of view. We will regard Aronhold's notation as a record of those tensor operations with whose aid invariants can be constructed in accordance with Theorem 17.4. Let us be concerned with an invariant I of the tensors 1

2

q

a, a, ..., a which is of degree d, (i = 1, 2,

..., q) in the components of a

195


18.3-18.4

tensor a; then the tensor (17.1) is equal to the product of d1 +d2 + .. . 1

+ dy tensors, the first d1 of which are equal to the tensor a, the following 2

d2 to the tensor a, etc. In Aronhold's symbolic representation of the invariant 1, each of the symbols a, b, etc., (imaginary vectors) refers to one of the indices of a definite one of the tensors whose product is (17.1) (by virtue of symmetry, the indices of the same tensor carry equal weight

and one need not distinguish one from the other); parallel symbols correspond to factors representing the same tensors. Therefore the num-

ber of different imaginary vectors in the symbolic representation of Aronhold is equal to d1+d2+ ... +dq, i.e., to the number of tensors whose product gives the tensor (17.1). Every symbol figures as often as there are units in the variance of the corresponding tensor. The factors of the first kind indicate the contractions, the factors of the second and third kinds the total alternations with respect to the sub- and superscripts which must be performed on the tensor (17.1), in order to construct the invariant 1. Thus, one can regard Aronhold's symbolic method simply as a convenient way of writing down the contractions and alternations which are required for the construction of the invariant in question. In this way, the formula (18.18) denotes that the discriminant D of the quadratic form (18.11) is obtained from the product a1Jak1/2 by two total alternations each of which is to be executed with respect to one of the subscripts of the first factor and with respect to one of the subscripts of the second factor (the symbol a represents the first factor a11, the symbol b

the second). The formula (18.6) shows that the tensor a1j1z...;, is to be contracted with respect to all its indices with a contravariant vector x. 1

2

n

By (18.15), the Jacobian (f, f, ..., f) is obtained from the product of the 1

2

n

tensors a, a, ... , a which correspond to the forms, if one performs on them a total alternation involving one index of each tensor and contracts

with respect to the remaining indices with a contravariant vector x (by virtue of the symmetry of the tensors, the choice of the index of each tensor which is to be placed under the sign of total alternation is of no consequence); in fact, the definition of the Jacobian reduces to this result [cf. the first part of (14.28) and (14.25)]. The formula (18.7) is simply

another way of writing the formula (14.25). One can decipher the remaining symbolic expressions of 18.2 and 18.3 in a similar manner. From this point of view, one can readily establish the legality of the

196


CHAP. IV

transformations executed above. For example, consider the expansion of the right-hand side of (18.18) in Example 5. On the basis of the interpretation given above, one has

D = 4[ab]2 = 2a(,(,

a21,1.

Expanding the first bracket [ab ] = a, b2 - a2 b, , we obtain

D = jaI b2[ab]-+a2 bl[ab],

(18.24)

which from our point of view (with the symbol a corresponding to the first, the symbol b to the second factor of the product a,jak,) denotes D = a,[, af212)-a2 a1,121;

(18.25)

thus, the expansion of the first bracket is equivalent to the expansion of the first alternation. Expanding in (18.24) the remaining bracket [ab], we find D = +a, b2 - a, b2-ja, b2 , a2 b, -Ja2 b, . a, b2+Ja2 b, . a2 b, , i.e.,

D = Ja1,a22-+aI2a21-. a21a12+Ja22all thus the remaining alternation sign in (18.25) has been expanded. Hence,

application of the ordinary rules for the expansion of brackets to the symbolic polynomials corresponds completely to the rules for the expansion of alternations, and it is perfectly justified. The legality of the application of the power law in (18.18) can be established in the same manner. Next, consider the expression (18.4). Expanding the first bracket (ax) = a, xl + a2 x2 + .... +a. x", we obtain

9 = a,x1(ax)2+a2X2(aX)2+ ... +a"x"(aX)2, i.e., 4Q = a,jkxIXiXk+a2jkX2X.Xk+

...

+ajkx"xjxk;

we see that we have written out in full the contraction with respect to the

first index i, employing the Einstein convention with respect to the remaining indices. Expanding the second bracket (ax), we find T = (a1)2(x1)2(ax)+a, a2 xIx2(ax)+ ... +(a")2(x")2(ax), or

(p =

allk(x1)2xk+a12kxIX2Xk+

...

+a,,,,k(X")2Xk

18.4-18.5


197

thus two contractions, i.e., those with respect to i and j, have been written out in full. Expanding the last bracket, we arrive at the complete expression for all three contractions. We see that it is completely justified to consider the right-hand side of (18.4) as a third power of the symbolic linear form (ax) = aIx' +a2x2+ ... +a"x, and the right-hand side of (18.6) as an r-th power of the same form; this explains completely why in (9.17) the numerical factors of a;,;2.. are polynomial coefficients.

As has been shown in § 14, Example 8, differentiation of the form (14.21) with respect to x` involves subsequent release of each of the indices from the contraction and its replacement by i; the same holds true for the differentiation of the symbolic power (18.6), regarded as a product

of identical factors from which it follows that the results (14.25) and (18.7) coincide.

The reader will readily explain in the same manner all the remaining transformations in Examples 1-6, and as a consequence their legality will be fully justified.

18.5 We will now consider some more complicated examples of the application of Aronhold's symbolic method, all the time giving interpretations from the point of view of 18.4. 7. The invariant (14.36) of the fourth order, binary form tji [cf. (14.35)] is of second degree in the coefficients of the form; therefore, in order to write it symbolically, we have to introduce two parallel imaginary vectors, setting

i = (ax)4 = (bx)4 In (14.36), four total alternations have been performed on the product aifkl apq.s ,

and hence one finds

i = .[ab]4

(18.26)

[four total alternations require the presence of the multiplier (2!)4 in

front with the factor 8]. Expanding the right-hand side of (18.26), we obtain i = j(a1 b2-a2 b1)4 = J((al)4(b2)4-4(a1)3(b2)3a2 b1 +6(a1)2(b2)2(a2)2(b1)2 -4a1 b2(a2)3(b1)3+(a2)4(b1)4} = M(a1111a2222 - 4a1112a2221 + 6a1122a2211 - 4a1222a2111 + a2222a1111)

198


CHAP. IV

or, in the notation (13.5),

i=

aoa4-4a1a3+3(a2)2.

As has been explained in 18.4, this calculation is completely equivalent

to the expansion of all four alternations; therefore it is not surprising that the same result has been obtained as in § 14, Example 10, where the expansion of the alternation was executed directly [cf. (14.37)].

8. The discriminant D of the n-nary quadratic form f, defined by (14.6), is an invariant of degree n in the coefficients of the form; in order to write it down symbolically, we must introduce n parallel imaginary 1

2

vectors a, a, ... , a, so that we must take 2

1

n

f = (ax)2 = (ax)2 =

... - (ax)2.

(18.27)

As has been stated in § 14 (Example 4), in (14.7) which defines the discriminant 1), one may assume that all second indices with the symbol a are included under the second alternation sign; hence we arrive at 12

D=

n

n1-! [aa ... a]2

(18.28)

[two total alternations require the multiplier (n!)2 in front].

9. In the same manner, we can convince ourselves that the joint invariant (14.12) of the quadratic form (18.27) and a covariant vector u can be written in the Aronhold symbols in the form 12

[aa

n-I

... a

u]2;

(18.29)

as a consequence, the tangential equation (14.13) of the hypersurface (14.15) assumes the form 12

n-1

[aa...au]2=0.

(18.30)

10. The Hessian H of the n-nary form f of order r [cf. (14.30)] is a covariant of degree n; reasoning as in Example 8, we let 1

2

e

f = (ax)' = (ax)' = ... = (ax)'

(18.31)

and take into consideration (14.26), which may be given the form

f;j = (ax)r-2a;aj.

(18.32)

18.5


199

We then obtain for the Hessian the symbolic representation 12

H = 1- [aa n!

n

1

n

2

... a]2(ax)'-2(axr-2

(ax)'

(18.33)

11. Next, consider the concomitant of the binary cubic form qp introduced in § 14, Example 9. For the purpose of its symbolic representation, we immediately introduce the five parallel imaginary vectors (P = (aX)3 = (bx)3 = (cx)3 = (dx)3 = (ex)3.

(18.34)

By (18.33), the Hessian H of the form cp may be written in the form

H = J[ab]2(axxbx).

(18.35)

A second order tensor hi j of weight 2 corresponds to H; by (18.35), one has

h. xix' _ J[ab]2aibjxix', whence follows (cf. 9.7)

hij = - [ab]2a(; ba).

(18.36)

It is readily shown that the symbol of symmetrization with respect to i and j on the right-hand side of (18.36) can be dropped; in fact, transposing the parallel symbols a and b in the symbolic representation of the tensor +[a b]2ai b; we obtain (since [ba]

,

(18.37)

[ab])

J[ab]2ai b, = J[ba]2bi a; = . [ab]2a, bi;

in the non-symbolic representation, this operation is equivalent to a rearrangement of the factors and, consequently, one finds 2a[1[llil a2]2]J - 2a[2[21i1 al]I]i

= 2a[1[11;1 a2]2]i (in replacing the first part of the last equality by the second part, we rearrange the factors in each term of the expression obtained as it result of expansion of all the alternations; this, of course, does not change its value; the second part is equal to the third, because transposition of the indices 1, 2 changes the sign of the alternation). Thus, the tensor (18.37) is symmetric in the indices i, j, whence (18.36) gives

hi; = J[ab]2ai b j;

(18.38)

200


CHAP. IV

using parallel imaginary vectors, we can write (18.38) in the form

h;J = 1[cd]2cidJ.

(18.39)

The discriminant D of the form q is of the fourth degree in the coefficients of the form, and one has to employ for its symbolic representation four parallel imaginary vectors; since D is four times the discriminant of the tensor hi j, we have D = 8h[I(I h2121 = {cf.

(18.38), (18.39)}

= 2[ab]2[cd]2a,, c2, btl d2l

,

i.e.,

D = 4[ab]2[cd]2[ac][bd].

(18.40)

The covariant Q is the Jacobian of the form qp and twice its Hessian H; since, by (18.34) and (18.38),

q, = (cx)2ci,

2H; = 2hiix' = [ab]2a;(bx),

one finds Q = 4cp[I H2] = 2[ab]2ctl a2) (cx)2(bx),

Q = [ab]2[ca](cx)2(bx).

(18.41)

Conversely, the symbolic representation of an expression makes it possible to assess its invariance. For example, let us construct for the form (18.34) the expression w = [a b]2[ca][bd][cd](cxXdx).

(18.42)

If aiJk is the symmetric tensor corresponding to the form (18.34), then the right-hand side of (18.42) shows that the expression co can be obtained

from the product ai,j,k, a12J2k2

ai3J3k.,

ai4J1k4

(18.43)

by means of five total alternations: With respect to the first indices of the first two factors, with respect to the second indices of the same factors (where these two alternations are indicated by the symbol [ab]2), with

respect to the first index of the third factor and the third index of the first factor (which is the multiplier [ca]), with respect to the third index of the second factor and the first index of the fourth factor (which is the multiplier [bd]) and, finally, with respect to the second indices of the

18.5


201

third and the fourth factors (the factor [cd]); the result of these total alternations is then contracted with respect to the remaining indices with the contravariant vector x [this corresponds to the factors (ex) and (dx)]. By Theorems 12.1 and 12.4, the expression w is a relative invariant of weight 5 of the tensor a,1 and the vector x or, in the classical terminology, a covariant of the form (18.34) of weight 5. It is readily shown that the covariant co is identically equal to zero.

For this purpose, in (18.42), we interchange the symbols a and b (an operation which we can indicate symbolically by a -> b) and c and d (c + d). As a result, we find [ba]2[db][ac][dc](dx)(cx) = -w,

since [db] = - [bd], etc. On the other hand, interchange of parallel imaginary vectors must not change an expression; consequently, we must have

w= -w,

w=0.

Just as in the case of the proof above of the symmetry of the tensor (18.37), the last reasoning is readily interpreted from the point of view of 18.4. 12. We compute now the Hessian H of the cubic binary form cp = (px)2(gx),

(18.44)

where p, q are real covariant vectors. The form (18.44) can be written as (p = pi Pj qk

x`x'xk,

consequently (cf. 9.7), it corresponds to the symmetric tensor (18.45) aijk = P(IPjgk) = J(Pipjgk+pigjPk+giPjPk) For the convenience of the ensuing manipulations, we will rewrite (18.45) with other indices:

arst =

P, Psgc+pgspt+q,pp1).

(18.46)

By (18.35), in order to derive the Hessian, one must subject half the product of the tensors (18.45) and (18.46) to complete alternations with

respect to the indices i, r and j, s and contract the result with x with respect to the indices k, t. By virtue of the equalities [pp] = 0,

[qq] = 0

202

CHAP. IV


seven of the ten terms of the product will vanish after the alternations; the product of the second term of the last part of (18.45) and the third term of the right-hand side of (18.46) gives ils [Pq] [q P] (Px)2

the same result follows from the product of the third term of (18.45) and [pq], the second term of (18.46). Taking into consideration that [qp] we obtain, finally, (18.47)

H = -' [Pq]2(Px)2. 13.

We will establish for the fourth order binary form

q = (ax)4 = (bx)4 = (cx)4 = (dx)4

(18.48)

the symbolic representation of its Hessian H and the covariant Q = (vi, 2H). By (18.33),

H = 1[ab]2(ax)2(bx)2

(18.49)

which corresponds to the symmetric tensor h,;k, = j[ab]2a(;aJbkbl).

(18.50)

We must now represent (18.50) in a different form. For this purpose, we reason as follows: If we separate from the 24 terms, obtained as the result of Eymmetrization of the tensor p;;k, with respect to all its indices, first those terms in which the index i occurs in the first position, then those with the index i in the second position, etc., we obtain P(iik,) = 4{P+(jk,)+P(jIiI

r)+P(jkiiI:)+P(iki)i}.

(18.51)

Applying the rule (18.51) to the right-hand side of (18.50), we find that hrik, _

(L.

DJ tai a(i bk b,, + [ab]2b) c(i ak b,)}

(since the first and secondteTms, and also the third and fourth terms, turn out to be the same). Finally, making the interchange a4-+ b of the parallel symbols in the second term in the braces, we see that it is equal to the first term, so that hi;k, = -[ab]2a; a(; bk bj) ,

(18.52)

Hi = hIJklx'xkx' = Z[ab]2(axxbx)2a;.

(18.53)

whence

The equality (18.53) may be obtained more simply, if we differentiate

203


18.5

(18.49) with respect to xi (the law of such a differentiation having been explained in 18.4), and then in the second term of the result make the interchange a 4- b. Further, by (18.7), Wi = (cx)3ci,

and

Q = (0, 2H) = 4/it1 H21 = 2[ab]2(ax)(bx)2(cx)3ci1 a21, (18.54)

Q = [ab]2[ca](ax)(bx)2(cx)3.

We will now find the symbolic expression for the Hessian HN of the Hessian H of the form i/i. Writing (18.52) in the form hijkl = 6 [ab]2{al a; bk bl+2a; b; ask bp}

(18.55)

and contracting the result with respect to the indices k and I with the vector x, we obtain H;; = 6 [ab]2{al a;(bx)2 +2a; b;(axxbx)}

(18.56)

(a result which we can also obtain by differentiation of (18.53) with respect to .xj). The reader must not be embarrassed by the bracketing of the factor [ab]2/2: this factor by itself has no meaning, since for either imaginary vector a meaning can be attached only to the products of four components aia;akal or bib;bkbl; consequently, the right-hand sides of (18.55) and (18.56) can only be understood as abbreviations of what is obtained from them after expansion of the brackets in observance of the rules of ordinary algebra. Using a set of parallel imaginary vectors, we may also write (18.56)

in the form HPq =

6

[cd]2{c,cq(dx)2+2c,, dq(CXXdX)}.

(18.57)

By the definition of the Hessian, we must now multiply the right-hand sides of (18.56) and (18.57) together, perform on all four terms of this

product total alternations with respect to the indices i, p and j, q, and multiply the result by 1/2; this gives HI{ = 2HtI1I H2121 =

,12[ab]2[cd]2{[ac]2(bx)2(dx)2

+4[ac][ad](bx)2(cx)(dx)+4[ac][bd](ax)(bxxcxxdx)}

(18.58)

(the expressions obtained from the alternation of the product of the first

204


CHAP. IV

term of the right-hand side of (18.56) and the second term of the righthand side of (18.57) and the product of the second term of (18.56) and the first term of (18.57) go over into each other after the interchanges a t-' c, b 1, q > 1, r > 1. 19. For the binary fourth'order form w express the Jacobian (Q, yr) in terms of i, H and yr, where the notation is the same as in § 18, Examples 7 and 13. 20. If a binary form f has the binary form q' as a factor, then the Jacobian (f,97) is also divisible by qr conversely, if the Jacobian (f, q) of two binary forms f and 9, is divisible by p and the form q- and its Hessian have no common factors, the form f is divisible by q. Prove these statements. 21. Show for the form (20.43) that (9', 9,)(2) = -

P-

p+q

V, 0121 - (F .J

9

p+q

Or, 9)12) . j

pq

(PT4)(p+4-1)

(, 9))1sr . V'

(20.45)

withp> 1,q> 1,r> 1. 22. Let f be a quadratic, qj a cubic binary form. Prove that the relation 3(f, q.)121 . f-4Dp = 0,

(20.46)

where D is the discriminant of the form f, represents the necessary and sufficient condition for the form f to be divisible by the form 97. 23. Let f be a binary quadratic form with discriminant D and tp a binary fourth order form; prove that the form Vr is divisible by the form f, if and only if 2D(J, pp) --f (f 2, V) (3) = 0.

(20.47)

24. Let f be a quadratic binary with non-zero discriminant D and +p a binary fourth order form; prove that the necessary and sufficient condition for rp to be divisible by f is given by 8D2V,_8D(4',f)1Yf+(Y',f2)141 . f2 = 0.

(20.48)

25. Find the first and second polar of the Hessian H of the binary fourth order form (18.48).

26. For the covariant Q of the binary fourth order form W [cf. (18.54), (18.48)1 find the first polar 27. For the binary forms

f = apx'+ra,x'-')' i ... .f a,V' = (ax)', hox'-;-rb,x*-,i+ ... +b,j.` = (h.r)' F obtain their joint concomitant given by Boole's Theorem 20.1 by selecting the form F as qr, f as rp, and rind the symbolic expression for this concomitant.

21.1

235

COMPLETE SYSTEMS OF INVARIANTS

28. For the binary forms (20.36) compute their joint concomitant a

all

fly av'

a_)9(X,3)

29. Compute the concomitant ((

a

a

Q l - ay' aX

T(.Y, y),

where 97 is the cubic binary form (13.8) and Q its covariant (14.34)30. Show that the concomitant (20.32) is given by 5!

(s-r)!

(9' 011),

and explain on this basis the results of 27, 28, 29 (cf. also 3 and 5).

§ 21. Complete systems of invariants On the basis of Theorems 9.1, 12.1, 12.4-12.6 (or, in symbolic form, Theorem 18.1), we can construct for a given system S of tensors a, b.... an infinite manifold of its invariants. If none of these invariants vanish identically, their number will also be infinite; this would be clear even from the fact that integral rational functions (with numerical coefficients) of several invariants of the system will likewise (under known conditions) be invariants of the same system; in any case, such a property belongs to products of powers of invariants. In this connection, there arises the concept of an irreducible invariant 21.1

of a system S, i.e., of an invariant which is not an integral rational function of several other invariants of the system. The manifold of all irreducible invariants of a system S will be called its complete system of invariants.

In other words, the invariants I1, 12, ... of a system of tensors S form a complete system of invariants, if every invariant of the system S is an integral rational function of the invariants I1, 12, . . and if, in addition, none of the invariants 11, IZ , ... is an integral rational function of the remaining (or of several of the other) invariants. Without additional study, it is, of course, impossible to assert that for any system of tensors S the number of irreducible invariants may not turn also out to be infinite. Therefore, from a basic point of view, great importance attaches to -

Theorem 21.1 (Hilbert's Theorem): A complete system of invariants of any finite system of tensors consists of a finite number of invariants.

The proof of this theorem requires the preliminary establishment of two lemmas.

CHAP. IV


236

21.2

The first of these lemmas concerns the theory of tensors with 1

2

h

constant components. Let h + 1 such tensors C, C, C, ... , C of weights g, 91, g2 , ... , gh, respectively, be linked by the relations 2

1

S(,,,, ... ib C b. II... i,)PI

?(1112... idX Cid+ tt... ix}P' +

.. .

h

+ Ui11112

... i,n

Cim+l1

1,)PA

CI 92 ... t, ,

(21.1)

where S, T, ..., U are tensors with the indicated variances and weights

, 9-9k. The notation in (21.1) is the same as in (16.23): all the r! permutations of 1, 2, ... , r are assumed to have been renumbered in a certain order and {il i2 ... i,}P, denotes that permutation of i, i2 ... it whose subscripts form the P1-th permutation of 1, 2, . . ., r; a similar

g -91, g -92,

1

2

h

convention applies to the superscripts. Assuming the tensors C, C, ..., C to be real, we have now to prove that, without violating it, in every relation

of the form (21.1) one may replace the tensors S, T, ... , U by tensors with constant components. For this purpose, we will regard (21.1) as a system of equations in the

components of the tensors S, T, ... , U. We will first assume that this system has only one solution. By virtue of its tensor character, the system will retain its form after a linear transformation of space with coefficients 1

2

h

pa. Since C, C, C, ..., C are tensors with constant components, for this transformation one will observe changes in neither the coefficients of the unknowns nor in the right-hand sides of the equations (21.1), and consequently the single solution will also remain unaltered. Therefore, if we denote by asterisks above the symbols the components of the tensors S. T, ..., U after the linear transformation pa, we have *

SXIX2 ... X. = CXIX2 ... X. i112 ... is JIIi2 ... 1b ,

*

UXIX2 ... X1 _ UXIX2... XI. 1,i2 ... i., , 1112 ... 1,

(21.2)

since the p; are arbitrary; the relations (21.2) show that S, T, ... , U are tensors with constant components. The system (21.1) cannot be incompatible, since, by assumption, there exist tensors S, T, . . ., U for which (21.1) is valid; therefore we have still to consider the case when this system is indeterminate. Let the rank of the

system (21.1) be N-q [cf. the footnote on p. 176], where q > 0 and N = na+h+n`+d+ ... +ni+M is the number of unknowns in the system.

237


21.2

In each of the equations (21.1) set the right-hand side equal to zero and denote the resulting system by (21.1h); the system (21.1h) will have q linearly independent solutions XIX2 ... X. Si,i2... ib ,

. .

.,

. .

.,

XIx2 ... x1

U11i2 ... Im s 1

1

Sx,x2... xa 1112 ... ib

'

XIx2... X1 I1t2 ... im f

(21.3)

2

2

.................... x1x2 .

. X.

SIli2 ... ib

>

. .

xlx2 ... X1

., Vq ifi2... I,

q k

2

1

By virtue of the fact that the tensors C, C, ... , C are real, the coefficients of the unknowns in the system (21.1 h) are real, and hence also the solutions (21.3) can be assumed to consist of real numbers.

Now, combine the system (21.1) with the q additional equations XaS

L ^' tt2 ... Ib

1121

1b

a+

. .

.+L

U11I2?

..

1,,,JUti2

... in,! = 0,

A= 1,2,...,q,

(21.4)

where in the first term the summation extends over every one of the indices i1, i2, ..., ii, x1, x2, ..., xQ independently of each other from I to n, and similar summations occur in the remaining terms. For the sake of brevity, we will write (21.4) in the form

SS+...+YUU=0,

2= 1,2,...,q.

(21.5)

By Lemma 16.1, the rank of the system consisting of the equations (21.1) and (21.4) is equal to the number N of unknowns; therefore this system will have the single solution xIx2 ... xa Si1i2...ib , 0

.

... x1 . y Ux1x2 I,I2...im

(21.6)

0

so that

SS+ ... + y UU = 0, AO

x0

2 = 1, 2, ..., q.

(21.7)

In order to find the solution (21.6), we must select from the system (21.1) N-q independent equations, augment them by q equations (21.4) and solve the system of N equations thus obtained by Cramer's rule. As we have already done above, we will mark the components after a linear transformation with coefficients p' of the tensors under considera-

CHAP. IV


238

tion by asterisks above the symbols, and by tildes above the symbols after a linear transformation with the coefficients

p; = p; .

(21.8)

Then, by reasoning analogous to that given at the beginning of the proof, Sx1X2

11'2 .

.

. xa . ib

. x,

XI X2

(21.9)

01,12 ... 1-

!

>

0

0

will be the solution of the system (21.1) and j)XIx2. .. XI

X,12. .Xa 1,12 ... 1b )

., V 1112 ... Ins ! 1

1

X.

x,X2

51,12..

ib

,

>

UXIX2 . . Xt

(21.10)

1112.. .

2

2

X,X2... X°

S1112... 1b

) . .

X,X2... XI ., V i,12 ... im

are the q solutions of the system (21.1h). Each of the solutions (21.10) is a linear combination of the solutions (21.3), and therefore it follows from (21.7) that

Y_SS+...+YUU=0, 20

A0

A= 1,2,...,q.

(21.11)

We will now consider in greater detail the first of the sums on the lefthand side of (21. l 1); as usual denoting the reduced minors of the elements p" and in( the determinants Ip'l and Ip'l by qx and q;, we find

SS A 0

"' i 12 ... 1b A

_

a"' Iti2 ... 1b

a

0

P,,P22 ... pbbg2,9 2 ... q,aSa,' 22' ' ;°.Sii22..i;a A

= [cf. (21.8)] _

Sz;s :::

11b

0

Pa; P1'2 ... Psb qX; qX; .

. .

q).' S ;FIZZ :: i; a

A

0

SAIIA2 ... A°SA,2 ... Ab _ a122 ...2b

a122 ... 2b

Y SS. A0

0

A

The same transformation can also be employed for the remaining sums after which (21.11) assumes the form

ySS+... +1: UU =0, 10

A0

A = 1,2,...,q.

239


21.2-21.3

Thus, we have verified that (21.9) also satisfies all the equations (21.4). Since the system obtained by addition of the equations (21.4) to the equations (21.1) has only a single solution, it follows from this that

-

,X2. X. 12 ... ib 0

X,X2.

Xa

1112... 1b f

-

X,x2. .sl_

.,

111112

0

... In,

a1s2. Ar 612 ... 1m f 0

0

i.e., that S, T, ... U are tensors with constant components. In this way 0

0

0

we have proved 1

h

Theorem 21.2:

2

If h + I tensors with constant components C, C, C, ... ,

C, the last h of which are real, are linked by the relation (21.1), one may replace, without violation of this relation, the tensors S, T, . . ., U in it by certain tensors with constant components (some of which may be equal to zero). It is understood, of course, that in the case of such a replacement the variances and weights of the tensors, equal to the order of the indices of each of them, must remain unaltered (cf. Exercises 1-5). 21.3

The second proposition required below is

Theorem 21.3 (Basis Theorem of Hilbert): One can always separate ,from every infinite manifold E of forms in N variables x', x2, .. , x" a finite number of forms

fI , f2 ,

.

.

.,fk

(21.12)

such that any form f of the manifold E can be represented as

f = F, f, + F2 f 2 +

... + Fk fk ,

(21.13)

where F1, F2, ..., F1, are forms in the variables x', x2, ..., x" (which, generally speaking, do not belong to E).

The forms (21.12) constitute the base of the manifold E. For the case N = 1, Theorem 21.1 is obvious: in that case, the base consists of the single form f, , where f, is the form of the manifold E which has the lowest degree in x'. We will employ induction to prove the theorem for any value of N. Assuming Theorem 21.3 to be true for a form in N-1 variables, we

consider some manifold f of forms in N variables. One can always assume that one has in T a form f0 of degree r for which the coefficient of (x')' is non-zero. In fact, if there is no such form, we will select for f0


240

CHAP. IV

any form of £ fo(xt, X2, ..., xN) = ai,i2 ... i,X'1Xi2 ... Xtl;

it

, I2 , ..., 1'. = 1, 2,

...,

N,

and subject the variables to the linear transformation

X'=pa z',

i,or = 1,2,...,N;

then the new coefficient of WY, equal to ar,t:... t,PI P" I

... P1 = fo(Pl , 1

P2I

>

P ",),

will be non-zero for a suitable choice of p'R. Then, if the theorem is true for

the transformed forms, it is obviously also true for the original forms. Expanding the forms of the manifold E as polynomials of decreasing powers of the variable x', we divide them into fo and collect in the remain-

der the terms containing x' to the degree (r- 1). Then every form f of £ can be represented as

f=

Qfo+cp(xt),-1+g,

(21.14)

where I p is a form in the N-1 variables x2, x3, ..., xN and g is a form in x', x2, . 'x xN of degree < (r-2) in x'. By the relation (21.14), each form

f of I is related to a form (p; all the forms p constitute a manifold £' whose elements are found to be in a mutually single-valued relationship to the elements of the manifold Z. *) By the assumption of the induction, Theorem 21.3 is true for the manifold £'; consequently, there exist in £' the k forms cpl, (P2, ..., which are equai to each other, but correspond to different forms j are considered to be different elements of the manifold E'.

21.3-21.4


241

By (21.14)-(21.16), one has

f = Qf0+P1f1+02f2+ ... +Okfk+9', where

Q' = Q-Q101-Q24'2- ... -Qk0k 9' = 9-91 'P1-9z'P2-

.

-9k'k,

and g' is again a form in x1, x2, ..., xN of degree < (r-2) in x'; separating in it the terms containing

(xI), - 2, we obtain

g' _ (x')'-2i/i+h.

The forms 0 depend only on x2, x3......; they form together a manifold E" whose elements correspond in a mutually single-valued man-

ner to the elements f of E. By the assumption of the induction, the manifold E" has as its base the forms i 1, 2 , ..., 0k'; denoting by fk+ 1, fk+2, ,fk+k the forms of v corresponding to the forms 01, 02, ..., lk', and applying a transformation analogous to that introduced above, we find

f=

+4kfk+T1fk+1+ ... +Tk'fk+k'+h,

where his a form in x', x2, ..., x" of degree < (r-3) in x'. Proceeding further in the same manner, we discover after (r-1) steps that for every form belonging to E

f = Fofo+F1f1+F2f2+ ... +Fmfm+w, where fo, fl,

..., f. are forms of I and w is a form depending only on

..., xN. Applying the assumption of the induction to the manifold of all forms w and making for the last time a transformation analogous to those above, we verify that Theorem 21.3 is also true for the forms in N variables. x2, x3,

X1.4 We now turn to the proof of Theorem 21.1. By the assumptions of Theorem 17.5, we can for a given finite system S of tensors a, b, . . . *)

restrict consideration to those of its invariants whose coefficients are integers. One can apply Theorem 21.3 to the manifold I of all such invariants, as a consequence of which one may separate from this manifold *) For the sake of simplicity, it has been assumed that all tensors of the system S are absolute; it is readily seen that this restriction is not essential.


242

CHAP. IV

k invariants I 1 1I2 , ... 'k with the following properties: for any invariant

I of the tensors a, b, ..., one has the equality

I = F111+F212+ ... +Fklk,

(21.17)

where F, , F2, ... , F. are integral, rational, homogeneous functions of the

essential components of the tensors a, b, ... We will assume that the manifold E has no base with less than k invariants; then the invariants Ix, I2, ..., Ik will be irreducible and none of them can be an integral rational function of the others. As has been shown in § 17,

1=C12.:.I,.

x',x2...xS,

where C is a tensor with constant components of the same weight g as the invariant I and B is a tensor which is itself a product of p+q+ .. . tensors the first p of which are equal to the tensor a, the following q to b, etc. In exactly the same manner, one forms I

I

I, - C 1.

2

Xf'

I

where C is a tensor with real (rational) constant components whose I

weight g1 is equal to that of the invariant 11, and B is the product of pl tensors a, q1 tensors b, etc.; the invariants 12 , 13, ... , Ik can be formed in a similar manner. The function F1 may be represented in the form I

I

xaj(, i2... F, = Cx*x2... 1112 ...1h x1x2 ... X. I 1

(21.18)

I

where a+f = s, h+b = r, the tensor L is the product of p-p1 tensors a, q-q1 tensors b, etc. (cf. 17.1), so that the product of the tensors 1

I

B and L is equal to the tensor B. We will apply an analogous representation to the functions F2, F31 . . ., Fk: 2

F2 -

2

11i2 ... Id cLtx,x2 ..' xc,

etc.

For a linear transformation of space, the components of the tensors a, b, ... change in accordance with well known rules; the components of 1

2

1

2

the tensors B, B, B, ..., L, L, ... will vary in a corresponding manner. If in (21.17) we express the old components of all tensors in terms of

21.4

243


I

their new ones and divide byyAY, we verify that the coefficients VI" $112

Ib

in (21.18) change as the components of a tensor of contravariance a, 2

as the compocovariance b and weight g - g 1, the coefficients S ;Z nents of a tensor of contravariance c, covariance d and weight g -g2, etc. 1

2

Therefore, we will assume that in (21.18) and analogous equations S, S,..

are tensors whose variances are indicated by their representations and whose weights are g - 91 g - 92, ...; then (21.17) will be preserved after any linear transformation of space. 1

As has been stated in §17, all relations of symmetry and skewsymmetry which are exhibited by the tensor B must involve corresponding

properties for the tensor C, where the symmetry of B with respect to superscripts corresponds to symmetry of C with respect to subscripts, etc. We must assume that similar conditions are also fulfilled by each pair of 1

2

2

1

1

2

1

2

tensors C, B; C, B; .. ;S,L,S,L;... Next, consider the equality (21.17). It involves on its left- and righthand sides polynomials in the essential components of the tensors a, b, .. . which are identical to each other; comparing their coefficients and taking

into consideration the remark just made, we obtain t

XIX2... XS C11i2

1

1

1

(XIX2... zq z°+1... XS) QM

... r - - L S(i,i2 ... ib

Cìt, f

t µ=1

+

1 ...

['

1

1r}Pp

2

2

S(iXl1t2

p=t 4-1 2

1

... td

(

I

x2...XcCxe+1 ...XS)QM+

ldr ,

.

i,jPM

...

(21.19)

2

In each of the tensors B, B, B, ... L, L, .. , the indices of any of the factors which are equal to one of the tensors of the system S occupy definite locations; therefore, by virtue of their definitions, every index 1

2

1

2

position in the tensors C, C, C, ... , S, S, ... , is linked to one of these factors. In the first sum on the right-hand side of (21.19), the permutations Pµ and Qµ must be chosen in such a manner that the indices corresponding to the factors equal to the tensor a or to the factors equal to the tensor b,

etc. will occupy the corresponding places. Generally speaking, this can be done in various ways (since the systems of indices corresponding to

factors which are equal between themselves can be interchanged); if the number of such selections is equal to t, all of them must figure in the


244

CHAP. IV

first sum. This type of explanation must also be applied to the other sums. I

2

If we consider S, S, ... to be tensors, then, as has been noted above, the identity (21.17) will not be violated after any linear transformation

of space; therefore the relation (21.19) has the same property. 1

By

2

Lemma 21.2, in (21.19) we can replace the tensors S, S, ... by tensors with constant components in such a manner that it will remain true. If we contract both sides of the equality obtained after this replacement with the tensor B, we find

I = KI1,+K212+ ... +Kklk,

(21.20)

where K1, K2, ..., Kk are already invariants of the system of tensors

a, b, ... (some of which may be zero). The same reasoning can also be applied to the invariants K,, K2 , ... , Kk; consequently, one has

... +Liklk, i = 1, 2, .. , k, (21.21) ..., k) are again invariants of the system

Ki = Li11,+Li2I2+

where the Lip (i, j = 1, 2, a, b, ... It follows from (21.20) and (21.21) that k

1 = I Lijlilj 1,j=1

Proceeding further in this manner, we show that I is an integral rational

function of Il , I2 , ... , Ik (with numerical coefficients). Thus, the invariants 4, 12, ..., Ik transform as a complete system of invariants of the system of tensors a, b, ... , and Theorem 21.1 has been proved. 21.5

In what follows, we present a study of several tensors of a given

type and of systems of tensors each of which are of a given type; the type of a tensor is determined by its covariance, contravariance and its conditions of symmetry and skew-symmetry. The first fundamental problem which arises here quite naturally involves the search for complete systems of invariants (for a given value of n); in this way we arrive at an exhaustive representation of all the invariants of the system of tensors under consid-

eration. The number of tensors of the systems which we will consider will always be finite; consequently, by Theorem 21.1, the complete system of invariants will also by finite. For greater clarity, the second fundamental problem will first be defined

21.4-21.5

245


for the case of a single tensor of a given type. Two tensors which may be reduced to each other by a linear transformation of space will be said to be equivalent; all tensors which are equivalent to each other will be combined into one type *). The second fundamental problem consists of the establishment of a complete classification for a tensor of a given type in a space of a given number of dimensions, i.e., of the search for all possible

representations of this tensor. In the case of a system of tensors, this problem can be defined in a similar manner. Generally speaking, the solution of both these problems runs into very serious difficulties, as a consequence of which complete answers have only been found for several comparatively simple cases. The most important of these will be studied in Chapters V-VII; it should be noted here that in what follows we will not always state that all tensors of a system-are assumed to be absolute. Exercises 1. What are the modifications which must be introduced into the proof of Theorem t

2

ti

21.2, so that one may drop there the requirement that the tensors C, C, ..., C should be real? 2. Prove that if every one of the tensors S, T, . . ., U in (21.1) satisfies certain conditions of symmetry or skew-symmetry, they may be replaced by tensors with constant components which satisfy the same conditions., 3. For n = 2, find all tensors S and T which satisfy the equation Situ ear-i-Ts'e,r = 2w , eti ear, w 0, (21.22) and then separate from them those which will also satisfy an equation of the form (21.4).

4. Find tensors S, T, U, V satisfying the relation

S,6,+T;6;+Ut6i+Vi6;=2w6;6',

c o:;60,

assuming n = 2. Having found all solutions, separate from them those which also satisfy an equation of the form (21.4). 5. Find tensors S and T satisfying the equation

S bk T;rI

2wb 6J 6k, to-AO,

(21.23)

for any n; then separate from them the solution which also fulfills an equation of the form (21.4). 6. Pursuing the method employed in the proof of Theorem (21.3) find the base of the system of forms j,. = xm+ yTM (m = 1, 2, 3, ...) in the two variables x and y. *) In certain cases, the concept of the type of a tensor can be defined in a somewhat

different manner; we will say more about this aspect later on.

CHAPTER V BINARY FORMS

§ 22 Linear and quadratic binary forms In the theory of algebraic invariants, special attention has always been given to binary forms. The reason for this is that binary forms, or, 22.1

from the point of view of tensor algebra, symmetric covariant tensors with n = 2, are the simplest objects to be investigated; an exceptional role is also played here by the close link which exists between binary forms and algebraic equations with a single unknown. In the classical theory of binary forms, a wide range of results have been derived; we will present the principal results of this theory in this chapter. Following the classical

model, we will give geometric interpretations, of these results (unless specially stated) only in one-dimensional projective geometry. The binary field (space of order n :6 2) has the following important feature: in this space, there exists no difference between covariant and contravariant vectors, since the formulae (12.6) and (12.7) relate uniquely

and invertibly every contravariant vector x to a covariant vector u. This is also related to the fact that both forms of vectors have the same geometric interpretation: points of a straight line (cf. 5.1, 5.2) correspond to both of them. In this connection, the vectors are assumed to be given exactly apart from a numerical factor, which makes it possible to neglect the difference between the weights of the vectors x and u in the formulae (12.6) and (12.7). For the same reason, the relation between a contravariant vector p' and a covariant vector p; which correspond to the same point on a straight line can be written as

P2 = -PP1 Pi = PP2. [cf. (12.6)] or, in invariant and more concise form, pi p` = 0.

(22.1)

This feature of the binary field justifies a study of forms which only

22.1-22.2

LINEAR AND QUADRATIC: BINARY FORMS

247

involve contravariant vectors or, in other words, first-order covariant tensors. Below, we will solve for binary forms both the fundamental problems

formulated in 21.5: given a single binary form of definite order (or a system of forms), 1. find a complete system of concomitants, 2. establish a complete classification. In the study of the first of these problems, we will only take concomitants of two types into consideration: invariant forms and their covariants with a single series of variables (i.e., invariant systems consisting of a single contravariant vector and the tensors corresponding to the forms

under study). As a basis for such a treatment of the problem, one has, first, the inessential character of the difference between the vectors of the two types, secondly, the fact, as will be shown in 24.4, that a covariant of several vectors can be reduced to a covariant of a single vector. As a rule, we will employ the classical notation (13.10) for binary forms; however, side by side with it, we will at times also resort to tensor notation. In particular, the components of a contravariant vector will be denoted

by x, y and by x', x2. The concomitants of forms (and also the forms themselves) will be written in non-symbolic form as well as in Aronhold's symbolic notation. In this section, we will study the simplest binary forms, i.e., linear and quadratic forms. 22.2 The case of systems of linear binary forms is easily dealt with. Here classification is trivial: two systems of linearly independent forms are equivalent, if they are comprised of the same number of forms. A complete system of concomitants is established directly by virtue of the fundamental theorem of invariant theory (Theorem 18.2): in the symbolic representation of invariants and covariants all vectors will be real in the case of linear forms, and consequently the symbolic factors become real factors. Hence we arrive at the following results. The single linear binary form (ax) has no invariants (since [aa] = 0); every covariant of such a form is a product of a numerical factor and some power of (ax). Thus, one has,

Theorem 22.1: A complete system of concomitants of the single linear binary form (ax) consists of one single concomitant, namely the form (ax) itself.

CHAP. V

BINARY FORMS

248

Every concomitant of a system h of linear binary forms (ax), (bx), ..., (kx), (lx) is, in accordance with Theorem 18.2, a linear combination of products of determinants of the form [ab] and several factors of the form (ax), etc. Theorem 22.2: A complete system of joint concomitants of the h linear binary forms (ax), (bx), . . ., (kx), (lx) consists of the forms themselves and of h(h - 1)/2 invariants which are equal to the determinants [ab], [ac], . . ., [kl]. The irreducibility of these concomitants follows from the fact that each of them is of the first degree in the coefficients of the forms participating in it. The identities (19.6) and (19.7) and other similar ones give integral and rational relations between the concomitants (ax), (bx), ...,

[ab], [ac], ...; however, not a single one of the concomitants can be expressed through these relations in the form of an integral rational function of the remaining ones. Relations of this type between the concomitants of a complete system are said to be syzygies. 22.3

Next, we will consider the theory of the single quadratic binary

form

f = aijxix' =

a11(x1)2+2a12x'x2+a22(x2)2,

aid = a11. (22.2)

First of all, we will give its geometric significance; by virtue of the homogeneity of the coordinates x1, x2 of the point x, such a meaning can only be attached to the case when the form is'equal to zero: ai,xixj = 0, (22.3) ais(x')2+2a12x'x2+a22(x2)2 = 0. As has been stated in 5.5, equation (22.3) determines two points on a straight line: if all 0 0, then x' also vanishes when x2 = 0; therefore we may divide both sides of (22.3) by (x2)2 and obtain two values for the ratio x'/x2 (which may be found to be equal). If a22 # 0, one again

obtains two values for the ratio of x1 to x2. When all = a22 = 0, a12 # 0, equation (22.3) gives the first and second coordinate points:

x1 = 1, x2 = 0, and x' = 0, x2 = 1. There remains only the case all = a12 = a22 = 0; but then the form vanishes identically and it is, of course, impossible to give it a geometric interpretation. The form (22.2) is closely linked to its polar bilinear form

Dx,f = aijx'yi = a11x'y'+a12(x'y2+x2y')+a22x2y2,

(22.4)

which is a joint absolute invariant of the tensor a1; corresponding to the

22.2-22.3

LINEAR AND QUADRATIC BINARY FORMS

249

form (22.2) and the two contravariant vectors x and y. The equality

aijx'y' . 0

(22.5)

relates every point x of a straight line to a point y of the same straight line. It follows from the symmetry of the tensor aii that this correspondence is invertible: if the point x is related to a point y, the point y corresponds to the point x. By (22.1), the equality (22.5) may be rewritten in the form

y; = a;tx',

(22.6)

where y; and yt are covariant and contravariant vectors corresponding to the same point y of a straight line (cf. Exercise 2). The representation (22.6) shows that the stated correspondence is projective (if the determinant Ial jI # 0). Thus, the equation (22.5) or, what is the same thing, the equation (22.6) determines (for air # 0) an involution on the straight line (cf. § 5, Exercises 3 and 9). Comparing (22.5) and (22.3), we see that the two points for which the form (22.2) vanishes will be double points of the involution. One readily arrives at (§ 5, answer to Exercise 8)

Theorem 22.3: In an involution, those and only those points which divide the double points of the involution harmonically correspond to each other.

The involution which is defined by the binary form (22.2) gives its second geometric interpretation (or, more exactly an interpretation of the tensor al j of the coefficients of the form). We will now solve the two fundamental problems stated in.22.1 for the form (22.2). We have frequently encountered above an invariant of the

form f, its discriminant

D = a1Ia22-ail

(22.7)

(the geometric meaning of which has been discussed in 5.5); in addition, the form f is a covariant of itself. Obviously, we cannot reduce the concomitant f. By Theorem 18.2, the form cannot have invariants of the first degree in the coefficients of the form, since such an invariant can only contain factors [aaj which are equal to zero; therefore the invariant D is also irreducible. We will prove

Theorem 22.4: A complete system of concomitants of the binary quadratic form f consists of the form itself and its discriminant D. For this purpose, we write the form f in Aronhold's symbolic notation

f = (ax)2 = (bx)2 = (cx)2 = (dx)2 = ...,

(22.8)

BINARY FORMS

250

CHAP. V

where a, b, c, d.... are parallel symbols. By the fundamental Theorem 18.2, we may limit ourselves to the consideration of concomitants which are equal to products of symbolic factors of the first kind of the form (ax) and symbolic factors of the second kind of the form [ab] (there cannot occur factors of-the third kind, since we have only a single contravariant tensor, the vector x`, and [xx] = 0); under these conditions every imaginary vector must be encountered twice and only twice in the products

above. If the symbolic expression for such a covariant K does not contain factors of the second kind, every factor of the first kind (ax) must occur twice, and the covariant K must be a power of the form f. However, if K contains the factor [ab], the imaginary vectors a and b must occur in it once more in one of the remaining factors. In this context, one may deal with only the following four cases:

1°. The imaginary vectors a, b occur the second time in one of the factors of the second kind. Then

is likewise a covariant of the form f (in particular, an invariant; invariants may be considered as covariants of zero order). 2°. The symbols a, b occur the second time in two factors of the second kind. Then K1

K = [ab][ac][bd] ... = (a -* b, arithmetic mean)

_ j[ab]{[ac][bd]-[bc][ad]} ... = {by (19.7)} _ +[ab]2[cd]

...

= D - K1 .

3°. One of the imaginary vectors a, b occurs in a factor of the second kind, the other in a factor of the first kind. We reason then as in the second case, except that instead of the identity (19.7) we must apply the identity (19.6).

4°. Both symbols a, b enter into factors of the first kind:

K = [ab](ax)(bx) ...; then interchange of parallel symbols a +-+ b changes the sign of the covariant K, and consequently K = 0. The same reasoning may be applied to the covariant K1; continuing in the same manner, we get rid of all factors of the second kind and in this

251


22.3-22.4

way arrive at the case considered above when K is a product of only single factors of the first kind. Thus, we see that every covariant of the form f is an integral rational function of the form f itself and of its discriminant D, and Theorem 22.3 has been proved.

22.4 A classification of binary quadratic forms is readily brought about by reduction to the canonical form; for this purpose, the use of geometric terminology is convenient. Assuming the form not to vanish identically, we select as first coordinate point any point x of the straight line whose coordinates do not make the form vanish; as second coordinate point, we take the pointy which is related to x by (22.5) (where, by virtue

of the condition a;;x'xi # 0, the points x and y differ). Then all # 0, a12 = 0 and the form (22.2) reduces to ;,(XI )2

2 # 0.

+ µ(i2)2,

(22.9)

If also µ is not equal to zero, by letting

x' = \14x',

12 = x/p x2,

A -' _

Ap # 0,

(22.10)

we bring the form into the canonical form (x' )2+(x2)2 (where the asterisks

have bean omitted for the sake of simplicity). However, if y = 0, we obtain after the transformation v = %!!;. x' ,

2

= Y2

A-'=,i#0

(22.11)

(again omitting asterisks): f = (x' )2. Obviously, forms which reduce to the same canonical form are equivalent to each other, and therefore belong to the same type. Also taking into consideration forms which vanish identically, we conclude that binary quadratic forms occur in three types corresponding to the canonical forms I

f = (x')2+(x2)2,

11

f = (x')2,

III

f = 0.

(22.12)

These three types differ from each other by the value of an arithmetic invariant of the form, i.e., its rank (cf. § 14, Example 4): for Type 1: p = 2, for Type II: p = 1, for Type III: p = 0. We will say that the value of the invariant p describes the arithmetic character of the type. One may also have for the three types of quadratic binary forms algebraic characteristics defined with the aid of the concomitants f and D: for Type I: D # 0, for Type II: D = 0, f # 0, for Type 111: f = 0 (where for such characteristics the vanishing of the covariant is understood in the sense that it vanishes for any values of x', x2).

252

CHAP. V

BINARY FORMS

These results reduce to the following table: Canonical form

Type

III

D:;6 0

p=2

(x1)2-(x')'

I

D=0,f:e-0 f=0

P = I

(x')2

II

Algebraic characteristics

Arithmetic characteristic

p=0

The form vanishes identically

The classification derived above refers to the complex domain: the coefficients of the forms and the coefficients pa of the linear transformation

are assumed to be complex. In the real domain, i.e., for forms with real coefficients and for real linear transformations, the classification is more complicated. A reduction of the form (22.9) remains possible in the real domain also. Further, it is obvious that, in order not to resort to linear transformations with complex coefficients, we must replace the transformations (22.10) and (22.11), respectively, by

x' = JjAjxt,

x2

= JIUIx2

and

xt = J1A1xI,

x2 = x2

whence we obtain instead of three canonical forms the six forms 1a f = (xt)2+(x2)2,

Il' f = (x1)2,

Ib f = -(xt)2-'(x2)2,

IIb f = -(xt)2,

1` f = (xt)2-(x2)2,

111 f = 0

(22.13)

(for Ap < 0, one can assume without loss of generality that A > 0, p < 0). Thus, in the real domain, quadratic binary forms occur in six types. Their algebraic characteristics are:

1' D > 0, f 0, 11' D = 0, j?0,

Ib D>0, f!50, I1b D = 0, f60,

I` D < 0, III

f = 0.

(22.14)

The fact that a form belongs to one of the types (22.13) is reflected in

the properties of the correspondence (22.6) for the form in question: in the cases I' and lb, the double points of the correspondence are imaginary and we have an elliptic involution, for the case I`, the double points are real and the involution is hyperbolic. For the types II' and IIb, equation

253


22.4-22.5

(22.6) assumes the form

Y,=±x',

Y2=0, i.e., it refers every point of the straight line (other than the second coordinate point) to the same point (namely, to the second coordinate

point), and thus it does not determine a projective transformation. Such a degenerate correspondence is often called a parabolic involution. 22.5 We will now dwell on the theory of pairs of quadratic binary forms

f = a;tx'x' = (ax)2 = (bx)2 = ..., cp = a;, x'x' _ (ax)2 = (fix)2 = ...

(22.15)

These forms are related to a pencil of quadratic binary forms (22.16)

Af+ U4p,

where A and it are arbitrary numbers; the forms f and qp form the base of the pencil. Obviously, any two forms of the pencil

f' = A1f+µ,w,

4,' = A2f+u2gv

(22.17)

can be selected as its base, provided only that the determinant

b = Atµ2-A2NI e 0.

(22.18)

We have considered in § 18, Examples 2 and 3 two joint concomitants of the forms f and cp: the invariant

D52 = (f,

rV)(2)

_ [ax]2

(22.19)

and the Jacobian

9 = (f, (p) = [aa](axxax);

(22.20)

in addition, the discriminants D, and D2 of the forms f and rp must likewise be regarded as joint invariants. If we combine these concomitants with the forms f and rp themselves, we obtain their complete system of concomitants; this result may be verified in the following manner. By Theorem 18.2, it is sufficient to consider products of symbolic factors of the form [ab], [x/3], [ax], (ax), (ax). If such a product K contains

the factors rab] or [afi], then, as has been established for the proof of Theorem 22.4,

K = D, K, or K = D2 K2 ,

BINARY FORMS

254

CHAP. V

where K1 and K2 are joint covariants of the forms f and (p. Further, let K contain the factor [ax]; then the symbols a and of must occur again in K. If they enter again into the same factor [ax], the factor D12 appears; if they figure in the two factors of the first kind (ax) and (ax), a multiplier

9 splits off from K. The following possibilities still remain:

K = [ax][afi][ab] ..., K = [ax][afl](ax) ..., K = [ax](ax)[ab] ..., where, applying the identities

[al][ab] _ [aa][f1b]+[ab][af], [a/3](o(x) _ [a2](flx)+[a/3](ax),

(ax)[ab] _ -[aa](bx)+[ab](ax) [cf. (19.7) and (19.6)], we divide K into two terms, the first of which has the multiplier D12, while the second becomes one of the cases already

considered. Continuing this factorisation further, we will in the end eliminate all factors of the second kind and arrive at covariants which are products of the factors (ax) and (ax) of the first kind only and are readily seen to be equal to products of powers off and (p. Thus, K is an integral rational function in D1, D2, D12, 9,f and cp, and we have proved

Theorem 22.5: The complete system of joint concomitants of two quadratic binary forms f and cp consists of the forms themselves, their discriminants D1 and D2, the invariant Dt2 = (f, (p)(2 ) and the Jacobian 9 of the forms.

The irreducibility of all six concomitants of the complete system is beyond doubt: for the first four, it has been established in 22.3; while the last two are of first degree in the coefficients of each of the forms. These concomitants are interrelated by {the syzygy 92 = -D2 f 2+D12fco-D1(p2 [cf. (19.21)]. The geometric meaning of the invariant equality D12 = 0 or, equivalent-

ly, of (p)`2'

(22.21) =0 (f, has been established in Theorem 5.6. If the discriminant of the form f

22.5


255

vanishes, it is readily shown (§ 5, Exercise 13), that, if the condition (22.21) is fulfilled, at least one of the two points for whose coordinates the form (p vanishes, coincides with a point determined by the form f Theorem 5.6 remains valid in this case, if we generalize the concept of harmonic separation in a way which follows from the result of § 5, Exercise 6: two points x, y harmonically divide two coincident points z, z, if one of the points x, y coincides with z. Note also that with such an interpretation of harmonic division Theorem 22.3 remains true for parabolic involutions (cf. end of 22.4). We will now explain a property of those points of a straight line whose coordinates satisfy 9 = 0.

(22.22)

It is easily seen that

(9, f)(2) = 0.

(9, (p)(2) = 0;

(22.23)

in fact, each of these transvectants is an invariant of weight 3 of theforms f and (p. But each non-zero invariant off and (p has even weight, since, by Theorem 22.5, it is an integral rational function of the invariants DI, D2, D12 all of which are of weight 2. Now the relations (22.23) are readily verified directly:

(9j)(') = [act][ab][ab] = (a -+ b, arithmetic mean) = 0, and similar results hold for (9, (p)(2) *). By Theorem 5.6, the relations (22.23) show that points determined by (22.22) divide the pair of points f = 0 and the pair of points (p = 0 harmonically. Hence we obtain by virtue of Theorem 22.3. Theorem 22.6: The double points of the involutions determined by the binary quadratic forms f and (p correspond to each other in the involution which is determined by their Jacobian 9. If the form 9 is not the square of a linear form, the involution deter-

mined by it is not parabolic and gives an invertible and single-valued correspondence; consequently, in this case the involutions f and (p cannot have common double points. However, if the form 9 is a perfect square,

the corresponding involution is parabolic and the two pairs of points determined by the equalities f = 0 and (p = 0 have a common point, *) It is readily seen that the relations (22.23) determine the form +9 apart from numerical factor, provided the forms l and ry are linearly independent.

it

256

BINARY FORMS

CHAP. V

namely that whose coordinates satisfy 9 = 0. Thus, we have Theorem 22.7: The Jacobian of two binary quadratic forms f and rp is itself a perfect square if and only if the forms f and q have the same linear

form as a factor (the square of this being 9). We have assumed so far that the Jacobian does not vanish identically; from the expanded expression for 9 [cf. (18.14)] there follows readily Theorem 22.8: The Jacobian of two binary quadratic forms f and cp vanishes identically if and only if the forms f and p are linearly dependent.

As has been shown in § 18, Example 5, the discriminant De of the Jacobian 9 of the forms f and qp is expressed in terms of their joint invar-

iants D1, D2, D12 by (19.19); the invariant R = -4D8 is called the resultant of the forms f and (p: R = D12-4D1D2.

(22.24)

On the basis of Theorems (22.7) and (22.8), we have Theorem 22.9: The vanishing of the resultant of two binary quadratic forms is necessary and sufficient for these forms to have a common linear multiplier (cf. § 19, Exercise 9). 22.6 It is more convenient to carry out the classification not of pairs of quadratic binary forms, but of the pencil of forms connected with them;

the results derived are then simpler and more distinct. We will dwell first on certain concepts related to pencils of binary quadratic forms. The number p is said to be the rank of the pencil (22.16), if the pencil contains a binary quadratic form of rank p, while the ranks of all the other forms in the pencil do not exceed p; obviously, the rank of a pencil is an

arithmetic invariant. In order to find the rank of a pencil, one has only to construct the discriminant D(A, p) of.the form Af+ prp. By (20.15), 2D(2, p) = (Af+pq', k{+1Up)(2) = A2(f,f)(2l+2Ap(f, (p)(2

+p2(q'r(p)(2),

i.e.,

D(Z, p) = D,)2+D12tip+D2µ2.

(22.25)

The form (22.25) in the variables A, p is said to be the characteristic form of the pencil. If the characteristic form does not vanish identically, the rank of the pencil is p = 2; if D(A, p) is equal to zero for any values of A and p, but some of the forms of the pencil do not vanish identically,


22.5-22.6

257

one has p = 1. Finally, if all forms of the pencil are identically equal to zero, then p = 0. By (22.24), the resultant R of the forms f and (p differs only by the multiplier -4 from the discriminant of the characteristic form D(A, A). If we change the base of the pencil in accordance with (22.17), we expose the variables ,1, it in (22.25) to a linear transformation with determinant d (cf. (22.18)]. Hence it is clear that the resultant of the forms f and rp' is

R' = b2 R.

(22.26)

The Jacobian 9 has similar properties. On the basis of the known properties of transvectants [cf. (20.14)-(20.16)], the Jacobian of the forms f and (p' is 9' = (A1f+p1(PIA2.f+p2(P) = (21p2-A2p1XJ,(P),

i.e., (22.27)

The covariants of the forms f and V which, like R and 9, are multiplied by a power of the determinant b under a change of the base of the pencil are called combinants. The combinants characterize the weight of a pencil

as a whole. Next, we will deal with the classification of pencils of quadratic binary forms in the complex domain. We will assume, first, that the resultant R of the forms f and 1p which constitute the base of the pencil is non-zero.

Then the discriminant of the characteristic form, equal to - R/4, is likewise non-zero; we now find from the equation D(A, p) = 0 two values of the ratio A/p for which the corresponding forms of the pencil (22.16) will be perfect squares. Two such forms corresponding to different values of the ratio A/p can always be selected as a new base for the pencil (cf.

the beginning of 22.5); by virtue of the fact that R * 0, these forms are linearly independent (cf. Theorem 22.9). By means of a linear transformation of space, we can reduce the forms of the new base to (x')2 and (x2)2 and the pencil will assume the canonical form A(x')2 + p(x2)2.

(22.28)

However, if R = 0, but 9 0 0, the forms f and cp of the base of the pencil will be linearly independent, by Theorems 22.7-22.9, and have a

common linear factor whose square is equal to 9. The discriminant D(2, p) of the characteristic form will vanish and the equation D(2, p) = 0

CHAP. V

BINARY FORMS

258

will give only one value for )/p; we will reduce the corresponding form of the pencil to (x' )2 and select it as the first form of the new base. All remaining forms of the pencil which do not depend linearly on it will be divisible by x' and will not be perfect squares. We will take one of them as the second form of the new base and bring it into the form 2x' x2. The pencil has now been reduced to the canonical form (22.29)

X'().x' +2px2).

Finally, if 9 = 0, we have by Theorem 22.8 that all forms of the pencil can be expressed linearly in terms of each other, and we can employ the classification of binary quadratic forms presented in 22.4. The arithmetic characteristics of the different types of pencils considered

above may be described by the values of two arithmetic invariants: the rank p of the pencil and the rank ps of the Jacobian 9. The results of the classification of pencils of binary quadratic forms in the complex domain are given in the following table:

Type

Canonical form ,.(Xl)2 -ti(X2)2

11

x'(7,x1-+-2µx2)

Ill

).{(x1)1+(x2)2}

IV

A(X')2

V

all forms of the pencil vanish identically

Arithmetic characteristics (22)

(21) (20) (10) (00)

Algebraic characteristics R # 0

R=0.$#0

t =0, D(A,p)#0 D(A,p)=0, 2f+pq?#0 of+N9' = 0

Note also that the five types may be grouped in three pairs which complement each other: (I, V), (11, IV), (1I1, 111) to which there applies the law of complements: if we write under the arithmetic characteristics of one type of a pair the arithmetic characteristics of the second in revese order

and add, we obtain the sum (22). Proceeding to the classification in the real domain, we note that for R > 0 the discriminant of the characteristic form is - R/4 < 0, and the equation D(J., p) = 0 gives two real values for the ratio A./p. In the sequel, we will achieve a reduction to the canonical form in the same manner as

for Type I in the complex domain; for this purpose, we must keep in mind that in case of need we can change the sign of any form of the base.

If R < 0, the ratios A/p which satisfy the equation D(A, p) = 0 will


22.6

259

be complex conjugates; the corresponding forms of the pencil may likewise be assumed to be complex conjugates. By a real linear transformation of space we will give them the form (x' + ix2)2

and

(x'

-

1.x2)2,

i2 = - 1.

We will construct a new base from the forms

)(X'+ix2)2+1(x1 -iX2)2 = (X')2-(x2)2 and .1

(x' + (x2)2

21

- _'2i (x' - ix2)2 = 2x'x2.

The pencil is then reduced to the canonical form A{(x')2 -(x2)2} +2Eix'x2

(22.30)

for which

D(t, p)

_ -22-p2,

(22.31)

so that all forms of the pencil for which at least one of the numbers 2, p is non-zero have negative discriminants. For R = 0, 9 0 0, it is readily seen that for pencils of real forms the reduction to the form (22.29) can be effected by real linear transformations, provided only that one changes the sign of the first form of the base if this should be necessary. Thus, in the real domain, Type I becomes the two types I° and Ib with the canonical forms (22.28) and (22.30), while Type II remains unchanged. For pencils of the types la and 11 in their canonical forms, the characteristic forms are, respectively,

P: D(1, p) = Ap,

II: D(2, p) = -i12.

(22.32)

For the remaining cases, we may utilize the results presented at the end of 22.4. Taking into consideration the possibility of adjusting the signs of the forms of the base, we arrive at types with the canonical forms

III* d{(x')2+(x2)2),

IIIb 2{(x')2-(x2)2}, IV 2(x')2,

(22.33)

and at Type V which consists of pencils all of whose forms vanish identically. We will now present a geometric interpretation of the results obtained; from this point of view, interest is attached only to pencils of the Types

BINARY FORMS

260

CHAP. V

lo, 1b and II. Under the assumption that the numbers A and u are not simultaneously equal to zero, the pencil of forms (22.16) (assuming A, p not to vanish simultaneously) corresponds to a pencil of involutions. By Theorem 22.6, the double points of the involutions of a pencil themselves form an involution determined by the Jacobian 9 of the forms of the base. It follows from (22.23) that (9 Af+u(P)r21 = 0,

i.e., points whose coordinates satisfy the equation 9 = 0, correspond to each other in all involutions of the pencil (22.16); since R = -4Dp, these points will be real for pencils of Type Ia, imaginary for Type Ib and they will coincide for Type ll. In the case Ia, the involutions of the pencil can be hyperbolic (Ap < 0)

well as elliptic (Ap > 0) and parabolic (Ap = 0); in the case 1b, all involutions of the pencil are hyperbolic and in the case II they are hyper-

bolic for p 96 0 and parabolic for u : 0 [cf. (22.31), (22.32)]. If the forms f and c correspond to involutions of which at least one is elliptic, the pencil (22.16) may belong only to Type Ia. Consequently, one has Theorem 22.10: Given two involution at least one of which is elliptic, there always exists a pair of real points which correspond to each other in both involutions. Exercises 1. Find a complete system of joint invariants of g + h vectors of which g are covariant and h contravariant in an n-th order space. 2. The correspondences of points on a straight line defined by (22.5) and (22.6) coincide completely with each other only if the rank p of the tensor a;i is equal to two.

What happens for p - 1? 3. For n 2 find a complete system of joint invariants of the symmetric tensor air = aia, = bib, and the two contravariant vectors x and y (where a, b are parallel imaginary vectors).

4. Find a complete system of joint concomitants of the quadratic binary form f = (ax)t and the linear binary form (px). 5. Given the three quadratic binary forms

f=(ax)2= (bx)2=..., g'= (°`x)2= (fx)2=..., V

(fix), = (µx)2 = ...,

(22.34)

prove that they have the joint invariant (axllazl(ax)(Ax) = 1'D12'V+JD139'-kD21f,

(22.35)

Die = U 9')(1) = [a«12, D,t = (f, TV)"I = la%12, D2a = (9', V)121 -- Ix).lt

(22.36)

where

CUBIC BINARY FORMS

22.6-23.1

261

6. For the invariant E12a = lax]IaAJl«AJ

(22.37)

of the binary forms (22.34) (cf. § 34, Exercise 14, and § 18, Exercise S) prove the equality E123 - ('412 , 012)where

fl,, is the Jacobian of the forms f and q'; on this basis, establish the geometric significance of the equality E1.3 = 0. 7. Find a complete system of concomitants of the three binary quadratic forms (22.34).

8. In the notation of 5 and 6, express E1Y3 in terms of the discriminants D1, D2, D, of the forms f, c, VP and their joint invariants D1Y, D33, D,3.

9. Find a complete system of concomitants h of the binary quadratic forms f,.

f2, ...,f, (h > 3). 10. In the notation of 9 and its solution, prove that E234f1- E13+f2 + E1s4f3 - E123f4 = 0-

11. By means of real linear transformations reduce to canonical form the pencil of form? Af+p(p, where f = 6x'x2r4(x2)2, p = 6(x')2T 18x'x2+(x3)2

§ 23. Cubic binary forms 23.1 For a study of the properties of a cubic binary form f, we will employ its classical representation (13.8), viz.,

f = aox3+3a,x2y+3a2xy2+a3y3,

(23.1)

where x, y are the components of a contravariant vector. The geometric interpretation of the case when f is equal to zero (under the assumption that some of its coefficients ao, a, , a2, a3 are non-zero) is readily established by the same reasoning as that used in the case of quadratic forms

(cf. 22.3): the equation f = 0 determines three points on a straight line whose coordinates satisfy this equation; some of these may be found to coincide. In this context, it is useful, as in 22.3, to distinguish the three cases

a00 0,

a3

0,

a0=a3=0.

This geometric interpretation of cubic binary forms permits one to obtain canonical forms for them immediately. First, assume that the points of the straight line determined by the form (23.1) are all different; then apply a linear transformation which transfers these points into the

first, second and unit coordinate points (cf. Theorem 5.1). Then the transformed form must vanish for x = 0, for y= 0 and for x = y, and hence it is equal to 3axy(x- y), a 0 0. A further linear transformation for which

CHAP. V

BINARY FORMS

262

13'xx and ti'xv become x and y, respectively, brings the form into the canonical form (23.2) f = 3x y(x - y).

Obviously, in the case under consideration, the form (23.1) may be transformed into any other binary cubic form, provided this last form determines three distinct points of a straight line; as a rule, one takes as canonical form

f = x3+y3.

(23.3)

If two of the points determined by (23.1) coincide and the third is different, we can, by means of a linear transformation, cause the coincident points to go over into the second coordinate point x = 0, and the third

point into the first coordinate point y = 0. The form then becomes 3xx2y, x # 0; after a second linear transformation which leaves x unaltered and changes ay into y, we find f = 3x2y.

(23.4)

By a similar reasoning, we verify that a cubic binary form which determines three coincident points on a straight line (i.e., a form which is equal to the third power of a linear form) can be reduced to

f = x3.

(23.5)

In the sequel, we will require expressions for the concomitants H, Q and D for all these canonical types of cubic binary forms; they are

easily obtained from the definitions of these concomitants (cf. § 14, Example 9). For the form (23.2), one has

H = -x2+xy-y2,

Q = (x+y)(x-2yx2x-y),

D = 3,

(23.6)

for (23.3)

H = xy,

Q = x3-y3,

D = -1,

(23.7)

D = 0,

(23.8)

for (23.4)

H = -x 2,

Q = 2x3,

and, finally, for (23.5)

H = 0,

Q=0,

D=0.

(23.9)

These results give the classification of cubic binary forms in the complex

domain; there are four types: I with canonical form (23.3) [or (23.2)),

23.1-23.2

CUBIC BINARY FORMS

263

If with canonical form (23.4), III with canonical form (23.5) and IV comprising all forms which vanish identically. Arithmetic characteristics

of each of these types are provided by the values of two arithmetic invariants: the rank p of the form (i.e., the rank of the symmetric tensor a;;k which corresponds to the form withal, I = ao, a, I z = a1, at 2, = a2, a,,, = a,; cf. § 14, Example 12) and the rank pI, of the Hessian H of the form (the rank of the covariant Q always coincides with the rank of 11, as can be seen from the expressions for H and Q given above). It is readily seen that the rank p of a binary form of order r is equal to unity, if the form is an r-th power of a non-zero linear form, that p = 0, if the form vanishes identically, and that p = 2 in all other cases. The results of the classification of cubic binary forms in the complex domain are summarized in the following table: Type

Canonical form

Arithmetic characteristics


x3- V3

(22)

D

II

3x1v

(21)

D- 0, H

111

X3

IV


I

0

(10)

0 H = 0,f = 0

(00)

f-0

Note that one has again here the law of complements (cf. 22.6); the follow-

ing pairs of types are complementary with respect to each other: (l, IV) and (11, 111) (cf. Exercise I and 3). Incidentally, the investigation above leads to Theorem 23.1:

A cubic binary form is a cube of a linear form, if and only if its Hessian H vanishes identically (cf. § 19, Exercise 17, § 14, Exercise 12). 23.2 The method described in 23.1 for the reduction of a cubic binary form to one of its canonical forms requires beforehand the solution of a cubic equation; Cayley has presented a method for the

same purpose in which the new variables are expressed directly in terms of the concomitants of the form. At the same time, this approach led to a new method for the solution of cubic equations. We will begin with the case of a form of Type I (when the discriminant of the form D 0 0). The covariant Q (Theorem 14.9) is then likewise a

CHAP. V

BINARY FORMS

264

cubic binary form. Construct now the pencil of cubic binary forms Q+Af,

(23.10)

where i is an arbitrary number, and find the Hessian HA of the form (23.10):

Hx - j(Q+2f, Q+Af)(2) = j(Q,

p2(f,f)`2'.

(23.11)

On the right-hand side of (23.11), the first term is the Hessian of the form Q and is equal to DH (cf. § 19, Exercise 4), where H is the Hessian of the form f; by (20.20), the second term vanishes, and finally, the last term is equal to 22H. Thus,

Hx = (i2+D)H.

(23.12)

We will now seek within the pencil (23.10) a form which itself represents

the cube of a linear form; by Theorem 23.1, this requirement will be ful-

filled only by those forms for which Hx = 0. Thus we arrive at the quadratic resolvent 22+D = 0,

(23.13)

whence 2 = ± - D. Letting

2`` -D(px)3, Q- -Df =

(23.14)

where (px) and (qx) are linear binary forms, we find

f = (px)3+(qx)3 Since f is not a perfect cube, the forms (px) and (qx) are linearly independent; the linear transformation

x' _ (px),

y' = (qx)

(23.15)

reduces the form f to the canonical form (23.3). Obviously, this method reduces the solution of a cubic equation to the

solution of the equation with two terms (X,

)3+ 1 = 0,

i.e., to the study of the cube roots of,( I). Note further that in this manner one can readily find a linear trans-

23.2

CUBIC BINARY FORMS

265

formation which brings (23.2) into the form (23.3) (cf. Exercise 4, c). For forms of Type 11 (D = 0, H zA 0), the reduction to the form (23.4)

presents no difficulties. Since the discriminant D of the form f differs

only by a numerical factor from the discriminant of its Hessian H, the last is the square of a linear form in the case of forms of Type II; by (23.4) and (23.8), the form f is divisible by H. In order to reduce the form f to the canonical form (23.4), it is sufficient to let X,

11,

y'=3f2.

(23.16)

In the case of Type III (H = 0), the form f is a cube of the linear form

ao x+a3 y and can be reduced to the canonical form (23.5) by the linear transformation

x' = /aox+

. a3Y,

y' = yx+8yi,

(23.17)

where the arbitrariness in the choice of the numbers y and b is only limited

by the condition 4o-ay :A 0. For a cubic binary form with real coefficients, this transformation can always be expressed as a real transformation except when the discriminant D > 0. Then we let

Q+i\lDf =

iv1D {(px)+ i(qx)}3,

i2

(23.18)

where (px) and (qx) are binary linear forms with real coefficients. The second of the forms (23.14) will be the complex conjugate of the first,

so that

Q-N/Df = -iN1D{(px)-i(qx)}3 and

f = 3{(px)+i(qx)}3+3{(px)-i(gx)}3 = (px)3-3(pxxgx)2. The forms (px) and (qx) are linearly independent (since f is not a perfect cube); the linear transformation (23.15) reduces f to the canonical form

f = x(x2-3y2)

(23.19)

(where the primes have been omitted). Thus, we see that the classification of cubic binary forms in the real

domain differs from that in the complex domain only in that Type I becomes two types: Type I' with the canonical form (23.3) (D < 0)

CHAP. V

BINARY FORMS

266

and Type 16 with the canonical form (23.19) (D > 0). Forms of Type 1b define three real points of the straight line, while forms of Type Ia define one real and two imaginary points. In the case of Type 1b, one may also employ as canonical form the form (23.2); by use of this method, one can readily find real linear transforma-

tions which transform the forms (23.19) and (23.2) into each other (Exercise 5, a). 23.3

We now proceed to the search for a complete system of conco-

mitants of the cubic binary form f. It turns out to be very difficult to employ for this purpose the method by which we solved the same problem in § 22 for quadratic forms and for pairs of quadratic forms: the objective can only be achieved by means of an improvement of this method which was proposed by Gordan and will be described in § 24. We will give now a proof which lies at the basis of another type of argument. Without reducing generality, it may be assumed that the discriminant D of the form f is non-zero (cf. remark at the end of 15.1). Let the symbol K denote some covariant of order r of the form f. Reduce the form f to the canonical form (23.3) and let the covariant K then assume the form

K = cox'+c,x'- 1}'+CZxr-2 y2 +

... +c,y',

(23.20)

where co, c, , cZ , ... , Cr are certain numbers. Under the linear transformation

x = wx ,

}

w-13",

w=

3,

iz = -1, ws = 1,

(23.21)

the determinant of which is equal to (+I), the coefficients of the form (23.3) do not change and, consequently, the coefficients on the right-hand side of (23.20) also remain unaltered. Under the transformation (23.21), the coefficient of the term cx'yk is multiplied by w'-k; hence we conclude

that on the right-hand side of (23.20) only the coefficients of those products x'yk for which the difference j-k is divisible by 3 can be nonzero.

We will first dwell on the case when the weight g of the covariant K is an even number. Applying an analogous reasoning to the linear transformation

x = y',

y = x',

(23.22)

the determinant of which is (- 1), we arrive at the conclusion that the coefficients of the products x'yk and xky' on the right-hand side of (23.20)

23.2-23.3

267

CUBIC BINARY FORMS

must be the same; two such terms can be combined in an expression of the form c(xy)'(x3p+y3P), where p is a positive integer. However, the expression x3p+y3° is a symmetric function in x' and y3 and therefore it may be represented in the form of a polynomial in x3 +y3 and x3y3; thus we see that the covariant K of the form (23.3) is an integral rational function in

xy and x3+y3, i.e., [cf. (23.3), (23.7)) in f and H: K = Y c,, f SH`,

(23.23)

3s+2t = r.

(23.24)

where in any term For each term on the right-hand side of (23.23) the relation (15.20) gives

3d-3s-2t = 2g, where d is the degree of the covariant K in the coefficients of the form f.

We conclude from the last equality that 2g-4t, and hence also g-2t is divisible by 3; however, taking into consideration the fact that g is even, we see that g-2t is divisible by 6. As a consequence, the relation (23.23) can be rewritten in the form

K = Y c f3H(-D)1e-2x)16 Sf

(23.25)

[since, by (23.7), D = -1 ]. But this equality remains true in the same form after any linear transformation: the weight of H is 2, the weight of D is 6 and both sides of (23.25) are covariants of the same weight g; hence (23.25) will be true for any cubic binary form.

It still has to be shown that on the right-hand side of (23.25) all exponents of the determinant D are non-negative. Let t attain its largest value in the term for which

t = to,

r - 2to

s = so = . - -3

[cf. (23.24)]. If (g-2to)/6 were negative, on multiplying both sides of (23.25) by Dh, where 2t h=-y`-°-], 6

and comparing the coefficients of x' on both sides, we would find [cf. (14.31)] that ao(a0a2-a;)'° is divisible by D; however, this is obviously

CHAP. V

BINARY FORMS

268

impossible, since D depends on a3 [cf. (14.33)]. Thus, (g-2to)/6 > 0, and therefore (g - 2t)/6 >_ 0 for all t on the right-hand side of (23.25). If the weight g of the covariant K is odd, the transformation (23.22) shows that the right-hand side of (23.20) is a sum of expressions of the form c(xy)'(x3p-y3p), where p is an integer. Since, by (23.7), the quantity

Q = X3 - y3, we conclude that

K=Q-K,, where Kl is a covariant of even weight (g - 3) which is integral and rational in x, y and is rational in each of the coefficients of the form f. We can apply to it the reasoning above by which it may be represented in the form (23.25). Thus, one finds cnfsHt(-D)(8-3-2t)l6.

K=QY The fact that in the last equality the powers of D in any of the terms on the right-hand side cannot be negative, can be established in the same way as in the case of even g; for this purpose, one has to refer to the circumstance that D is of second degree in a3 and that the coefficient Of X3 in the covariant Q contains a3 only in the first degree. In this way, we have proved Theorem 23.2: A complete system of concomitants of a cubic binary form f consist of the form f itself, its Hessian H, its discriminant D and the Jacobian Q of the forms f and 2H. A second proof of Theorem 23.2 will be given in 24.3. Since Q2 is a covariant of even weight, it must, on the basis of the

result proved above, be an integral rational function of f, H and D. In fact, for the form (23.3), one has Q2

= (X3-y3)2 = (X3+y3)2-4X3y3 =f2-4H3,

or

Q2 = -Df2-4H3. The covariants on the left- and right-hand sides of the last equality are of weight 6, and hence the equation is invariant with respect to any linear transformation. Thus, one has for any binary cubic form

Q2+Df2+4H3 = 0.

(23.26)

This relation interlinking the basic concomitants of the form f is called Cayley's syzygy.

23.3-23.4

CUBIC BINARY FORMS

269

We have still to remove any doubts regarding the irreducibility of the concomitants H, D and Q (since that off is obvious). By virtue of the reasoning leading to the proof of Theorem 23.2, those covariants which can be expressed in terms of H, D, Q by means of integral rational functions must be polynomials in f, H, D, Q, and, consequently, if they are reducible, H, D, Q must also be polynomials of this type; each term of such a polynomial must be of the same degree in x and y and in the coefficients of the form and of the same weight as the concomitant which is expressed as a polynomial. From this follows directly their reducibility of D which is the only invariant among the concomitants under consideration. The covariant H cannot be equal to a polynomial depending on D and Q, since the weights of D and Q are equal to 6 and 3, respectively, and the weight of H is 2, nor to a polynomial in f, since the degree off is 3 and that of H is 2; therefore H is also irreducible. By virtue of the fact that the weight of Q is odd and those off, H and D are even, all terms of any polynomial expression for the covariant Q would have to depend on Q itself, which is obviously impossible. Hence the irreducibility of the concomitants f, H, D, Q has been established. 23.4 In conclusion, we will now consider the geometric interpretation of the concomitants of the complete system. The geometric significance of the equality f = 0 has been stated in 23.1; if we multiply all coefficients of a form by the same number, we do not change its geometric meaning. Hence it is clear that for a geometric interpretation of the remaining concomitants, they must also be set equal to zero; this follows from the fact that all these concomitants are relative. The geometric meaning of the equality D = 0 is obvious: among the three points defined by the form f, some coincide (since in such a case the form is not of Type I). In order to establish the geometric significance of the concomitants H and Q, we limit ourselves to the case of forms belonging to Type I (it being trivial for the other types). The equality H = 0 defines two distinct points E and F of the straight line, since D # 0; their relation to the three points A, B, C, defined by the form f, is readily explained. For this purpose, reduce f to the canonical form (23.2); then this equality assumes the form

x2-xy+y2 = 0 [cf. (23.6)], and hence

CHAP. V

BINARY FORMS

270

x

-= -w, X--= -w. 2

y

y

By virtue of Theorem 5.4, we have {ABCE } = -co, {ABCF} = -co", i.e., (cf. 5.4) Theorem 23.3: Each of the two points defined by the equality H = 0, where H is the Hessian of a cubic linear form f with discriminant D # 0, forms with the three points defined by fan equi-anharmonic tetrad. In an analogous manner, one may establish the geometric meaning of

the equality Q = 0. Reducing f to the form (23.2), we find from the equation Q = 0 [cf. (23.6)J C Y = - 1, y

- = 2, y

y

Denoting the points whose coordinates satisfy the equation under consideration by A', B', C, we find (cf. 5.4): {ABCA'} -1, {ACBB'}

_ -1, {BCAC'} = -1. We thus arrive at Theorem 23.4: Let A, B, C be three points of the straight line corre-

sponding to a cubic binary form f with discriminant D # 0, Q the Jacobian of the form f and 2H twice its Hessian. From the equation Q = 0 we obtain the coordinates of those three points each of which together

with one of the points A, B, C divides the remaining two in a harmonic manner. Exercises In the following equations, as throughout § 23, the symbol f denotes a cubic binary form, H its Hessian, D its discriminant and Q the Jacobian (f, 2H). 1. Find for a given cubic binary form f a cubic binary form q' satisfying the conditions

(f

)(2) = 0,

(f, 99) 13) = 0.

(23.27)

Show that the highest possible type of the form 99 is always complementary to that type

to which the form f belongs (where of cubic binary forms of two types that one is higher in which the sgtt r_t- , of the arithmetic characteristics is 1ar^-r). 2. Two types of qua -atic Ninary forms will be said to be complementary, if the sum of their ranks is equal t..2;: pairs of complementary types are (1, 111) and (11, 11). Determine for a quadratic binary form F a quadratic binary form 45 which satisfies the conditions

(F, 0) = 0, (F, 0)«' = 0.

(23.28)

Show that of the types possible for b the highest in rank will always be complementary to the type to which F belongs.

CUBIC BINARY FORMS

23.4

271

3. Let a,rk and a; rk. be the symmetric tensors corresponding to the forms f and 9' of 1,

and b;r and fl, the symmetric tensors corresponding to the forms F and 0 of 2. Show that the two relations (23.27) are equivalent to the single condition a;;i[iaz]z?r = 0,

and the two relations (23.28) to the condition brrl flz)i = 0.

4. Employing Cayley's method, reduce the forms a) f = 7x3 +15x2y+3xy2 +2y3, b) f'= x3+ 3x2I + l 2xy2 - l6y3, c) f = 3xy(x y) to the canonical form (23.3). 5. By means of real linear transformations reduce the forms

a) f = 3xy(x--y), b) f = 2x3--6xzy4 3xyz-2y3 to the canonical form (23.19).

6. Reasoning as in the proof of Theorem 23.2, show that a cubic binary form cannot have (non-zero) invariants of odd weight. 7. Give a proof of Theorem 23.2 for the case of the invariant l of a cubic binary form. 8. Using the method employed for the proof of Theorem 23.2, express the transvectants (f, Q), (Q, 2H), (Q, Q)'2l/2, (f, Q)(t', (H=, f) and (H=, f )'_) in terms of the concomitants of the complete system [cf. § 13, Example 3 and (19.14), Example 4 and (19.16), Exercise 4; § 20, Example 5 and (20.20)).

9. Applying arguments analogous to those employed for the proof of Theorem 23.2, show that for a cubic binary form of Type III only those concomitants which are equal to a product of a numerical factor and a power of the form itself can be non-zero. Generalize the result to the case of a binary form of any order which is a power of a linear form (cf. § 15, Exercise 3 and § 17, Exercise 6). 10. Derive Cayley's syzygy [cf. (23.26)] starting from (20.38). 11. Express the transvectants (H', f2)'b', (Hz, Qz)(s), (H3, f') in terms of the basic concomitants of the form f.

12. If D; 0, the equality of+µQ = 0 determines for different values of the ratio .1/p an infinite set of point triplets on a straight line (Clebsch's point triplets). Show that, if two points coincide in such a triplet, the third also coincides with them and all of them coincide in one of the points whose coordinates satisfy H = 0; for. what value of d/la does this take place? 13. Let A, B, C be one of Clebsch's point triplets (cf. 12), and E, F points whose coordinates satisfy H = 0. Show that the point quintuplets E, A, B, C, F; E, B, C, A, F; E, C, A, B, F are projectively equivalent. 14. Let a,Jk be the symmetric tensor connected with the cubic binary form f. The equation 0 (23.29) relates two points x' and y' of a straight line to the third point z'; this correspondence is

said to be involutionary, if the point z' corresponds to the points x', y', the point y' corresponds to the points x', z' and the point x' to the point y', z'. Let t' be the point which together with z' divides harmonically the pair of points for whose coordinates the Hessian H vanishes. Show that in the case of a form f of Type I one has

qukx'yttr = 0, where q,tk is the symmetric tensor corresponding to the covariant Q of the form f.

CHAP. V

BINARY FORMS

272

15. The equation (23.29) can also be interpreted in the following manner: the form f relates to every point x' an involution on a straight line (defined by the tensor By varying x° we obtain a pencil of involutions (cf. 22.6); show that in all cases

its arithmetic characteristic coincides with that of the form. 16. Under the conditions of 15, assume the form f to belong to Type I. Show that in this case any point y' of a straight line will be a double point in one of the involutions of the pencil defined in 15. Let z' be the second double point of the same involution; show that the correspondence y' --> z' is an involution given by the Hessian H of the form .f The Exercises /6 and /4 give a new geometric interpretation of the covariants H and Q of the form J:

§

24.

Gordan's method for finding complete systems of concomitants of binary forms. Gordan-Clebsch series

24.1

So far, two methods have been applied to the search for complete

systems of concomitants of binary forms: one based on a study of the symbolic representation of concomitants - for quadratic forms and for pairs of quadratic forms (cf. 22.3, 22.5); another based on the application of linear transformations which leave the coefficients of the forms unchanged - for cubic forms (cf. 23.3). In the case of forms of higher order, the second of these methods leads to insurmountable difficulties; but after introducing improvements proposed by Gordan, the first method turns

out to be very useful for the stated purpose. We will present Gordan's method in this section. First of all, certain properties of transvectants must be considered. Let it be required to construct the h-th order transvectant of two covariants 2

1

k

X = A(axXax) ... (ax),

2

1

t

to = B(bx)(bx) ... (bx),

(24.1)

where A and B are products of certain factors of the second kind. The tensors corresponding to these covariants will be 1

2

k

Aat;, a;2 ... aht ,

1

2

Bb(;, b12

1

.

.

. bjt) .

(24.2)

In order to construct the transvectant (X, co)(''), we must expand the symmetrization signs in (24.2) and carry out h total alternations on the products of each term of the first of the sums obtained with each term of the second sum; each of these is to be performed with respect to one

of the first h indices of the first factor and with respect to the corresponding index of the second (cf. 20.2). The remaining indices must be

273

GORDAN'S METHOD

24.1

contracted with the vector x. Thus we obtain k! 1! terms of the form 11

22

AB[ab] [ab)

h+1

hh

h+1

k

I

... [a b] ( a x) ... (ax) ( b x) ... (bx);

(24.3)

Ih)Terms such a term will be said to be a term of the transvectant (X, co). ij

which have the same factors [ab] of the second kind and the same factors.

of the first kind are equal; hence we obtain each term of the transvectant h!(k -h)!(I -h)! times and the number of distinct terms of the transvectant is k!I!

(k-h)!(I-h)!h! In the expansion of the symmetrization (24.2), we will have in front

of the corresponding sums the multipliers Ilk! and 1/1!; due to the presence of equal terms in the brackets, we arrive at the factor (k-h)! x (I-h)!h!. Thus, the transvectant (X, w)(6) is obtained in the form of a sum of its terms with the multiplier

(k-h)!(l-h)!h!

1

k!1!

N

in front, so that 1

(X, W)') =

N

(T1 +T2+

...

(24.4)

+TN),

where T1, T2, ..., TN are terms of the transvectant. 1

2

k

1

2

1

If some of the imaginary vectors a, a, ... , a or b, b, ... , b turn out to be the same, certain of the terms of the transvectant will be equal to each other without the validity of (24.4) being affected. Two terms of a transvectant are said to be adjacent, if they are obtained 1

2

k

from each other by transposition of two of the symbols a, a, ... , a or 1

2

of two of the symbols b, b, the three possibilities:

1

..., b. Thus, as two adjacent terms, we have

C[a'b'](a "x),

C[a"b'](a'x),

(24.5)

C[a'b'](b"x),

C[a'b"](b'x),

(24.6)

C[a'b'][a"b"],

C[a"b'][a'b"],

(24.7)

274

BINARY FORMS

CHAP. V

where C denotes the product of the remaining symbolic factors. By the identity (19.6), the difference between adjacent terms (24.5) is given by

C[a'a"](b'x) and represents a term of the transvectant (X, VV''-'), where j is the covariant obtained from X by symbolic total alternation with respect to the ideal vectors a' and a"; a similar situation applies with respect to the difference between the terms (24.6). By the identity (19.7), the difference between the terms (24.7) is given by

C[a'a"][b'b"] which is a term of the transvectant (X, w)«-21, where j and w are obtained from X and co by symbolic total alternation. Since each permutation is a product of transpositions, the difference between any two terms of a transvectant can be represented in the form of a sum of differences of adjacent terms. Hence, one obtains Theorem 24.1: The difference between any two terms of a transvectant of two covariants X and co of a binary form is a sum of terms of transvectants with lower index, formed from y, w, and covariants obtained from X and co by symbolic total alternation. It follows from (24.4) that

Ti -(X.,(0)", = N {(T-T,)+(T-T2)+ ... +(T;-TN)); on the basis of Theorem 24.1, we now conclude that each term of a transvectant is equal to the sum of the entire transvectant and terms of transvectants with lower indices formed from the covariants X, co, X, F0, .

. ., where j, w are obtained by total symbolic alternations of X, co.

For the same reason, each of these terms of transvectants is also a sum of a complete transvectant and terms of transvectants of still lower index. Continuing in this manner, we arrive at the following theorem which is the foundation for Gordan's method: Theorem 24.2: Every term of the transvectant (X, w#) is equal to a sum of the entire transvectant and a linear combination (with numerical coeffi-

cients) of transvectants of lower order of certain pairs of covariants X, co and covariants derived from X and to by one or several symbolic total alternations.

24.1-24.2

275

GORDAN'S METHOD

In applying Theorem 24.2, one must keep in mind that all transvectants whose linear combinations are equal to terms of the transvectant (X, c))t", must have the same weight; hence in each of them the sum of the index,

the number of symbolic total alternations performed on X and the number of such alternations performed on co is equal to h. Theorem 24.2 immediately permits one to reduce the study of all covariants of the binary form 24.2

f=(ax)'=(bx)'=...

(24.8)

to a study of transvectants of a special type. Let there be given a covariant K. of the form (24.8) which is of degree m in its coefficients. In order to represent it symbolically, we require m parallel symbols a, b, . . ., k, 1; in this representation, separate out those symbolic factors which contain the imaginary vector 1

a+f+ ... +K+!t = r,

K. = P[al]°[bl]fl ... [k1]"(Ix)z,

where P is a product of factors containing only the symbols a, b, Obviously, the covariant K. is a term of the transvectant (Km-I,f)("1

..., k.

h = r-2,

where Km_, = P(ax)Q(bx)B

... (kx)"

is a covariant of the form f of degree (m - 1) in its coefficients; the concomitant Km _, obtained from Km_, by symbolic total alternations will

have the same degree. Performing symbolic total alternations on the form f itself, we obtain the result zero; therefore, by Theorem 24.2, the covariant K. is a linear combination of transvectants of the form

(K.-I ,f)I"l,

(Km-1,f)ce),

g < h.

(24.9)

Theorem 24.3: Every covariant K. of a binary form f of degree m in its coefficients is equal to a linear combination of transvectants of the form

(24.9), where KmKmare covariants of the same form of degree (m -1). From this now follows Gordan's method for finding a complete system of concomitants of a binary form f of order r. Consider consecutively covariants of degree 1, 2, 3, ... in the coefficients of the form. In order to find irreducible covariants of degree m, one must construct for all

276

BINARY FORMS

CHAP. V

linearly independent covariants Km_ 1 of degree (m-1) all transvectants

of the form (Km-I,f)`h',

(24.10)

setting consecutively h = 1, 2, . . ., r and omitting all those transvectants which can be expressed in terms of the concomitants already found. By

virtue of Hilbert's Theorem 21.1, starting with a certain value of m, all covariants of the form (24.10) turn out to be reducible and a complete system is obtained. The study of the covariants (24.10) is simplified by employment of the following two propositions:

Theorem 24.4: If a covariant K. the covariant (24.10) is reducible.

contains an invariant I as a factor,

Theorem 24.5: If K. -I is a product of two covariants 45 and 0 of which the first is of order greater than or equal to h and the second is not a constant, the transvectant (24.10) with index h, constructed with the aid of K.- 1, may be excluded from the investigation.

Theorem 24.4 is obvious [since I enters inside the brackets in each of the transvectants (24.10)]. The truth of Theorem 24.5 can be established by the following reasoning. Since the order of 4s is not less than h, and its degree is smaller than (m -1), one can construct the transvectant (0, f)(*) whose degree will be less than m. Let T be one of the terms of this transvectant; it is obtained from the product Of with the aid of h total symbolic alternations. Performing the same symbolic alternations on 450f (where Iii remains unaffected), we obtain the covariant 1T which is thus a term of the transvectant

(40,f)(h' = (Km-I,fyh).

(24.11)

By Theorem 24.2, the transvectant (24.11) is equal to the sum of k/iT and a linear combination of transvectants

(K. _ I , f )(°),

g < h,

(24.12)

where K._ I is obtained from K.-, by total symbolic alternations. If in our search for a complete system of concomitants we follow the order

proposed above, the covariants i/i and T as well as the transvectants (24.12) will already have been considered: the first two, because their degree is less than m, the remaining ones, because g < h. Hence the study of the concomitant (24.11) turns out to be superfluous.

GORDAN'S METHOD

24.2-24.3

277

In order to clarify the preceding results, we will apply Gordan's method to a cubic binary form (p. Obviously, all covariants of first degree in the coefficients of this form will be equal to c(p, where c is a number. Therefore we may limit ourselves in the search for irreducible second degree covariants to the study of the transvectants h = 1, 2, 3. (24.13) (cp, 24.3

For h = 1 and h = 3, these transvectants vanish identically [cf. (20.16)]; for h = 2, we obtain twice the Hessian H of the form. The irreducibility of the Hessian H follows from the fact that it has weight 2 and all covariants of lower degree have zero weight. Linearly independent covariants of second degree will be (p2 and H. The first of these comes within the range of Theorem 24.5, since the order of the factor qp is 3 and h does not exceed 3 in all the transvectants under consideration. Therefore in our search for irreducible concomitants of degree 3 we need only study the transvectants (H, (p)l"l,

h = 1, 2

(24.14)

(since h must not be larger than the order of H).

For h = 1, we obtain - Q/2, where Q is the covariant of the form f which we have encountered many times before; for h = 2, the formula (24.14) gives zero (§ 20, Exercise 5). The irreducibility of the covariant Q

follows from the fact that its weight is 3 and all concomitants of lower degree have even weight.

Among the third degree covariants, we will have the linearly independent ones (p3, cpH and Q; by Theorem 24.5, there is no need to construct transvectants of the form (24.10) for the first two (taking p to be the

factor 0). We will form the transvectants (Q, (p)'"),

h = 1, 2, 3.

(24.15)

For h = 1, we obtain the irreducible concomitant 2H2 [cf. (19.14)]; for h = 2, the transvectant (24.15) vanishes [cf. (20.20)]. Finally, for

h = 3, it is equal to -2D, where D is the discriminant of the form (§ 20, Exercise 5). The discriminant D is irreducible, since there are no invariants among the concomitants of lower degree. Now it is not difficult to show by induction that all the concomitants whose degree is equal to or greater than 5, are reducible and that they are polynomials in gyp, H, Q and D. Let m z 5; assume that it has already been

CHAP. V

BINARY FORMS

278

established that all covariants of the form cp of degree (m -I) are integral .rational functions of cp, H, Q, D; among these one will have the linearly independent covariants

K.-, = cp°H'QyD-,

0, 1, 2, ...,

a, i, y,

a+2f+3y+4A = m-1. It follows from Theorem 24.4 that for the examination of the trans1 one must set the exponent A = 0; vectants (24.10) in the covariant since the covariants cp and Q are of order 3 and h does not exceed 3, one must also set a and y equal to zero (Theorem 24.5). For fi ? 3, one can divide out from HO the term Hz whose order is 4 > h, so that, by Theorem

24.5, this possibility for a study of the transvectants also disappears. There remain the transvectants h = 1, 2, 3. (H2 , (p)("', (24.16) If h = I or h = 2, the factor H again presents an opportunity for an application of Theorem 24.5; for h = 3, the transvectant (24.16) is equal to zero [cf. (20.14)]. Thus, all covariants of degree m are likewise polynomials in cp, H, Q and D and we have obtained a new proof of Theorem 23.2.

24.4 The above work still contains an essential gap: the reasons why

complete systems of concomitants of binary forms comprise only covar-

iants of one contravariant vector still remain unexplained. In order to close this gap, one has the series of Cordan-Clebsch by the aid of which covariants of several sets of variables (i.e., invariants of a system formed from the tensors corresponding to the forms and several contravariant vectors) can be reduced to covariants in one set of variables.

First of all, we will present several simple formulae relating to the properties of the polar operation D;y (cf. 20.4). By virtue of the definition

of the first polar equality D,,,, K = I y` s

vK

(24.17)

.

axi

where s is the order of the covariant K with respect to x', x2 [cf. (20.29)], we have (24.18) D=y[xy] = . [YYJ = 0 and

D x,, {cp 0 j =

p+q

>1iD

:r rp +

q

p+q

4,

W D xr

(24 19) .

279

GORDAN'S METHOD

24.3-24.4

where p and q are the orders of the forms rp and 0 with respect to x', x2;

setting 0 = [xy], we obtain, by (24.18), Ds,{p[xy]} =

(24.20)

[xy] Ds, (P

p+ 11

Further, we find DX,{cp[xy]}

P

P

p+1

p+1

Dx,{w[xY]} = p+

.

p1 [xy]DF,co, p

[xy]Dz,cp. 1

By induction, we show readily that Dx,{cp[xY]} =

p-h+1 P+1

(24.21)

[xy]DXgcp.

Next, consider the derivation of a Gordan-Clebsch series. Letting the quantity K = (ax)"(bx) in (24.17), where a and b are imaginary vectors, we find Dx.,{(ax)"(bx)} =

i

p+1

-

(ax)"-'(ayxbx)+

P+1

(ax)"(by),

and hence

(ax)(by)-Dxy{(ax)"(bx)) _ -p (ax)"-'{(axxby)-(ayxbx)}; (24.22) P+1 by virtue of the third fundamental identity (19.8) which for binary forms

takes the form

[ab][xy] = (ax)(by)-(ayxbx),

(24.23)

the equality (24.22) can be rewritten

(ax)"(by) = D.,{(ax)"(bx)}+ -+1 [xy][ab](ax)"-'.

(24.24)

The factor [ab](ax)"-' in the second term on the right-hand side of (24.24) is obtained from the left-hand side in the following manner: one of the indices contracted with x and the index contracted with y are released from the contraction, after which a total alternation is performed.

CHAP. V

BINARY FORMS

280

For example, if we had applied (24.24) to the form (ax)'(bx)(by), we would have after such an exclusion from the contraction

,'

bj,) x'x

. .

. x 6J2,

and after the total alternation with respect to jl and j2 [cf. (18.51)] P+ i [ab](ax)'-'(bx)

(since [bb] = 0). Next, in the relations (24.24), replace the term (ax)'(bx) by (ax)'(bx)(by) (ax)p(by)2 = Dxs,{(ax)'(bx)(by)) +

p+

1

[xy][ab](ax)'-'(by)

and transform by means of the same formula the expression following Dx,

in the first term on the right-hand side of (24.24) and the coefficient of [xy] in the second term to obtain (ax)P(by)2 = D,{D.,((ax)P(bx)2)+

+

p

p+l

p+ l [xy]

p

p+l

p+2

[ab](ax)P-'(bx))

[xy] {Dx([ab](axY'(bx))+ p-1

[ab]2(ax)P-21

p

applying (24.20) to expand the first of the braces, we obtain after some simple manipulations (ax)P(by)2 = A'Ds,{(ax)'(bx)2} +A2[xy]DxJ,{[ab](ax)P-'(bx)} +A2[xy]2[ab]2(ax)-2,

where

AZ = 1,

A2

=

2p

p+2'

A22

=

p-1 p+1

.

(24.25)

We will now show by induction that for any q < p i=9

(ax)I(by)9 = Z A9[xy]DX,'{[ab]'(axy-'(bx)'-'},

(24.26)

I=O

where the A' are numerical coefficients depending on p, q and i. We find from (24.26)

281

GORDAN'S METHOD

24.4

i: =

(ax)'(byr+1

A'[xY]iD=:'{[ab]`(ax)p-'(bx)q-'(by)}

i=o j=q

{by (24.24)} _

A'[xY]'Dxy

:

(D,([abT(axy_1(bx)q-i+i) i

=O

p-i

p+q-2i [xy]

+

p+q-2i

p+q-2i+ 1 i=q

= {cf. (24.21)} =

(bx)q-'} 1

(xy]'Dxy' -`{[ab]`(ax)p-'(bxp+ 1-i)

Aq

=o

[xy]i+1Dxs i{[ab]i+t(ax)°-i-1{bx)q-i))

(P-1)z

+

[ab]'+'(ax)c-i-'

(p+q-2i+lxp+q-2i)

,

i=q+1

(ax)°(by)q+' =

Aq+1[xy]`DXy 1-i{[ab]'(axy-'(bx)q+ i=o

where z

(p+q-2i+3)(p+q-2i+2} i = 1,2,...,q, A0+ q1

q+1 =

A0 j

(24.27)

_Pq Aq A.

p-q+1

Thus, the formula (24.26) has been established; Clebsch and Gordan gave the following expression for the coefficients A9: Ai A' - _

CvC9

C,+q-i+ I

=

p!q!(p+q-2i+1)!

i!(p-i)!(q-i)!(p+q-i+l)!

(24.28)

The truth of (24.28) is easily proved by induction. For q = I and q = 2, the formula (24.28) is correct (cf. (24.24) and (24.25)]. We find from (24.27) and (24.28) (for i = 1 , 2, ... , q)

+ (p-i+1)z P!q!(P+q-2i+1)! ql - i!(p-i)!(q-i)!(p+q-i+l)! (p+q-2i+3xp+q-2i+2)

A'+

X

p!q!(p+q-2i+3)!

(i-1)!(p-i+1)!(q-i+l)!(p+q-i+2)!

BINARY FORMS

282

CHAP. V

p-i+l 1 + p!q!(p+q-2i -I)! (i-1)!(p-i)!(q-i)!(p+q-i+1)! t I (q-i+lXp+q-i+2)1 _ p!(q+l)!(p+q-2i+2)! _

i!(p- i)!(q+ 1 -i)!Qp+q- i+2)! which is equal to the result obtained by replacing q by (q+ 1) in the last part of (24.28). It follows from this that the formula (24.28) is correct for

any q g, Theorem

26.3 is true for variance s. The case h < (g - 1) is easily settled; after striking out the first twos in the terms of the sum S2, the number of twos remains larger than the number of ones, and, by the assumption of the induction, S2 = 0. Next, striking out the first ones in the terms of SI , we see on the basis of the same inductive assumption that also S, = 0, so that S = 0. There still remains one possibility; h = (g - 1). Reasoning as above, we verify that for this case S2 is a linear combination of expressions of the

form (26.22), the indices of which now do not contain ones outside the alternations. If S2 # 0, QS2 will be a non-zero sum of terms of the form 0AI[1 ... 2)[12) ... [I ... 21...;

by (26.11), QS, will have the same appearance. Striking out the first indices in the terms of S, which are ones, we obtain a polynomial S, for which QS, is a non-zero linear combination of expressions of the form (26.3), which is impossible by Theorem 26.2; consequently, S2 = 0,

whence we conclude, as in the case h < (g -1), that also S = 0. Thus, Theorem 26.3 has been proved. It is not difficult to formulate a proposition for the operation 0 which is analogous to Theorem 26.3; as a consequence, we find Theorem 26.4:

If h ? (g -1), then all the forms (26.17) are linearly

independent.

In fact, let S be an expression of the form (26.10) of weight (h + 1, g -1); write it in the form of the right-hand side of (26.16). Then OS will be equal to the right-hand side of (26.18), where the vQ are defined by (26.19). Since (h + 1) > (g - 1), the expression OS cannot vanish, unless S is identically zero. Therefore the system of equations YàN,O = 0;

or, what is the same thing [cf. (26.20)],

Y µp0'=0,

1,2,...,N,

fl= N+1,...,M,

can only have zero solutions, and this means that the forms (26.17) are linearly independent. 26.3

Reverting to the original formulation of the problem, let the tensor A be a product of d tensors each of which is equal to the tensor

CHAP. V

BINARY FORMS

302

a,,,= ...,, which corresponds to the form (26.1), and suppose the equality (26.11) holds true identically with respect to the components of the tensor a. Then the tensor A exhibits a number of symmetry relations; by virtue of these relations, certain of its components Aa turn out to be equal to each other; we will assume that the corresponding AQ in S are also the same. For example, in the case of (22.20) for s = 4. the tensor Aijkl = a,faki

will be symmetric with respect to the indices i, j and k, 1, respectively; in addition, this tensor remains unaltered on interchanging the first pair of

indices with the second. Correspondingly, we must assume that in

(26.12) Al ='6,1.2 =A3=A4=A5. It is not difficult to understand that under these conditions in (26.16) all the p, which occur with the same A,, will be expressed by the same linear forms for different A2. On requiring that QS is to vanish identically with respect to the coefficients of the form (26.1), we find again that all

go = 0 (fi = N+1, N+2, ..., M). Thus, the A (x = 1, 2,

...

N) are

subject to the same linear relations as in the cases of 26.2, and consequent-

ly Theorem 26.3 also remains true for the case under consideration. The same arguments may be repeated for the case of several forms. Let the forms f1, f2, ... , f9 be written in Aronhold's symbolic notation

fl = (ax)" = (bx)" = .

.

f2 = (cx)'2 _ (dx)'2 = ..., (26.23)

then the results obtained may be formulated as Theorem 26.5: Let S be a homogeneous polynomial in the coefficients of each of the binary forms (26.23), isobaric of weight h with respect to x' and of weight g with respect to x2 and satisfying the relation (26.11). If

h < g, S = 0; if h z g, then S is a linear combination of expressions of the form

[ab][ac]

... [cd] ... a1 b1 ... d1 ...,

(26.24)

where the number of factors of the second kind is equal to g and the number

of indices which are ones and are outside the total alternations is (h-g). If d, is the degree of the polynomial S with respect to the coefficients

the forms fi, one has

h+g = > rid,, i=1

(26.25)

26.3-26.4

DIFFERENTIAL. EQUATIONS FOR INVARIANTS

303

whence the condition h >_ g is equivalent to the inequality q

I r; d; - 2g >_ 0.

(26.26)

+=I

By the fundamental theorem, S will be an invariant, if and only if

h = g, i.e., if r; d; - 2g = 0

(26.27)

[cf. (15.20)]. In the classical representation, the-form (26.27) of this condi-

tion is more convenient, since the weight with respect to x' as distinct from the weight with respect to x2 cannot be seen directly from S.

Next, consider the fundamental proposition underlying the CayleySylvester method: Theorem 26.6: In order for a polynomial I, homogeneous in the coefficients of each of the binary forms f, , f2 , ... , fq and isobaric of weight g (with respect to x2) to be a joint invariant of the forms, it is necessary and sufficient that it satisfy the differential equation (26.4), in which the operator

0 is defined by (26.8), and that the relation (26.27) hold true, where r; is the order of the.form f; and dI is the degree of the polynomial I with respect to its coefficients (i = 1, 2, . . ., q).

It is readily seen that in the formulation of Theorem 26.6 one can replace the. differential equation (26.4) by the equation (26.6), where 0 is defined by (26.9).

Following Cayley, a polynomial which satisfies the conditions of Theorem 26.5, is called a semi-invariant of the forms f, , f2, . . ., ./q (semi-invariants may also be defined as invariants of a certain subgroup of all linear transformations, cf. Exercise 2). Thus, Theorem 26.5 gives the general form of all semi-invariants. Semi-invariants of binary forms are closely linked to their covariants. 26.4

In fact, the semi-invariant (26.24) is the coefficient of the lowest power of

the variable x' in the covariant

[ab][ac]

...

[cd]

... (axXbx) ... (dx) ...;

analogous results apply with regard to any semi-invariant. This correspondence between semi-invariants and covariants is obviously singlevalued and invertible; the order p of the covariant corresponding to the

CHAP. V

BINARY FORMS

304

given semi-invariant is given by q

p = h-g = > r,d;-2g.

(26.28)

I=1

Cayley has shown how one can find from the leading coefficient of a covariant all the other coefficients. Let there be given a semi-invariant

S = Pa, b, ... k1 l1,

(26.29)

where P is a product of factors of the second kind; the quantity S is the coefficient of (x' )p in the covariant

K = P(ax)(bx) ... (kx)(Ix),

(26.30)

where p is defined by (2.6.28) and is equal to the number of factors of the first kind on the right-hand side of (26.30). The coefficient of (x')p-'x2

in K will be pPa(lb,

... k112);

it is readily seen that the last expression is equal to OS. In a similar manner, the coefficient Of (XI)p- 2(X2)2 is

P(P-1) 1 -2 Pa(lb,...k212)=

1

202S,

etc. The final step leads to the relation Op+'S = 0.

(26.31)

Thus,

K = S(x')p+OS(x')p-'x2+ -1-

02S(x')p-2(x2)2

2!

+ 1 03S(x1)p-3(x2)3+ 3!

... +

1

O°S(xz)p.

(26.32)

P!

The semi-invariant S thus defines all the coefficients of the covariant; therefore S is called the leading coefficient of the covariant K. Taking into consideration that 0 can be represented in the form of the differential operator (26.9), we draw from (26.32) the following conclusion: if a semi-invariant S vanishes identically, the corresponding covariant also vanishes identically with respect to the coefficients of the form and the variables x', x2; naturally, the converse is also true. If the covariants K1, K2, ... , Kq have leading coefficients S1, S2 1 ... , Sq,

26.4

DIFFERENTIAL EQUATIONS FOR INVARIANTS

305

respectively, the covariant F(K1, K2, ..., Kq), where F is an integral rational function, is readily seen to have leading coefficient F(SI, S21 ., SQ). From this follows: Theorem 26.7: Every integral rational relation between semi-invariants is equivalent to the same relation between the corresponding covariants. By virtue of the above, a study of the completeness of covariants can be reduced to a study of semi-invariants.

As an example of the application of Theorem 26.7, we give a new derivation of the syzygy (23.26). The leading coefficients of the covariants f, H and Q are ao, a0a2 -a2, a02 a3 - 3aoala2+2a;, respectively (cf. (14.31),

(14.34)]; the order of the invariant D is equal to zero, so that the corresponding semi-invariant is D [cf. (14.33)]. From the relation

(aoa3-3aoala2+2ai)2+a0D+4(a0a2-ai)3= 0

(26.33)

which is readily established, the syzygy (23.26) follows by Theorem 26.7.

In conclusion, we will acquaint ourselves with a rule, due to Cayley, for the computation of the number of linearly independent covariants of a binary form (26.1) of degree d in its coefficients and of weight g; by (26.28), the order p of such a covariant is rd- 2g and it must, of course, be non-negative [cf. (26.26)]. By virtue of Theorem 26.7, this problem can

be reduced to a search for the corresponding semi-invariants. Each term of such a semi-invariant has the form

caoai`...a;',

00+01+... +0,=d;

since the weight of the term (with respect to x2) is equal to g, we have the relation 0.00 + 1 .01 + 2. 02 + ... + r - 0, = g. (26.34)

Thus, the number of possible terms of the unknown semi-invariant is

equal to the number of representations of g in the form of sums of d integral non-negative terms each of which does not exceed r (where we allow some of the terms to be the same); let this number of representations be denoted by (g; d, r). Writing down all these terms with undetermined coefficients, we subject them to the operation Q; as a result, we obtain a polynomial of the same degree d and of weight (g - 1). As above, we verify that the number of its terms is (g - 1; d, r); equating their coefficients to zero, we find (g -1; d, r) linear relations between the undetermined coefficients of the semi-

BINARY FORMS

306

CHAP. V

invariant. The reasoning which led us to Theorcm 26.4 also remains applicable in this case, if in (26.16) and (26.18) we give different numbers only to those pp and v,, which occur in essentially different terms of the sums S and OS. Therefore all the relations above will be linearly independent and we have proved the following proposition stated by Cayley

and proved with full rigour by Sylvester: Theorem 26.8: The number of linearly independent covariants of weight g of a binary form of order r which are of degree din its coefficients is for rd-2g Z 0 equal to the difference

(g; d, r)-(g-1; d, r), where (g; d, r) is the number of possible representations of g in the form of sums of d integral non-negative terms not exceeding r some of which may be identical; however, if rd- 2g < 0, there are no such covariants. For example, the number of linearly independent covariants of a cubic binary form of degree 6 and weight 6 is (6; 6, 3) - (5; 6, 3) = 7 - 5 = 2. By Theorem 23.2, such covariants will be Df 2, H3, Q2; they are interrelated by the relation (23.26) whose existence is revealed in this way. Exercises 1. What happens to an expression S of weight (h; g) [cf. (26.10)] when it is subjected

to the operation OS1? Further, prove that OQS-QOS - (g-h)S. 2. Show that every semi-invariant of a system of binary forms is an invariant with respect to the group of transformations TP:

x =- x'+py, y , y'

(26.35)

and that, conversely, every isobaric polynomial in the coefficients of several binary forms, homogeneous with respect to the coefficients of each of the forms and invariant under the transformations (26.35), is a semi-invariant. On the basis of this result give a new (independent of § 17) proof of the fundamental theorem for the case of invariants of binary forms. 3. Prove that every linear transformation of space of second order can be represented

as a product of several transformations of the form Tp ]cf. (26.35)), TI, : x = x',

y= px'±y' and Sk: x = x', y =ky'. 4. On the basis of the results of 2 and 3, prove the following theorem: an isobaric and homogeneous polynomial I in the coefficients of the binary form (13.10) is an invariant, if it satisfies the equation (26.4), where 9 is defined by (26.5), and if it only acquires a numerical factor on replacing at by ar_k (k = 0, 1, 2, ..., r). Generalize to the case of invariants of several binary forms. 5. If everywhere in a semi-invariant of weight g of the binary form (13.10) we replace ao by f and ak by

(r-k)!

i3kf

r! _ (a)

k = 1,2. ,r (x'=x,xt=y),

26.4

DIFFERENTIAL EQUATIONS FOR INVARIANTS

307

we obtain the covariant which corresponds to the semi-invariant multiplied by (x')o. Prove this and verify it for the discriminant of a quadratic form and for the Hessian of a cubic form. 6. On the basis of Theorem 26.6, find for the binary quadratic form (13.7) its second degree invariants (with respect to the coefficients of the form). 7. Starting from Theorem 26.6, find for the binary cubic form (13.8) its invariants of fourth degree. 8. On the basis of Theorem 26.6, find for the binary form (13.10) all its second degree invariants. 9. Show that a binary quadratic form and a binary fourth order form do not have a joint invariant which is linear in the coefficients of both forms. 10. Find all joint invariants of the quadratic and cubic binary forms w and f which are of first degree in the coefficients of the form q, and of second degree in those of the form f; write 'heir expressions in the symbols of Aronhold. 11. For a binary cubic form find the covariants of weight 4 and degree 4 in the coefficients of the form, proceeding as stated at the end of 26.4. 12. For the covariants H and Q of a cubic binary form (cf. § 14, Example 9) and the Hessian of the fourth order binary form (13.9) find all coefficients, starting from the leading one, on the basis of the formula (26.32); as a check, use the relation (26.31). 13. On the basis of the definition of a semi-invariant (cf. beginning of 26.4) for the binary form (13.10) find its semi-invariants of weight 3 and degree 3. Show that the

corresponding covariants differ only by numerical factors from the Jacobian of the form and its Hessian. 14. Verify that S = aoa, -4aoala3-r6aoaia2-3a; is a semi-invariant of the binary form (13.10), if r z 4. For the case r - - 4, express the corresponding covariant in terms of the complete system of concomitants (cf. Theorem 25.1). 15. The last part of (14.3), as an invariant of a cubic binary form, will only be a semi-invariant for a fourth order form. Find for the last case an expression for the corresponding covariant in terms of the complete system of concomitants (cf. Theorem 25.1); use the result for a new derivation of the syzygy (25.5). Explain also the relation

to § 24, Exercise 10. 16. On the basis of Theorem 26.7 for a cubic binary form, express the Jacobian (Q, 2H) in t ms of the complete system of concomitants [cf. Theorem 23.1, (19.16) 1. 17. Starting from Theorem 26.7 for a fourth order binary form f, express the Jacobian (Q, f) in terms of the complete system of concomitants (Theorem 25.1, cf. § 20, Exercise 19).

18. Show that in the formulation of Theorem 26.8 one may replace (g; d, r) and (g-l; d, r) by (g; r, d) and (g-1; r, d); hence derive the reciprocal law of Hermite: the number of linearly independent covariants of order p and degree d of an r-th order binary form is equal to the number of linearly independent covariants of order p

and degree r of a binary form of order d. Explain by use of this law the result of 8 (an invariant is a covariant of order zero). 19. Prove that a binary form of order r has a single linearly independent invariant of degree 3 or none at all depending on whether r is or is not divisible by 4. 20. Show that a binary form of order r has a single linearly independent covariant of order r and degree 2 or none at all depending on whether r is or is not divisible by 4.

21. Prove that a covariant of a binary form of degree 2 cannot be of odd weight.

CHAPTER VI

TERNARY FORMS. SECOND ORDER TENSORS

§ 27. The quadratic ternary form Beyond the limits of binary forms we come into contact with a part of invariant theory which has received considerably less attention. The two fundamental problems (cf. 21.5) have been solved for the case of any it only for second order tensors; in addition, ternary forms and skew-symmetric tensors (polyvectors) have been subjected to systematic study. In the remaining areas of the theory, progress has been casual and bears a disorderly character. We will become acquainted with the 27.1

fundamental and not too complicated results in these directions in Chapters VI and VII; with this object in mind, it will be convenient to begin with a study of quadratic ternary forms which will serve us simultaneously as an introduction to the theory of ternary forms and to the

theory of n-nary quadratic forms (i.e., of symmetric second order tensors in a space of arbitrary order n). 27.2 First of all, we will seek a complete system of concomitants of the quadratic ternary form

f = ai jxlxj = (ax)2 = (bx)2 = (cx)2 = (dx)2 = ... , art = a j1; (27.1) we will understand here by the concomitants of a form its invariants, covariants (i.e., the invariants of the tensor aij and of the contravariant vector x), contravariants (invariants of the tensor a, j and of the covariant vector u) and mixed concomitants (invariants of the tensor a,1 and the vectors x and u). By the fundamental Theorem 18.2, every such conco-

mitant is a linear combination of products of factors of the form (ax), (ux), [abc], [abu]. Consider one of the products of the type stated, de-

noting it by K, and assume at first that K contains the factor [abc]; in that case, the three symbols a, b, c must occur again in this product. If two of these symbols occur together in a square bracket, the quantity K

27.1-27.2

QUADRATIC TERNARY FORM

309

may have the following two forms:

K = [abc][aba][cfly] ...

(27.2)

K = [a bc] [a ba](cx) ...,

(27.3)

or

where the place of the symbols a, fi, y, ... is taken either by symbols d, e, ..., parallel to the symbol a, or by the vector u. On the right-hand side of (27.2), perform an exchange of the parallel symbols a and c and then the exchange b4-+c; taking the arithmetic mean of the three expressions for K, we obtain

K = }[abc]{[aba][cfy] - [cba][aby] - [aca][bPy]} ... = {by (19.4) for n = 3) = j[abc]2[afly]... Thus, a factor equal to the discriminant of the form D = f[abc]2

has been split off from K [cf. (18.28)]. The same factor is also split off from the concomitant (27.3), if we subject it to the same operation. However, if a, b, c enter the second time in three different factors, we have K = [abc][aa/3][by6]... or

K = [abc][aai6](bx) ...

(where the symbolic expression [abc] (ax)(bx) vanishes identically); the symbols a, fi, y, 6.... have here the same significance as in (27.2) and (27.3) above. Executing in each of the products the transposition a.--- b,

taking the arithmetic mean and applying again the identity (19.4), we arrive at the case already considered.

Splitting off from K a factor equal to the discriminant of the form in this manner, we can get rid of those factors of the second kind in which all vectors are parallel to a; if K after this still contains factors of the second kind, they must all be of the type [abu]. Thus, we have still to consider products of the form K = [abu][acu][bdu]...

(27.5)

K = [abu][acu](bx)...

(27.6)

and

310

TERNARY FORMS

CHAP. VI

We will apply the same method once more: transpose the parallel symbols a and b and take the arithmetic mean of the expressions obtained. In the case of (27.6), this process gives

K = +[abu]{[acu](bx)-[bcu](ax)) = [cf. (19.4)] = J[abu]{[abu](cx)- [abc](ux)).

In the first term, the contravariant

F = I[abu]2;

(27.7)

has been separated off; the second term contains the factor [abc] and can be subjected to the operation described above. The same effect is obtained by the method stated in the case of the expression (27.5), the only differ-

ence being that there is no second term. Thus, splitting off from the product K under consideration the factors

D and F, we get rid of all factors of the second kind; the residue will contain only factors of the form (ax), (ux) and is easily seen to represent a product of powers of the form f and powers of the concomitant (ux). Thus, every concomitant of the form (27.1) is a polynomial in f, D, F and (ux).

The irreducibility of the invariant D is obvious: since [aba] = 0, the representation of every invariant which does not vanish identically requires at least three different parallel symbols ind, consequently, its degree in the coefficients of the form cannot be less than three. A similar reasoning leads to the irreducibility of the contravariant F, and we arrive at

Theorem 27.1: A complete system of concomitants of the ternary quadratic form (27.1) consists of the form itself, its dicriminant D, the contravariant F [cf. (27.7)] and the mixed concomitant (ux). The projective geometric meaning of the equalities D = 0 and F = 0 has been given in § 14, Examples 4 and 5: under the condition D = 0, the second order curve Q with the equation

(ax)' = 0,

(27.8)

has a singular point; F = 0 is the tangential equation of the same curve (for D 0). 27.3

Next, we will establish the canonical forms of quadratic ternary forms. We begin with the case when the rank of the form p = 3; it will

311


27.2-27.3

be convenient to employ here the geometric results studied in 6.6. Select as first and second coordinate points two arbitrary points x, y which do not lie on the curve (27.8) and are conjugate to it, and as third coordinate point the pole z of the straight line joining the points x and y. Then the three points x, y, z will be conjugate in pairs; they are said to form a polar triangle of the curve (27.8). The condition of conjugacy

(6.33) gives a12 = a13 = a23 = 0, so that (27.1) assumes the form a.1(x')2+2(X2)2+3(X3)2,

D = A11223 # 0.

(27.9)

= VA.,X1

(27.10)

The linear transformation

x1 =

X1,

x2 = J22X2,

z3

now reduces it to the canonical form

I (x')2+(x2)2+(x3)2.

(27.11)

If the rank p < 2, the curve (27.8) has one or several singular points; if y is such a point, one has (cf. (14.8)) a; j y' = 0.

(27.12)

Let y be the third coordinate point; then we obtain from (27.12) a13 - a23 = a33 = 0; the form (27.1) becomes a binary form in the variables xt, x2. By virtue of the results of 22.4, we conclude from this that for p = 2 and for p = I a quadratic ternary form may be transformed into the canonical forms

II (XI)2+(X2)2,

III

(x1)2,

(27.13)

respectively; for p = 0, the form vanishes identically. Next, consider the forms assumed by the polar and tangential equations as well as the equation F = 0 in the case when the left-hand side of the

equation of the second order curve belongs to Types II and III. For p = 2, the form (27.1) can be rewritten

j = (px)(qx),

(27.14)

where p, q are real coordinate vectors which are not collinear; the curve (27.8) degenerates into two straight lines p and q. For the singular point y, we find from (27.12) p1(gy)+g1(py) = 0,

TERNARY FORMS

312

CHAP. VI

or, by virtue of the linear independence of the vectors p, q,

(py) = 0,

(qy) = 0,

i.e., the singular point is the point of intersection of straight lines p and q.

The equation of the polar of the point x0 with respect to the curve (27.8) will be (27.15) (axo)(ax) = 0 [cf. (6.34)]; for (27.14), this equation assumes the form

(Pxo)(gx)+(gx0)(Px) = 0.

(27.16)

If x0 is a singular point, the left-hand side of (27.16) vanishes identically;

for singular points, the concept of the polar loses its meaning. However, if x0 is not a singular point, it is seen from (27.16) that the polar is always

a straight line which passes through the singular point of the curve (cf. Exercise 9). In the case when the point x0 lies on the curve (27.8), (27.15) is the equation of the tangent at the point x0. If the curve degenerates into two straight lines p and q and the point x0, not being singular, lies on the straight line p, i.e., if (pxo) = 0, (qxo) # 0, the equation (27.16) becomes

(px) = 0. Thus, we have Theorem 27.2: If a second order curve Q degenerates into two different straight lines p, q and the point x0 lies on one of these straight lines, but not on the other, the tangent to the curve Q at the point x0 coincides with the one of the lines p, q on which the point x0 lies.

For the form (27.14), the contravariant F will be

F = - j [Pqu]Z (cf. § 18, Exercise 19), as a consequence of which F = 0, i.e.,

[pqu] = 0, denotes (cf. 6.2) that the straight lines p, q, u pass through one point; in other words, the straight line u passes through the singular point of the curve Q. The quadratic ternary form of rank p = 1 is given by

f = (px)2,

(27.17)

where p is a real vector; the curve f = 0 represents two coincident straight


27.3-27.4

313

lines p, p. As is readily seen, every point of such a second order curve is singular. For a non-singular point x0, i.e., for (pxo) # 0, the equation (27.15) becomes (px) = 0, i.e., the polar of the point coincides with the straight line p. For the form (27.17), the contravariant F vanishes identically.

The results above provide all the necessary information for the classification of quadratic ternary forms in the complex domain and specification

of the arithmetic and algebraic characteristics (the first of these being given by the rank p) for each type: Type

C

a.- -f- .

1

I

Arithmetic characteristics

p=3 p=2

(x')'-l- (x2)' f (x')2 (x')'+ (xs)2 lI IV

(xl)2

p-1


p = 0


D*0 F=0,f* 0

D=0, Fj6 0

f=0

In proceeding to the classification in the real domain, we note that we can restrict consideration to quadratic ternary forms with real coeffi27.4

cients of rank p = 3; if the rank of the form p S 2, the corresponding curve has a real singular point and the form can be converted into a binary

form by means of a real linear transformation (cf. 27.3); however, the classification of binary forms in the real domain has already been given in 22.4. A form of rank 3 can be reduced to the form (27.9) by a real linear transformation; for this purpose, it is sufficient to select a coordinate polar triangle with three real vertices, and such triangles obviously will always exist. The transformation (27.10) may turn out not to be real; therefore we replace it by the transformation

x3 = V IA3IX3, as a result of which the form will be of one of the following types: XI = V IAIIXI,

Ia 1°

X2 = Y JA2Jx2,

(Xt)2+(X2)2+(X3)2,

(X1)2-(X2)2-(X3)2,

Ib

(XI)2+(X2)2-(X3)2,

Id

-(X')2-(X2)2-(X3)2

(27.18)

(after changing, if necessary, the numbering of the coordinates).

It is not difficult to see that the number of plus (+) and minus (-) signs in each of the canonical forms (27.18) is an invariant for linear real

CHAP. VI

TERNARY FORMS

314

transforma*ions. In fact, for the forms I' and I` the discriminant D is positive and for the forms 1b and Id it is negative; since the weight of D is equal to 2, the sign of D is invariant with respect to linear real transformations. The forms P and P differ from each other by the fact that the first

is zero only for xt = x2 = x3 = 0, while for the second there exist nonzero real values of the variables xt, x2, x3 which cause it to vanish; the same distinction, exhibiting an invariant character in the real domain, also holds between the forms Ib and 1d. Thus, a classification of quadratic

ternary forms in the real domain has been established. The duality principle permits us to transfer all the results to the case of quadratic ternary forms 27.5

rp = a'lu;u; = (au)2 = (bu)2 = ...,

a'1 = a-',

(27.19)

involving the covariant vector u. In fact, if the rank of the form rp is equal to 2, then tp = (p`,u)(qu), where p, q are real, non-collinear, contravariant vectors. The straight line u, satisfying the equation qp = 0, either passes

through the point f or through the point q; therefore it is said in such a case that the curve of the second class 4p = 0 degenerates into two distinct

points fl, q. If the rank of rp is 1, then rp = (Pu)2 and the curve of the second class V = 0 represents two coincident points. Exercises In Exercises 1-17, the symbol! denotes a quadratic ternary form whose tensor and symbolic representations are given by (27.1). 1. Show that a form f of rank 3 can be reduced to the form 2xlx3+(x=)=. What will be the position of the coordinate triangle with respect to the curve f = 0, if f is reduced to this form?

2. Let

(x'): = (ax)a:, (u')' = i[abu][ab]', so that x is the polar of the point x with respect to the curve f = 0 and u' is a pole of the straight line u with respect to the same curve, fix, y) = (ax)(ay) and D is the discriminant of the form f. Prove that

f(x, u') = D(xu),

[xyl' _ [x'y'], [x'y'z'] = D[xyz],

and explain the geometric meaning of these relations. 3. Let there be given four points 1, 2, 3, 4 three of which are independent. Show that the equation of any second order curve through these points has the form

R[x23][x14]+µ[x13][x24] = 0.

(27.20)

On the basis of (27.20), establish that one and only one second order curve passes through five points no four of which lie on one straight line.

27.4-27.5


315

4. Show that six points x, y, 1, 2, 3, 4 lie on one second order curve, if and only if [x231 [x141 [y13)[y241-- [x13][x241 [y23][y14] = 0

(27.21)

(the lack of symmetry of this condition with respect to all six points is only apparent). Represent this condition in the form of an equality between two cross-ratios { [xl ], [x2], [x31, [.v41} _ { [yl 1, [y21, 1y31, 1y41}

and hence derive Steiner's Theorem: every second order curve can be regarded as the geometric locus of the points of intersection of corresponding straight lines of two projective pencils').

5. In the notation of 4 and § 12, Exercises 4-6, let p = [131, p' = [241, q = [3y], r'=[xl]. Show that [[pp'J[gq'J[rr'1] is equal to the left-hand side of (27.21), whence follows Pascal's Theorem: A hexagon I3y42x is inscribed in a second order curve, if and only if the three pairs of its opposite sides p, p'; q. q'; r, r' intersect in three points which lie on a straight line.

q'= (2x], r

6. Assume the points y, 1, 2, 3, 4 in (27.21) to be fixed and the point x to be variable.

Find the discriminant of the form in x on the left-hand side of (27.21) and the conditions under which the curve (27.21) degenerates into two straight lines. 7. Construct the equation of a second order curve passing through the four points 1, 2, 3,4 and touching the straight line a at the point 1; in addition, find the discriminant of this curve. 8. Formulate the propositions which are dual to those stated in 3, 4, 5. 9. A second order curve Q has the equation f = 0, where f is defined by (27.14). Now are the point x and its polar it with respect to the curve Q located with regard to each other? 10. Let the coefficients of the form f be real; a straight line u is said to be a secant with respect to the curve f = 0, if its points of intersection with this curve are real and different, and it is said to be external, if they are imaginary. Under what conditions will the straight line it bea secant and when will it be external?

11. In the notation of 10, a point is said to be external with respect to the cure:

f = 0, if one may draw from it two real tangents to this curve, and internal. if the stated tangents are imaginary. Under what conditions will the point x be internal and external, respectively?

12. What is the character (in the sense of 10 and 11) of the vertices and the sides of any polar triangle of the curve f = 0' Exercises 13-16 deal with affine properties of second order curves: the improper straight line will be denoted by I. 13. Write down the equation of a pair of improper points of the curve (27.8). 14. Given two diameters g and h of the curve (27.8), write down the condition under which they will be conjugate.

IS. Find the cross-ratio formed by ttft two diameters g and h of the curve f - 0 and its asymptotes. 1,6. Given a point y and a central non-degenerate second order curve f = 0 with Centre at the point z, find the ratio in which the polar of the point y divides the segment yz. ') Two pencils of straight lines are said to be projective, if one can establish relation.

ship between them such that the cross-ratio of four straight lines of the first pencil is always equal to the cross-ratio of the corresponding straight lines of the second pencil.

316

TERNARY FORMS

CHAP. VI

17. For the case of a circle in metric geometry, establish the link between the results of 16 and § 1, Exercise 7. Exercises 18-29 present several deductions of the theory of pairs of quadratic ternary forms

f = (ax)' -_ (bx)2 = (cx)2 = ... , 9, _ (ax)' = (fix)' = (yx)' _ ..., (27.22) where the following notation will be employed: D,11 = *[abc]',

D11, _ f [aba]',

D122 = J [aafi J',

D,,, = [apy J',

F11 = k[abu)',

Fu = [azu]', F,, = f(afJuJ'

(27.23)

(27.24)

For many of these exercises, the application of (12.15) is essential. 18. Show that in the pencil of second order curves

f4 AT = 0 only those curves for which A satisfies the equation

(27.25)

DI11+D31IA+"D,,,R'+D,,,J.' = 0 (27.26) degenerate into two straight lines; the equation (27.26) is called the characteristic equation of the forms f and W. 19. Prove that if for A = 2, a curve of the pencil (27.25) degenerates into two straight lines, its singular point has one and the same polar with respect to all curves of the pencil. 20. Prove that if all roots of the characteristic equation (27.26) are different, the curves f = 0 and q) = 0 have a common polar triangle.

21. Letting F11 = (du)' = (bu)' _ (eu)' =

. .

., F

prove that

[daxJ(c4)(au)(cx) = 0, }[dbc]' = Di,1, f[dbbl]' = D11, . D:,,. 22. Express the discriminant of the form F1, in terms of the i nvariants (27.23) of the forms f and q'. 23. Show that the Jacobian of the forms F11, F,,, F. is equal to J = }[abcl [afiyl [bcu][)3yu][aocuj,

(27.27)

and explain the projective geometric meaning of J = 0, under the assumption that all roots of the characteristic equation of the forms f and 9' are distinct (cf. 18).

24. Show that the point equation of the curve [aau]' = 0 is

D'1,9'+Da,f-}flab] [ari]xJ' = 0. 25. Prove that the envelope 1' of the polars of the points of the curve i = 0 with respect to the curve f = 0 has the tangential equation

D1aFii-DIIIFis = 0. 26. Explain the projective geometric significance of the equalities [abaJ[abu]((xx) = 0,

(27.28)

[aauJ(ax)(ax) = 0,

(27.29)

[a-jjbcaj[bcu](ax) = 0.

(27.30)

GEOMETRIC INTERPRETATION OF CONCOMITANTS

28.1

317

27. The affinor k1 = (1/2)[abx][ab)'a; corresponds to a concomitant of the left-hand side of (27.28); how are the collineations determined by it related to the curves f --- 0

and q) = 0? Express the invariants S 13, /, of the affinor k (cf. (14.39), (14.51)] in terms of the invariants (27.23) of the forms f and ¢.

28. Prove that if the poles of the straight line u with respect to the curves f = 0 and [aau]' = 0 lie on the same straight line as a point x, then [arulIbca][bcu](ax) = 0 (cf. (27.30)].

29. Show that the discriminant of the left-hand side of (27.29) is equal to J/4 [cf. (27.27)] and explain the geometric meaning of the result [under the assumption that all the roots of the characteristic equation of the forms f and (cf. 18) are distinct].

§ 28. Geometric interpretations of the simplest concomitants of ternary

forms of order r 28.1

Setting the ternary r-th order form

f = (ax)' _ (bx)' = (cx)'

(28.1)

equal to zero, we obtain the equation of an r-th order curve in projective coordinates. A study of the projective properties of this curve leads to the geometric interpretation of certain concomitants of the form (28.1). First of all, we will find the intersections of the curve

(ax)' = 0

(29.2)

with a straight line determined by two of its points x0 and x. A contravariant vector xo+Ax corresponds to any point of this straight line (different from x); the values of A for the unknown points of intersection are given by the equation (a(xo+Ax))' = 0 or, in expanded form, by

(axoY+Ar(axo)'-t(ax)+A2 +jr- 2

r(r2 1)

(axjy-2(ax)2+ .. .

r(r2 1) (axo)2(axY - 2 +, r- t r(axoxax)' - t

+ 2'(ax)' = 0. (28.3)

If the straight line under consideration does not form part of the curve (28.2), one can assume, without reduction of generality, that (ax)' # 0. Therefore (28.3) leads to Theorem 28.1: A curve of order r intersects any straight line, which does not form part of it, in r points (some of which may coincide or be imaginary).

CHAP. VI

TERNARY FORMS

318

The coefficients of the different powers of A in (28.3) are directly related

to the polars of the form f (cf. 20.4): The coefficient of A'-', with the numerical factor r omitted, represents the first polar Dsso f, the coefficient

of A", omitting the factor r(r-1)(2, is the second polar Dxxof, etc. From equation (28.3) we readily find the equation of the tangent to the curve (28.2) at the point x0. Among the r points of intersection of the tangent with the curve (28.2), two or more must coincide with x0. Therefore, if x is a point on the tangent, the equation (28.3) must have a root A = 0 with multiplicity > 2; this will be true, if and only if

(axo)'-'(ax) = 0,

(28.4)

since (axo)' = 0: thus, (28.4) also represents the equation of the tangent to the curve (28.2) at the point x0. This conclusion is true under the condition that the left-hand side of (28.4) vanishes identically with respect to x, i.e., if (ax0)'- 1 a1 = 0,

i = 1, 2, 3,

(28.5)

the point x0 is said to be a singular point of the curve (28.2) [the point x0 lies on the curve, since, obviously, it follows from (28.5) that (axo)' = 01. For the simplest of the singular points one will have (ax0)r-2(a.Y)2 q 0

for any x; they are called double points of the curve. As is shown by (28.3), any straight line through a double point x0 intersects the curve (28.2) in points at least two of which coincide with x0; if the number of points coinciding at x0 is three or more than three, the straight line is said to be a tangent to the curve (28.2) at the double point x0. In this case, the

root i = 0 of the equation (28.3) must have multiplicity not less than three; the point x lies on the tangent at the point x0, if (axo)r-2(ax)2 = 0.

(28.6)

(ax)2 = (axo)'-2(ax)2;

(28.7)

Introduce the notation by (28.5), one has

(xxo)xj _

(axo)'-,a`

= 0,

where xo is a singular point of the second order curve (28.6). Consequently

(cf. 27.3), the curve (28.6) degenerates into two straight lines through

28.1-28.2


319

the point xo; they will be tangents to the curve (28.2) at the double point x0 (cf. Exercise 2).

28.2 We will now derive the tangential equation of the curve (28.2); for this purpose, we employ the following Principle of Clebsch: Theorem 28.2:

Let the vanishing of the invariant

[ab]'[ac]B

...

(28.8)

of the binary form (p = (ax)' = (bx)' = (ex)' = ... denote that r points whose coordinates satisfy the equation 4p = 0 possess a certain projective geometric property (E). Then the contravariant

[abu]'[acu]B

...

(28.9)

of the ternary form (28.1) vanishes for those and only those straight lines u which intersect the curve (28.2) at points possessing the property (E). In order to prove Clebsch's Principle, it is sufficient to select the straight line u as the third coordinate line (u1 = u2 = 0, u3 = 1). Then the points of intersection of the straight line with the curve f = 0 will have the coordinates x', x2 satisfying the equation cp = 0, and the contravariant (28.9) will become the invariant (28.8) (cf. § 14, Example 5).

For the sake of simplicity, let first r = 3. The straight line u touches the third order curve (aX) 3 = 0,

(28.10)

if some of the three points common to the straight line u and the curve

(28.10) coincide. One or two of the points determined by the cubic binary form (18.34) will merge with the third, if and only if the discrimi-

nant of the form vanishes (cf. 23.1); by Theorem 28.2, we obtain the condition for tangency of the straight line u to the curve (28.10), i.e., the tangential equation of this curve, if in the symbolic representation of the discriminant we replace all factors of the form [ab] by [abu] and equate it to zero. Consequently [cf. (18.40)], the tangential equation of the curve (28.10) will be +[abu]2[cdu]2[acu][bdu] = 0,

(28.11)

where a, b, c, d are parallel symbols. If the third order curve (28.10) has a singular point x0, every straight line u through this point likewise intersects the curve in three points, two (or all three) of which coincide and, consequently, satisfy the equation

CHAP. V1

TERNARY FORMS

320

(28.11). Therefore the left-hand side of (28.11) contains in each case a factor of the form (uxo); we obtain the real tangential equation of the curve by omitting this factor. The degree of the tangential equation of a curve is said to be its class (§ 10, Example 3); thus, the class of a third order curve does not exceed six.

In order to transfer this reasoning to the case of curves of any order, one must employ the discriminant of a binary form of order r; this is thu name given to the function of the coefficients of the form which vanishes when all points determined by the form and the straight lines coincide. If a form is given in terms of an expansion in linear factors 1

2

(Px)(Px) ... (px), its discriminant D will be 12

13

r-1r

D = [PP]2[PP]2... [ p p]2, whence it is seen that D is an invariant of the form of weight r(r- 1) (cf. § 19, Exercise 11). Since the discriminant is a symmetric function of 1

2

r

the vectors p, p, ..., p, it is an integral rational function of the coefficients of the form (28.1). Applying Clebsch's Principle to the symbolic represen-

tation of the discriminant D, we obtain the tangential equation of the curve (28.2). The degree of the tangential equation with respect to u is then equal to the weight of the discriminant; by the principle stated above, this degree decreases in the presence of singular points on the curve. Thus, one has

Theorem 28.3: The class of a curve of order r is not larger than r(r - 1)/2. In order to find the tangential equation of a curve given by an actual equation, it is often more convenient to proceed in the manner illustrated by the following example. 1. Find the tangential equation of the curve

x3+y3+6xyz = 0

(28.12)

(where, for the sake of simplicity, we have written x, y, z instead of x1,

x2 x3). We will seek the points of intersection of the straight line

ux+vy+wz = 0

(28.13)

28.2-28.3


321

and the curve (28.12); letting u # 0, we eliminate x between (28.12) and (28.13) to find (p = 0, where cp =

(1,

'Y+ wz)3 -u3y3 +6u2yz(vy+ wz).

First, compute the Hessian h of the form gyp:

h = -u2{(vw+2u2)(uw+2v2)y2

- w(uw2+4u2v+4v2w)yz+2w2(vw+2u2)z2}. Four times the discriminant of the Hessian h will be the discriminant D

of the cubic binary form cp (§ 14, Example 9); having found D in this manner, by equating it to zero and dividing by u6 # 0, we obtain w2{24uvw2 +48u2v2 +32w(u3+v3)- w4} = 0.

The factor w2 corresponds to the singular point (0, 0, 1); consequently, the required tangential equation will be

24uvw2+48u2v2+32w(u3+v3)-w4 = 0, and the class of the curve (28.12) is four. The limitation u 0 is not essential (cf. the Note in 15.1). The compu-

tations would have been simpler, if we had eliminated the variable z from.(28.12) and (28.13); however, in that case we would have lost the factor w2. 28.3

In conclusion, consider the problem of points of inflection of algebraic curves. A point x0 is said to be a point of inflection of the r-th order curve (28.2), if x0 is not a singular point of it and if the tangent at the point x0 to the curve (28.2) has at least three points in common with the curve which coincide at x0 *). If the point x lies on the tangent at the point of inflection x0, the equation (28.3) must have a root A = 0 of multiplicity ? 3. It follows from this that for any x satisfying equation (28.4) the form (axo)'-2(ax)2 must also vanish, and this can only happen when it is divisible by the left-hand side of (28.4). A quadratic ternary form, which is divisible by a linear form, either vanishes identically or is itself a product of two linear forms; in either case, its discriminant must *) From the point of view of Euclidean geometry, this means that the curvature of the curve vanishes at the point x0; thus, the definition of points of inflection presented here is somewhat different from that in differential geometry.

CHAP. VI

TERNARY FORMS

322

be equal to zero (cf. 27.3). Thus, we see that the relation o[abc]2(axo)r-2(bxo)r-2(cxo)r-2

=0

(28.14)

must be satisfied by x0.

In other words, the points of inflection x0 must lie on the curve H = 0, where 11 =

*[abc]1(a.x)'-2(bx)'-2(cx)'- 2

is the Hessian of the form (28.1) [cf. (18.33)]. We will call the curve H = 0 the Hessian of the curve (28.2). Arguments similar to those in 28.1 show that for singular points one also has the relation (28.14), so that singular points of a curve likewise lie on its Hessian. Conversely, let the point x0 of the curve (28.2) lie on its Hessian, so that the relation (28.14) applies to x0. Then the second order curve (28.6) degenerates into two straight lines p and q and, by virtue of the equality (axo)' = 0, the point lies on this curve. If x0 is a common point of the straight lines p and q, then [cf. (28.7) and 27.3] (axo)a; = 0,

(axo)'- Ia1 = 0,

i = 1, 2, 3,

(28.15)

and [cf. (28.5)] x0 is a singular point of the curve (28.2). *) However, if x0 lies on one of the straight lines p, q, say on p, and not on the other, the relation (28.15) does not apply, i.e., the point x0 is not singular; the tangent to the curve (28.6) at this point

()rx0)(ax) = 0

(28.16)

[cf. (27.15) and Theorem 25.21 coincides with the straight line p. But, by (28.7), (axo)(a.x) _ (axo)'-l(ax)

and, by (28.4), the straight line (28.16) is tangential to the curve (28.2) at the point x. Consequently, one will have for any point x the tangent (28.4) and the equality (28.6), and the multiplicity of the roots A = 0 of the equation (28.3) will be not less than 3; three or more of the points common to the tangent (28.4) and the curve (28.2) will coincide with x0, and this means that x0 is a point of inflection of the curve (28.2). Thus, we have ) We arrive at the same conclusion, if the left-hand side of (28.6) vanishes identically

with respect to x.

28.3


Theorem 28.4:

123

Any of the points of intersection of a curve of order r

with its Hessian is either a singular point of this curve or a point ofinflection.

Conversely, every singular point or point of inflection of an r-th order curve lies on its Hessian. Exercises 1. What is the curve (28.2), if the rank of the form (28.1) is less than three? 2. A singular point x0 of the curve (28.2) is said to be of multiplicity p, if !;;1- 0 for some x; show that the set of p tangents to (ax)v-' = 0 for any x and (ax0)r_D

the curve (28.2) at the point xo with multiplicity p is determined by the equation (axp)r_.D(ax)y = 0.

3. In the notation of § 27, Exercises 18-29, give a geometric definition of the curve

[aauJ2 = 0 in its relationship to the curves f = 0 and q' = 0. 4. Given the two second class curves (du)2 = 0 and (bit)" = 0, find the equation of the geometric locus of points from which the tangents drawn to the first curve divide harmonically the tangents drawn to the second curve. 5. In the notation of § 27, Exercise 18-29, show that the geometric locus of points whose polars with respect to the curve q' = 0 intersect the curves f == 0 and qV = 0 at harmonically conjugate pairs of points has the equation Dtttf-D,ttpp = 0.

6. If two ternary forms f = (ax)e and F = (bu)t satisfy the relation (ab)2 == 0, the curves f = 0 and F = 0 are said to be apolar. Show that apolar curves have the following properties: if it is a tangent to the curve F 0 and x the pole of the straight line It with respect to the curve f 0, the tangents drawn from the point x to the curve f = 0 divide the tangents drawn from the same point to the curve F - 0 harmonically.

7. Let f - 0 and F = 0 be apolar curves of second order (cf. 6). Prove that the polar with respect to the curve F = 0 of any point x lying on the curve f = 0 intersects the curves f -= 0 and F = 0 in two harmonically conjugate pairs of points. 8. Given three second order curves (ax)2 = 0, (ax)2 = 0, ()x)2 = 0, find the envelope of straight lines which intersect them in three pairs of points corresponding to each other in the same involution.

9. Given the fourth order curve f = 0, where f = (ax)4 = (bx)4 = (cx)' _ ..., state the geometric definition of curves (K) and (L) with tangential equations (abu]4= 0

and [abu]t[acuJ1 [bcu]t = 0, respectively, in'their relationship to the fourth order curve above. What can be said about a straight line It, if it touches both curves (K) and (L)? 10. A concomitant 0 of the form (28.1) is defined by the equation

0 = 4 [abu]t(ax)'-t(bx)'-t; show that

0=-

fit

fi3

UI

f33

Ut

At At 133

U3

Ut

0

fee

At ftt Ul

U3

(28.17)

where theft, are the same as in (14.24). If the concomitant 0 vanishes for some u and every x, what property has the straight line u with respect to the curve (28.2)? 11. A ternary form f represents the r-th power of a linear form, if and only if its concomitant 0 (cf. 10) vanishes identically with respect to u and x. Prove this statement.

TERNARY FORMS

324

CHAP. VI

In Exercises 12 and 13, the coordinates of points are denoted by x, y, z (instead of by superscripts), the coordinatcs of straight lines by u, r, w. 12. Find the tangential equations of the curves:

a) x'+ y'+z'+6lexyz = 0, b) x'+3y2z - 0. 13. The straight line ux--Ly+wz == 0 touches the curve x3+y'+z'-' 6,uxyz = 0; apart from this point of contiguity, it has also one other common point N with this curve. Find the coordinates of the point N. Exercises 14-31 give fundamental deductions of the theory of the cubic ternary form

f = (ax)3 = (bv)3 =- (cx)' = (dx)" = ...

(28.18)

In expanded representation, the coordinates of contravariant and covariant vectors will be denoted by x, y, z and u, v, ww, respectively, and the Hessian of the form f by H;

in addition, if

0 - (9,x)°(4'«)°, '1' = (1Vx)r(,, u)'

;ire two concomitants of the form (28.18), then (0, W)["1 =

(28.19)

14. The contravariant P of the form (28.18) is defined by the equality P = [abc][abu][acu][bcu].

(28.20)

show that P may be obtained from the mixed concomitant

0 = j[abu]z(ax)(bx) = ($u)'($x)'

(28.21)

of the same form (cf. 10) in the following manner: polarize 0 with respect to a and replace x by [uu'], to obtain

0° _ ($ux$u')[$uu']z; P is then obtained from 20, by contraction with f with respect to u': P = 2(8u)($c)[8uc]2. 15. The invariants S and T of the form f are given by S = =14 (f p)lal,

T = } (H, P)['l

[cf. (28.19)]. Write down their symbolic expressions.

16. Show that the Hessian of the curve f = 0 is the geometric locus of those points x, of the plane for which the first polar curve

(axe)(ax)' = 0

(28.23)

with respect to the curve f = 0 degenerates into two straight lines, and that it is the geometric locus of the singular points z of the curves (28.23) corresponding to all possible points x0 which have the stated property. In addition, prove that all straight lines x0 touch the Cayley curve with tangential equation P = 0 (cf. (28.20)). 17. If the curye f = 0 has no singular points, it has, by Theorem 28.4, at least one point of inflection. Starting from this fact, prove that in such a case the form f may be reduced to the canonical form

I f= ax'+3#x2y+y3+3xz2, (28.24) where a2+4 f1' :P4_ 0. What will be the position of the coordinate triangle with respect to the curve f = 0 and its Hessian after such a transformation? 18. Study the location of the points of inflection of the curve f = 0, where f is given by (28.24), under the assumption that a2+4fi' 96 0.

28.3


325

19. Prove that if a curve f = 0 has no singular points, the form f can be reduced to the canonical form

f= x'±y3+z'+6uxyz,

814'

I

0.

20. Let the curve f = 0 have a singular point, but not be degenerate; show that in this case the form f reduces to one of the two canonical forms

11 f= x3+y'-6xyz,

III f== x3+3y=z.

Assuming such a transformation to have taken place, find the points of inflection of the curve f .- 0; what is the location of the coordinate triangle with respect to each of these two curves? 21. Under the conditions of 20, show that the form f can be reduced to the canonical form I [cf. (28.24)] in which a2- 4j#' = 0. 22. In the case in which the curve f -- 0 degenerates and the rank p of the form f is equal to 3, prove that the form f may be reduced to one of the following canonical forms:

IV f =

6x)'z-.e23,

V f = 3x2y 3xz',

(28.25)

where a is equal to zero or unity.

23. For forms f, reduced to the canonical forms I, IV, V (cf. (28.24), (28.25)], compute the concomitants H, P, S and T [cf. (28.20). (28.22)]. 24. Prove that (8, 8);2} = S (ux)' [cf. (28.21), (28.19), (28.22)]; on the basis of this formula, verify the value of the invariant S found in 23. 25. For the curve f = 0, where f is defined by (28.24), find the tangential equation of the Cayley curve starting from its geometric definition (cf. 16). As a check use the value of the contravariant P, found in 23.

26. If a cubic ternary form f has rank p < 2, it can be transformed into a binary form; prove that for such ternary cubic forms all the concomitants H, P, S and T are

equal to zero. 27. Show that the curve f = 0 degenerates into three straight lines, if and only if the Hessian H of the form f differs from it by a numerical factor only. 28. Prove that a curve f = 0 has one or several singular points, if and only if the discriminant T2-4S3 of the form f is equal to zero. 29. Show that under the condition T2 -4S' # 0 a form f can be reduced to the form (28.24) by a linear transformation whose determinant is equal to 1. On the basis of this result, establish that each invariant of the form f is an integral rational function of S

and T. 30. Denoting by HA the Hessian of the form ft = 3H+Af, prove that HA = (2' - 3SA- T)H+ (SSA' -r Td+ S=)f. This relation leads to the following result: in the pencil of curves 3H+Af = 0, those

and only those curves degenerate into three straight lines for which A satisfies the equation

A'-6SA'-4TA-3S' = 0.

(28.26)

31. If the discriminant T2-4S3#0, all the roots of the equation (28.26) are distinct. Prove this statement and find its relationship to the results of 18. 32. The non-symmetric ternary bilinear form (ax)(by) corresponds to a correlation of the point x to the straight line u, = a;(bx); what corresponds in this correlation to the straight line u? 33. Given two correlations the first of which is defined by the ternary bilinear form (ax)(by), the second by the ternary bilinear form (cx)(dy) (cf. 32), show that their product is a collineation and find the affinor corresponding to it.

CHAP VI

TERNARY FORMS

326

§ 29. The quadratic n-ary form 29.1

We will begin the study of the quadratic n-ary form k

it = 1, 2, 3, ..., aij = aj1, (29.1)

f = aljx'xj = (ax)2 = (ax)2,

by establishing a complete system of invariants of the form f and of the q linear forms

i = 1, 2, ..., q.

(ax) = 0,

(29.2)

By virtue of the fundamental Theorem 18.2, consideration may be restricted to products of factors of the second kind having the form 12

[a a

12

... u u ... ]. Let I be one such product and kij

12

[aa

I

... auu ...

0 1, it is necessary to employ a formula which will be very useful in Chapter VII. Let an alternation with respect to all indices have been performed on some tensor a1, iZ , ,1k; subdivide the expanded expression of this alternation into those terms in which the index it occupies the first location, then those in which it occurs in the second place, etc. Then we obtain ,

a[j,,,...ik)

{ai,[1213... ik7 - a[I:Ii, I6 ...ik7 - aL13121,, IN ... 1k7 -

- a[1kt313 ...ik-111!)

) Read: A to the oblique power k.

(30.16)

CHAP. V1

TERNARY FORMS

342

[cf. (18.51), and likewise (19.1) and (19.2)]. We will say that we have on

the right-hand side of (30.16) an expansion of the alternation on the left-hand side with respect to the index it ; the analogy between this opera-

tion and the expansion of a determinant with respect to the elements of a row or column is obvious. We will apply the rule (30.16) to the right-hand side of (30.15), expanding it with respect to the index j; ('41k])i =

(-1)k{AL8,A22

... AQkl6i-AiÀtQ2... AakSa1)

-A[21 131 ... A2ks22]

...

-A ["MI A2

ak]

On the right-hand side of this equality, the third term is equal to the second; we verify this fact by replacing the summation indices al by a2 and a2 by at . In an analogous manner, we ascertain that the remaining terms are also equal to the second (to see this, for example in the last term, we replace a1 by a2, a2 by a3, ..., ak_ t by ak and ak by al); taking into consideration the definitions of AIk-11 and Ik, we see that (Alk1)ì

ai+(Alk-t])i

= (-l)k1kSi

Aa1 ,

i

i.e., that A[k]

= A A[k-t1+(- 1)klkE.

Starting from the expression found earlier for Alt], we obtain by induction Ak-I1Ak-t+12Ak-2-

Alt] =

. .

. +(-1)kIkE.

(30.17)

For k = n, the right-hand side of (30.15) contains an alternation with respect to (n+1) indices, and therefore it vanishes identically. Thus, AI"' = 0, i.e. +12A"-2- . . (30.18)

. +(-1)"I"E = 0;

A"-I1A"-

this is the so-called Hamilton-Cayley equation *). If we introduce the notation cp(A) =

A"-111t"-1+12A"-2-

(30.19)

then (30.18) can be written in the form cp(A) = 0.

(30.20)

') This derivation of the Hamilton-Cayley equation and the concept of the concomitant Alki was given by Professor A. M. Lopshitz (cf. "Proceedings of the seminar on vector and tensor analysis", Parts 11-I11, p. 9., abstract of the lecture by A. M. Lop-

shitz: "Some problems of affine calculus ...").

30.2-30.3

M[XED SECOND ORDER TENSORS

343

The polynomial (p(A) defined by (30.19) is called the characteristic polynomial and its roots A1, A2 , ... , 2", the characteristic roots of the affinor A; since the coefficients of (p(A) are invariants, A1, - - ., A,, are irrational invariants of the affinor A. We also arrive at the polynomial (poi) when seeking the vectors of invariable direction of an affinor A. This is the name given to a vector which retains its direction under the influence of the affinor A. If x is such

a vector, then Ax = Ax;

(30.21)

consequently, A2x = AAx = )!x, A3x = AA2X = Aix, etc., and cp(A)x = (p(A)x. However, by (30.20), one has (p(A)x = 0 and, since x A 0, cp(A) = 0. Thus, in (30.21), A must be a characteristic root of the affinor A. We will see in 30.5 that to every characteristic root these corresponds one or several invariable directions. Null directions will also be included in the invariable directions; they correspond to the characteristic root

A=0. Since (30.21) can be rewritten in the form (A-2)Ex = 0, the invariable

direction of the affinor A corresponding to the characteristic root A will be a null direction for the affinor (A - AE); therefore the invariant In of the affinor (A - AE) must vanish for such a value of A. Thus, any root

of the equation 9(,l) = 0 also satisfies the equation jA) - A611 = 0; the degrees of these equations are the same, consequently the left-hand

sides can differ from each other only by a constant multiplier (this conclusion is obvious, if all the roots of (p(A) are distinct; on the basis of the note in 15.1, it is always true). Comparing the coefficients of A", we arrive at the result

;(30.22)

(cf. Exercise 6). 30.3

Proceeding to the solution of the first fundamental problem for the affinor A, we restrict our search for a complete system to those joint invariants of the affinor A, a contravariant vector x and a covariant vector u which have one and the same degree in the components of each vector. In accordance with the formula (15.20), such invariants must necessarily be absolute. For this purpose, write the affinor A in the Aronhold symbolic notation

Aj=a1a'=b16`=c3c'=...

(30.23)

CHAP. V1

TERNARY FORMS

344

By virtue of Theorem 18.3, we can confine the study to invariants which are products of factors of the first'kind; in the case under consideration, these factors can only be of one of the following types: (ux),

(ax),

(au),

(ab).

The factor (ux) is by itself an invariant. If the above symbolic product K contains a factor (ax), the ideal vector a must be encountered in another one of its factors. Assume first that a occurs in some factor with u; then one can split off from K the invariant (ax)(au) which is equal to the scalar product of the covariant vector u and the affinor A multiplied by

the vector x; this invariant will be denoted by (uAx). However, if a enters into a factor of the form (ab), the product must also contain somewhere the symbol b; in the case when 6 occurs in a factor (6u), one can split off from K the invariant

(ax)(ab)(6u)

(uA2x).

In the presence of a factor (6c) in K, we can reason in an analogous manner, and so on. These results show that the presence of the factor (ax) in K ensures the presence as a factor of the invariant (uAkx). Since the degrees of K in x and u are the same, the process of removing all factors containing x will also remove u; therefore we need only consider the case

when the product K contains neither u nor x. An imaginary vector a must correspond to every imaginary vector a in the product; if they occur together in the same factor, we can separate from K the invariant (aa) = S1. However, if we have a factor (ab), we must also encounter the symbol 6 either in a factor (ba) or in a factor (6c). In the first case, we factorize the invariant (a6)(ab) = S2, in the second case, the product contains the symbol e', etc. It follows from this reasoning that K is a product of invariants of the form (uA'x) and S1, where k is equal to one of the numbers

0,1,2,... andiisoneofthenumbers 1,2,3,.... Multiplying both sides of (30.18) by A", we obtain A"+h-IlA°+h-1+I2A"+r,-2- ...

0,

h = 0, 1, 2, ...;

(30.24)

if we contract with u the result of operating with the left-hand side of (30.24) on the vector x, we find

(uA"+"x)-Il(uA"+h-1x)+

... +(-I)"II(uA"x) = 0,

h=0,1,2,...

MIXED SECOND ORDER TENSORS

30.3-30.4

345

The invariants I, , I2 , ... , I are polynomials in S,, S2, . . ., S. (cf. 30.2); thus, we see that the invariants (uA"x) for k ? n are integral rational functions of the invariants (uA'x), Si,

1 = 0, 1, 2, . . .,

n-1,

i = 1, 2, . . ., n.

In an analogous manner, taking the traces of both sides of (30.24) (for h > 1), we verify that the invariant Si when i > n is a polynomial in S1, S2, ... , S. (cf. Exercise 4). The irreducibility of the invariants (ux), (uAx), . . ., (uA"-'x), S1, S2, ., S. follows from the fact that in their tensor representations the numbers of superscripts and of subscripts do not exceed n; as a consequence, the representation of the corresponding tensor with constant components in the form (16.24) will be unique (cf. 16.5). Thus we have arrived at

Theorem 30.1: A complete system of those invariants of the affinor A, a contravariant vector x and a covariant vector u which have the same degree in the components of both vectors consists of 2n invariants S1, S2, ... S,,, (ux), (uAx), (uA2x), ..., (uA"-'x) [cf. (30.12), (30.8)]. Theorem 30.2: A complete system of invariants of the affinor A consists of then invariants S1 , S2, ... , S.. Since the invariants S1, S2, ..., S. are integral rational functions of

I,, 12, ..., I and conversely, in the formulations of Theorems 30.1 and 30.2 the invariants S1, S2, ... , S. may be replaced by I1 , 12, ... , I.. 30.4 The classification in the complex domain of affinors in an ndimensional space turns out to be substantially more difficult than for any of the tensors considered above; in return, the results of this classification have great signifficance not only for geometry and algebra, but also for many other areas of mathematics (differential and integral equations, theory of continuous groups, etc.). For the sake of greater clarity, we will begin with the particular case when all the roots of the characteristic equation of the affinor A are the

same. Then 9(A) = (1_2)N and the Hamilton-Cayley equation [cf. (30.18)] assumes the form B° = 0,

(30.25)

A - A, E = B.

(30.26)

where for the sake of brevity

TERNARY FORMS

346

CHAP. V1

By (39.25), we can find a number h 5 n such that for m = h the equality

B'x = 0

(30.27)

is true for any vector x of space, and for m = h -1 this equation does not apply for certain vectors x. Let R,,, denote the manifold of all vectors satisfying (30.27); if the vectors x and y belong to Ax and x+y also have same property. Obviously, R, is contained in R2, R2 in R3, etc., and R,, coincides with the manifold R of all vectors of space. The number

of dimensions of a manifold Ris the maximum number of linearly independent vectors in it; such vectors will be said to form a base of the manifold

If h = 1, then B = 0, A = 11 E, and the affinor A has already been reduced to canonical form. We will dwell on the case h = 2. Let the number of dimensions of R1 be s, and r = n-s. Select in R = R2 the r vectors x1i X2, . . ., x, *) such that together with the s vectors of R1 they form a base of the manifold R; then no linear combination of x1, x2 , ... ,

x, can be contained in R1 (and, obviously, the converse is also true: if r vectors have this property, we arrive at a base of R by combining them with s linearly independent vectors of the manifold R1). Consider the vectors y1 = Bx1,

Y2 = Bx2,

...,

y, = BX,.

(30.28)

Since B2x1 = B2x2 = ... = B2x, = 0, all the vectors (30.28) belong to R1; in addition, they are linearly independent: if at Yt +a2 y2 + .. .

+a,y, = 0, then B(a1x1+a2x2+ ... +a,x,) = 0, and the vector ... +a,x,) is in R1, which is only possible for a1 = a2 = = a, = 0. Hence we see immediately that s r; if s > r, we combine the vectors y l, ... , y, with (s- r) vectors Y,+21 ... , y, in such a manner that all the vectors y form a base of R1. Altogether we obtain a base of R consisting of n = r+s vectors distributed over r links of length 2: x1, Yi; x2, Y2; . . . ; X, y where (0(1 x1 +a2 x2 +

Yt=BX1i BYt=0; Y2=Bx2, BYi=0; ...; Y,=Bx,, BY,=0 and (s-r) links of length 1: Yr+1 ; Yr+2; .

. .

; ys, where

BYr+t=BYr+2=...=Bys=0. ) The subscript denotes here the number of the vector, and not that of its compo-

nent.


30.4

347

The further reasoning employs induction: suppose we have proved that in the case when R coincides with Rh - 1, but not with R.- 2, there exists in

R a base distributed over links of length S h-1, where we may take as initial vectors of these links any vectors of Rh_, no linear combinations of which belong to Rh _ 2 . Now assume that R = R. and that we have in R vectors not contained in R. - I; denote the number of dimensions of R,, again by s and let r = n - s. I n the manifold R, select r vectors x, , x 2 . ..., x, from which we cannot form linear combinations belonging to Rh-, and form the vectors (30.28) all of which enter into the composition

of R,,_,, since Bhx, = B''x2 =

... =

Bhx, ='0. If the vector 11Y1+12Y2+ ... +x,Y,. = B(0(,x,+0(2x2+ ... +xxr)

lies in Rh_2, the vector (0(,x1 +0(2x2+ ... +0(,x,) lies in Rh_1 and, by virtue of the condition imposed on the choice of x, , x2 , ... , x one has

a, = a2 = ... = 0(, = 0. Applying the induction now to Rh_,, we conclude that one can combine the vectors y,. y2, . . ., y, with (s-r) vectors such that all of them form a base of Rh_,, distributed over links of lengths < h-1, where y,, Y2, ..., y, are the initial vectors of certain of the links of length (h - I). Adding now the vectors x,, x2, ... , 0(r, we obtain a base of R in which X,, x2 , ... , 0(r themselves begin links of length h. Thus, we have proved the possibility of the choice of a base formed from links of these types for any h. Now select the vectors of one of the links of length g 5 h as first g coordinate vectors e, , e2, . . ., eo; then Be, = e2,

Be2 = e3,

...,

Bea-1 = eo,

Be. = 0,

or [cf. (30.26)]

...,

Ae1 = A,e,+e2, Ae2 = A1e2+e3,

Aee = A, e,,,

Aeo_,

_

eo-1+ee, (30.29)

and the submatrix formed from the first g rows and columns of the matrix of the affinor A will have the form A,

I

k1

1

................ 21

I

A,

CHAP. V1

TERNARY FORMS

348

(where the empty spaces are occupied by zeros). As next coordinate vectors we again select the vectors of a certain link, etc. As a result, the matrix of the affinor assumes the canonical form

where each of the matrices V; is of order < h; its elements along the principal diagonal are all equal to A1, while all those in the diagonal immediately above the principal diagonal are ones, all other elements being zeros; all the elements outside the matrices V1 are also zeros. 30.5

The general case is readily reduced to the particular case studied

in 30.4 by means of the following reasoning. Let the characteristic polynomial 9(2) = cp10')cp2(2), where (p, and 92 are mutually prime polynomials. Then there exist two polynomials 'P1(A) and 02(A) for which

(pl(AA1M+102MO2{A) = 1, and hence

cp1(A)01(A)+cp2(A)i2(A) = E.

(30.31)

Introduce now two manifolds R1 and R2; let the first of them consist of

those vectors x for which cp, (A)x = 0, the second of those for which cp2(A)x = 0. The manifolds R, and R2 do not have common non-zero vectors: if a vector x belongs to both these manifolds, we have, by (30.31),

x = Ex =

0.

In an analogous manner, we show that an arbitrary vector x of space can be represented in the form of a sum of two vectors one of which is contained in R1, the other in R2x = Ex = cP2(A)02(A)x+lpl(A)01(A)x. Denote the terms on the right-hand side of the last equality by y and z,

30.4-30.5

349


respectively; we have for these vectors cpl(A)y = cp1(A)cp2(A)12(A)x = cp(A)b2(A)x = 0, cp2(A)z = cp(A)-P1(A)x = 0

[cf. (30.20)), i.e., y belongs to R1 and z to R2. If (p(A) = !p1(A)lp2(A) ... (pk(1.), where the polynomials (pl, q

,

, Pk

are prime in pairs, we verify by dividing consecutively by each factor, starting from the above, that the manifolds R, of the vectors x for which (p,(A)x = 0 (i = 1 , 2, ..., k) do not have pairwise common non-null vectors and that every vector of space can be represented in the form of a sum of k vectors belonging to R1, R2, ... , R., respectively. It follows

from this that, by combining the bases of the manifolds R1, R2, ..., Rk, we obtain a base for the manifold R of all vectors of space. Now let the expansion of a characteristic polynomial in linear factors have the form cp(A) = (A-A1)°i(A_),)P1 ... (A-Ak)°k,

PI+P2+ ... +pk = n.

Include the vectors x which satisfy the equation (A - A, E)°1 x = 0 in the manifold R; (i = 1, 2, . . ., k); the reasoning of 30.4 is applicable to each of these manifolds, as a consequence of which we may select in R; a base with the properties described in 30.5. Then the last vectors of the

links will be the vectors of invariable direction corresponding to the characteristic numbers A. As has already been stated, we obtain a base of R by combining the bases of all the R,; select the vectors of this base as coordinate vectors, numbering

them in such a way that the vectors of the links follow each other. Then the affinor A is reduced to the form (30.30), where each of the matrices V; has the structure described at the end of 30.4, the only difference being that in certain of these matrices the elements along the principal diagonal will be equal to A,, in others the same elements will be equal to A2, etc. In such a case, one says that the affinor A has been reduced to Jordan's canonical form. Evaluating the characteristic polynomial cp(1.) for an affinor A represented in Jordan's canonical form by use of (30.22), we see that the sum of the orders of the matrices V corresponding to the characteristic root .t, is equal to p; or, in other words, that the number of dimensions of the manifold R, is p,; incidentally, it follows from this that at least one matrix V corresponds to each characteristic root.

CHAP. VI

TERNARY FORMS

350

Denote by si9 the set of matrices V which correspond to the characteristic number 7.i and are of order g, and by qi,, the number of linearly independent solutions of the equation (A-;E)mx = 0. Then thedifferq;,. _ I is equal to the sum of those s,9 for which g > m, so that rence the numbers q;are determined by the numbers sig and, conversely, the latter determine the former. Since the q;are given by the affinor A in an invariant manner, the same is true for the sig. All possible Jordan canonical forms are distinguished by their Weierstrass characteristics which are formed in the following manner: first write down in non-increasing sequence of their orders the matrices V corres-

ponding to the characteristic root A,, then in the same sequence of their orders the matrices corresponding to A2, etc.; if more than two matrices correspond to the same characteristic root, their orders must be

joined in round brackets. The Weierstrass characteristics of affinors given by the canonical matrices

00 0 A, 0 0 .....................: A,

0 0A, 0 0

A,

1

0

0

A2

o

0

0

0

A,0 0

................

o

0

A2

0

o

0

0

A2

will be [(21) 1], [4], [(11)(11)], respectively. Assembling under one type of the affinors which reduce to canonical

Jordan forms with the same Weierstrass characteristic *), we obtain the required classification of affinors of n-dimensional space in the complex

domain (Weierstrass characteristics which go over into each other with a change of the ordering of the characteristic numbers, are, of course, not regarded as different). Note that the numbers which enter into the Weierstrass characteristics are not the values of the arithmetic invariants (cf. p. 146); it would be of interest to state for different types of affinors A the arithmetic characteris-

tics composed of the values of certain of its arithmetic invariants (cf. Exercise 10 and 11). *) Two such affinors are equivalent if only their characteristic roots are the same; thus, we include here under one type not only affinors which are equivalent to each other, but also those which reduce to canonical forms which are similar structurally; cf. 25.4, Type I and § 28, Exercise 19.

30.5


351

Exercises In the following exercises, we use the same notation for concomitants and the characteristic polynomial of the affinor A as in § 30 [cf. (30.12), (30.13), (30.15), (30.19)]. 1. Show that

1k = -kf I ± Cala2 ... ak S;' Ss' ... Skk,

(30.32)

where the summation is extended over all values of the non-negative integers z1, a2, ... , ak for which the relation a,+2x2+2x2+ ... -4 kak = k is satisfied and cola, ak is equal to the number of those substitutions of k symbols whose expansion in cycles contains a, cycles of one symbol, a2 cycles of two symbols, ..., ak cycles of k symbols; a plus (+) sign occurs in front-of c, when the number of substitutions is even, a minus (-) sign, when it is odd.

2. Starting from the results of 1, derive Waring's formula (_1)21+a2+ ... +a1.+k 1a1 . 2a2

... k kalla2! .

. .

aki

where the summation is extended over the same values at, x2, ..., ak as in (30.32). On the basis of this formula or of (30.32), verify the result of § 14, Exercise 15, and also

express 1, and 1, in terms of S1, S2, ..., S5. 3. Establish that {Alk,) = (--1)k(n-k)1k and derive Newton's formula

Sk-11S4+12S1_2- ... +(-1)k 'Ik-1S,+(-1)kklk = 0, k = 1, 2, 3, ...; with the aid of this formula express S21 S3 and S. in terms of 1,, 11, f2 and 1,.

4. For n = 3 express the quantities S. and f6 in terms of S S2, S. 5. Verify the Hamilton-Cayley equation [cf. (30.18)] for the affinor A given in a certain coordinate system by the matrix 1

0

-1

0

2

1

1

0

111

.

6. Verify by direct computations that the coefficients of the powers' of k on the righthand side of (30.22) coincide with the coefficients of q (d). 7. Prove that every affinor which is a concomitant of an affinor A is a linear combi-

nation of E, A, A2, . . ., A"-' with coefficients which are equal to integral rational functions of S1, S2, ..., S. 8. Let cu(rl) be some polynomial in A; find w(A)e1, w(A)e2,

. . ., w(A)e,, where . ., e, are the vectors of the links (30.29), and prove on this basis that the characteristic roots of w(A) are equal to w(A,), w(A.), ..., w(A,), where A1, A2...., A.

e e,, .

are the characteristic roots of the affinor A. 9. Express the invariants Sk (k = 1, 2, 3, ...) of an affinor A in terms of its characteristic roots. 10. For n = 2, there exist three types of affinors with Weierstrass characteristics I [11], II [2), 111 [(11)]. Show that one can use as arithmeticcharacteristicofeachtype the value p of the rank of the affinor q-'(A) and that the algebraic characteristics are as follows: I : [q (A)]2 :A 0; 11 : (92'(A))' 0, T'(-4) # 0; 111 : g1'(A) = 0.

11. For n = 3, there exist six types of affinors with Weierstrass characteristics: I [I11], 11 [211, 111 [(11)1), IV [3], V [(21)], VI [(I11)]. Denote by p, the value of the rank of the affinor 91'(A) and by p, the larger of the values of the ranks of the affinors

CHAP. Vt

TERNARY FORMS

352

Show that p1 and p, give q'(A) andq""(A) and let H = arithmetic characteristics of each of the types and that one can use as their algebraic characteristics the relations I

11 Hq?(A)=0,

Hq7'(A)=0;

III H=0, [97'(A)]'#0;

0;

0. V q,'(A) = 0, T"(A)te 0; VI 12. Reduce to Jordan's canonical form the affinors given by the matrices

IV [4p'(A)]2 = 0, 42'(A) # 0;

i

a)

1

1

1

1

l

l

1

1

1

2 -1 b)

0 0

1

0 2

c)

3

0 0

0 0

2

0 -1 1

0

0

1

0 -1

1

0

0

0 0

1

1

1

0 -1

d)

00

-1 -1

-1

1

-1

-1

0

1

13. Find the Weierstrass characteristics of the affinor given by the matrix

a

Al (X, a, a4 ... 0

Al a, a,...a

0 0 Al a, .

.

. at-3

0 000 0000...

X11

0 0 0 0 ... 0 in the cases a) m, # 0, b)

a, Al

0, at 7-1-- 0.

14. Starting from the projective geometric interpretations of an affinor (§ 10, Example 6), give a classification of the projective transformations of straight lines and planes. 15. Show that every involutory projective transformation of the plane *) is a nonparabolic homology (cf. Answer to 14).

') A projective transformation which is not an identity is said to be involutory, if in it points correspond to each other in pairs, i.e., if M --3o. M', then M' -+ M.

CHAPTER VII

POLYVECTORS

§ 31. Certain general properties of polyvectors

The concept of a covariant or contravariant polyvector has been defined in 9.4 and 9.5; at the same time, the number of essential 31.1

components of an r-vector (polyvector of order r) was stated. The number of linearly independent vectors of the form wi,12...irxi2 x `2 ... Xit,

where x, x, 2

3

(31.1)

..., x are arbitrary contravariant vectors will be said to be r

the rank of the covariant polyvector wi,i2... it (cf. § 14, Example 12) and will be denoted by p. It is easily seen that p is itself the rank of the matrix IIw,,12...irll of n rows and Cr-1 columns, where the row is specified by the value of the index i,, the column by a combination of different values of the indices i2, i3, ..., i,. It follows from this that the system of equations (31.2) wai2i2...irx - 0,

where x is an unknown vector, has exactly n-p linearly independent solutions.

Select among the vectors (31.1) p linearly independent ones 1

p

2

P1 p, ..., p,

(31.3)

and use them as the first p coordinate vectors; then (31.1) will vanish for it > p. However, the vectors x, ..., x are arbitrary; consequently, all 2

r

the components of the polyvector wi 112

,

, ,1,

with index i, > p are found

to be equal to zero, and, by virtue of the equivalence of all indices, the same will also be true for i2, i3, ..., ir. Thus, only those components of w,,12.. ,,r for which none of the indices exceed p can turn out to be non-

CHAP. VII

POLYVECIORS

354

zero; we have reduced iv to a polyvector in a space of order p. This space is called the space of the polyvector iv; for this choice of coordinate system, the polyvector w is contained in its space. It is said of the vectors (31.1) and (31.3) that they belong to the space of the r-vector w. One can also say that the rank of the po!yvector is the order of its space ). The geometric significance of this terminology will be explained in 32.3. From the above follows directly

Theorem 31.1: If the rank p of a r-vector is different from zero, then

p ? r. In fact, place the r-vector Ir in its space; if p < r, the order of the space will turn out to be smaller than the order of the polyvector and, conse-

quently, iv = 0 and p = 0 (cf. 9.4). By virtue of the duality principle, all the above results may be extended

also to the case of contravariant polyvectors. The same principle also permits one to restrict the following study to covariant polyvectors, where, for the sake of brevity, the word "covariant" will often be omitted.

31.2 A covariant polyvector w1112 ...I, is said to be simple, if it is equal 1

r

2

to the alternated product of the r covariant vectors p, p, 1

2

r'

w;,j,...r, = r!p[1,p ... A,] or, in the abbreviated notation, 12

..., p: (31.4)

r

(31.5) w = [pp ... P] (cf. 9.9). Obviously, each of the components of the polyvector (31.4)

is one of the minors of order r of the matrix formed by the components 1

2

of the vectors p, p, ..., p. Hence a simple polyvector is non-zero, if and 1

2

only if the vectors p, p, ..., p are linearly independent; in that case, one 1

2

r

of the vectors (31.1) is a linear combination of p, p, ..., p and the rank p = r (cf. Theorem 31.1). The polyvector of variance n (n-vector) e11 2...;,, is always simple: if its only essential component e12.... = a, it is equal to the alternated product of ae,, e2, . . ., e,,, where e1, e2, . . ., e are the coordinate covar) It is readily seen that the above observations are also true for asymmetrictensor.

31.1-31.3

355

GENERAL PROPERTIES ,

iant vectors. Therefore a r-vector of rank p = r will necessarily be simple; in order to verify this, one need only place such an r-vector in its space

(cf. 31.1). Thus, one has Theorem 31.2: A polyvector of variance r will be simple, if and only if its rank p is equal to r or zero. 31.3 In what follows, we will require the following simple properties

of the operation of alternation: Theorem 31.3: An alternated product does not vary on transposition of its factors together with their indices. In fact, P[tgj'*k] = gjPirk], V[i)WkimPs] = W[klmPsVij],

because both sides of the equalities are easily seen to consist of the same terms, provided with the same plus (+) or minus (-) signs. Hence one has

Theorem 31.4: In alternating an expression already containing. an alternation sign, one can abolish this sign. Thus, for example, W[ij[k Vim] Ps] ` W[ijk VIm Ps] ,

(31.6)

if on the left-hand side the external alternation extends to all six indices i,j, k, 1, m, s: expanding the internal alternation, we see that the left-hand side 1of{{ (31.6) is equal to 31 W[iJk Vim Ps] - W[i Ji Vkm Ps] + W[ i jI Vmk Ps] - W[ijk Umi Ps]

+ W[i jm Vki Ps] -W[i jm Vik Ps]}

If in the second term we interchange the indices k and I without changing the order of the factors w, v, p, the only effect is to change its

sign and this term is found to be equal to the first term. In a similar manner, we can verify that the remaining terms are also equal to the first, so that both sides of (31.6) are actually equal to each other. Taking into consideration Theorem 31.4, w;. will, in addition, always assume that the sign of the external alternation refers only to the indices outside the internal alternation signs. In order to study the properties of polyvectors, we will also apply in a systematic manner the formula for the expansion of an alternation with

POLYVECTORS

356

CHAP. V11

respect to one of its indices [cf. (30.16)1; in the case of an alternation of a product of polyvectors, this becomes considerably simpler. For example,

let there be given the alternated product of the tri-vector wijk and the bi-vector vim; by (30.16), one has W[ijkV,m] =

S{Wi[jkVlm]-W[jlilkVim]-W[kjlilVim]-W[IjkVlilm]-W[mjkV,]i).

By virtue of the skew-symmetry of the tensor wijk, we can interchange the indices i, j and i, k, respectively, in the second and third terms on the right-hand side; this will also change their signs; it then appears that the

first three terms are identical. In the same manner, we can verify that the last two terms are also equal to each other. The expansion thus assumes the simpler form (31.7)

W[ijk VIm] = 3 {3wi[jk vim] - 2w[mjk v,]i}.

It is readily seen what will happen in other, analogous cases. 31.4 In what follows, an important role will be played by the concept

of the divisibility of a polyvector by a vector. If the polyvector

w,2i2 ... r,

can be represented in the form of an alternated product of a certain (r-1)-vector and a vector p, the polyvector w is said to be divisible by the vector p. In an analogous manner, w is divisible by an alternated product of linearly independent vectors p, q, if wi112...I, = u[,02...1,_2Pr,_,q,,]

[where u is a polyvector of order (r-2)], etc. Theorem 31.5: In order for a polyvector w to be divisible by a given p # 0, it is necessary and sufficient that the condition WU,12...,,Pj] = 0

(31.8)

be fulfilled.

For the sake of simplicity, we will present the proof for the case r The necessity of (31.8) is obvious, since it follows from

4.

Wijki = u[ijkPl]

that (cf. Theorem 31.4) W[ijk, Pm] = u[,jk Pi Pm];

however, the last expression is equal to zero, because, on the one hand, it must change its sign under an interchange of the indices I and m and,

31.3-31.4

GENERAL PROPERTIES

357

on the other hand, it remains unaltered by such a transposition. Conversely, for the 4-vector w, let W[ijkl Pm] = 0;

(31.9)

expand the alternation on the left-hand side with respect to m to obtain wi jk1 pm - 4w,,,[ jkl Pi] = 0

[cf. (31.7)). Contract the left-hand side of the last equality with respect to

the index m with some vector x' which satisfies the condition Pmx" = 1;

introducing the notation 4w1klxm = ujki, we find W4Jkl = u[Jk1 P1]

as was required to be proven. Theorem 31.6: If a polyvector w is divisible by s linearly independent vectors, it is also divisible by their alternated product. Theorem 31.6 may be proved by induction with respect to s; we will confine ourselves to the case when the variance of the polyvector is equal to 5 and s = 3, under the assumption that the theorem is true for s = 2. Following this example, the reader will easily produce the reasoning for the general case. Thus, let the 5-vector wiJklm be divisible by the three linearly independent vectors p, q, r; by Theorem 31.5, w[ijklmPh] = 0,

x'[ijklmgk] = 0,

w[1Jklmrh] = 0.

(31.10)

In the three equalities (31.10), expand the left-hand sides with respect

to the index h and contract the result with respect to the same index with the vector x" satisfying the conditions

rkxk=0 gkxk=0, (such a vector x exists by virtue of the linear independence of p, q, r). pkxk= 1,

Introducing the notation 5w1, jklmx

= u jklm s

we then obtain MAIM = u[JklmPi] s

u[Jklmgl] = 0,

u[Jklmrl] = 0.

(31.11)

The second and third equalities (31.11) show that the 4-vector u is

CHAP. VII

POLY VECTORS

358

divisible by the vectors q and r; by virtue of the inductive assumption, the vector u is divisible by their alternated product U jklm - v[ jk gi rm] ,

where Vik is some bi-vector. The first equality (31.11) now gives (cf. Theorem 31.4) Wijklm = v[jkglrpt],

which proves Theorem 31.6 for the case under consideration. By virtue of Theorem 31.6, an r-vector which is divisible by (r- 1) or by r linearly independent vectors, is necessarily simple. Hence follow Theorem 31.7: A polyvector of variance (n - 1) is always simple. Theorem 31.8: The rank of a r-vector cannot be equal to (r+ 1). In fact, the condition of divisibility of a (n-1)-vector w by a vector p 0

represents a single equation between the n components of the unknown vector p, which obviously has (n - 1) linearly independent solutions. Theorem 31.8 is readily proved from Theorem 31.7 by placing the polyvector in its space (cf. Theorem 31.2). Note further that for w o 0 the number s entering into the formulation of Theorem 31.6 obviously cannot exceed the variance r of the polyvector w. 31.5

The results of 31.4 make it possible to derive easily the condition under which a polyvector will be simple. If an r-vector is simple and can

therefore be represented in the form (31.4), each of the vectors (31.1) 2

1

r

is a linear combination of p, p, ..., p, as a consequence of which 2

1

r

jxj2xh.

P[i1 Pt2 ... Pi,Wj,]j2... 2

.. xj' = 0, r

3

or, by (31.4), j2 js wti02...irwj11j2...j.x x ... Xj. = 0. 2

Since the vectors x, .r, 2

3

3

(31.12)

r

..., x are arbitrary, it follows from (31.12) that r

W1,2 ...1 Wj,)j2...j. = 0.

(31.13)

Conversely, let the relation (31.13) hold true for an r-vector w; then

31.4-31.6

359

GENERAL PROPERTIES

(31.12) will be true for any choice of the vectors x, x, ..., x, and this 3

2

r

means that the polyvector w is divisible by any vector of the space. If the rank p of the r-vector w were to exceed r, one would find that w was divisible by all the linearly independent vectors (31.3) of which there are p > r, and this is impossible (cf. end of 31.4). Consequently, we have either

p = r or p = 0, and the polyvector w is simple (cf. Theorems 31.1 and 31.2). Thus, we find Theorem 31.9: The equality (31.13) is necessary and sufficient for the polyvector w;,;,...;,. to be simple. Theorem 31.9 is a particular case of the stronger 2

I

Theorem 31.10: If two polyvectors It, and w of the same variance are linked by the relation 1

2

wUlj2...j,Wji]j2...j. = 0,

(31.14)

either one of them is equal to zero or both are simple and they differ from each other only by a scalar multiplier. 2

1

In fact, let w 0 0; then (31.14) denotes that the polyvector w is divisible 2

by all vectors of the space of the r-vector w (cf. the second part of the 2

proof of Theorem 31.9). If the rank p of the polyvector w is larger than r, I

then w = 0, by virtue of the remark made at the end of 31.4; however, if 2

2

r

2

1

1

2

p = r, then w is a simple r-vector: w = [pp ... p], where p, p, I

r

..., p

are linearly independent vectors and the polyvector w which is divisible 12

.

by all these vectors must be equal to a[p p,

.

. p) (cf. Theorem 31.6).

31.6

The condition for a polyvector to be simple can be 'represented in yet another form; for this purpose, we will first establish Theorem 31.11:

If a polyvector w;,;2

j

is simple, every polyvector

of the form ww,2j, . j, xa will also he simple; conversely, if for any contravariant vector x the polyvector 11'2;2!3. _;,x4 turns out to be simple and if .

_

r > 2, the polyvector will likewise be simple. The first part of the theorem is almost obvious. One can assume that w 0 0; place the polyvector w;,;2., into its space. Then the polyvector

CHAP. V11

POLY VECTORS

360

of order r; ... 1,x. also turns out to be contained in the same space applying Theorem 31.7 to this polyvector, we see that it is simple. In order to prove the second part of the theorem, for the sake of simplicity, let r = 4. If the 4-vector Wi jk1 has the property that for any x the tri-vector wajkixa is always simple, then, by Theorem 31.9, Hya1=;,

xawa[jki Wa]bcQ X 8 = 0,

or, by virtue of the arbitrariness of x (cf. 9.7), (31.15)

Wi[jki Walbcd+ Wd[jki Wa]bci = 0.

Again assuming w to be non-zero, select a vector y such that (31.16)

wajkiya 0 0.

By the conditions of the second part of the theorem, the trivector (31.16)

is simple; therefore its rank is equal to 3, and the equation (31.17)

yaWajk,x1 = 0,

where x is an unknown vector, has [cf. 31.1, (31.2)) the (n-3) linearly

independent solutions: x` = y', x' = x', h = 1, 2, 3, ..., (n-4). Conk

tract the left-hand side of (31.15) with respect to the indices c and i with the vectors y and x; taking into consideration the fact that x satisfies the equation (31.17), we find 1 yaWa[jkiWa]bdpXP - 0, 1

whence it follows, by Theorem 31.10, that K'ajkixa

= aI Wajkly

I

In an analogous manner, we show that Wajklxa

,a

= QkWajkr. ,

h = 2,3,...,n-4.

h

Thus, the equation Wajkixa = 0

has the (n-4) linearly independent solutions: x' = x-Qky, h = k

1, 2,

..., (n-4); therefore the rank of the 4-vector w is equal to four,

i.e., (cf. Theorem 31.2) the vector w is a simple 4-vector. The second part of Theorem 31.11 can be proved in the same manner.

361

GENERAL PROPERTIES

.31.6

Now we can prove Theorem 31.12: A polyvector w 112...1, will be simple if and only if (31.18)

w11112...j, w1112]i3... J, = 0.

The necessity of the condition (31.18) can be proved as follows: if w is a simple polyvector, the relation (31.13) holds true (cf. Theorem 31.9). Alternating the left-hand side of (31.13) with respect to the indices it , i2 , ... , i, , jl , j2 and taking into consideration Theorem 31.4, we obtain (31.18). For r = 2, the sufficiency of (31.18) can be established very easily: expanding the left-hand side of the equality (31.19)

wlj j wkj) = 0

with respect to the index 1, we find WIIj wkJI + WIjjII Wjk7 = 0.

Interchanging the factors in the second term together with their indices (cf. Theorem 31.3), we reduce it to the form w(lkwj]I; the last expression is easily seen to be the same as the first term. Thus, it follows from (31.19) that w[,i wk]1 = 0;

by Theorem 31.9, the bi-vector wig is simple. Assume now that the sufficiency of the condition (31.18) has been

proved for polyvectors of variance < (r- 1) and let the relation be valid for the r-vector ww,j,... j, (where r > 2). Expand the left-hand side of this relation with respect to the index i, to obtain

i. =

0.

(31.20)

Denote by xa an arbitrary contravariant vector; employing this vector to contract the left-hand side of (31.20) with respect to the indices it and

j we find Xawa[2 .

Gwl1l27J3... J- 1BX

B

= 0.

By the assumption of the induction, the last equality denotes that the polyvector

wj,j3...j,xa

is simple for any x; therefore it follows from

Theorem 31.11 that the polyvector wj1j,... j, is also simple. Theorem 31.12 was first established by Weitzenbock *); it will be required in § 33. *) cf. [161, 111, § 8.

CHAP. VII

POLYVECTORS

362

Exercises 1. Show that the concomitant Watb, w;;lk of the tri-vector w,rk is itself a tensor which

is symmetric with respect to the indices a and k. 2. Generalize I to polyvectors w of arbitrary variance r. 3. Prove that the tri-vector w,rk, satisfying the relation W,ab',i Wa.'rI WU;k ° 0.

(31.21)

Wfabc "'a,, Wifk = 0

(31.22)

also satisfies the equality

and that, conversely, the relation (31.21) follows from (31.22). 4. Prove that, if the variance r of the polyvector w is an odd integer and the order of the space n = r4 2, there always exists a vector p 0 by which w is divisible. 5. Prove that a polyvector w of variance r can be represented in the form of a sum . 2 z 1

of s r-vectors the first of which is divisible by p, the second by p, etc., where p, p, ... , p are linearly independent vectors, if and only if a

2

1

x'ii 112... i,Pj IPj2 ... pj'i = 0 12

(or, in abbreviated form, [wpp ... p)

(31.23)

0).

2

1

6. Two polyvectors w and w of variances r and s are linked by the relation 2

1

wlili2...i, Wj 11j2... j. = 0, 1

2

where s > r. Prove that one of the polyvectors I. and w is equal to zero. 7. Prove that the relation Wabe 14'i15 = 3W,ab[i Wik;.)

(31.24)

is a necessary and sufficient condition for the tri-vector wick to be simple. 8. Let Wick be a simple tri-vector for which the component wl :;6I 0. Show that the vectors p, = wig., qc - w,,,, r, - w,12 are linearly independent, that all of them belong to the space of the tri-vector w and that Wilk - 3!(W123) 2Wfi1131 w, 311 Wk;12

(31.25)

9. Let there be given in a space of order 5 the simple tri-vector w with components W123

W124 = W125 = W346 = 0, W134 - 2, w13s = 1, w146 = -3, W234 = 2, W236 - I,

w,,, _ -3. Find vectors p, q, r whose alternated product is equal to the tri-vector w, and hence verify that it is actually simple. 10. Generalize the results of 8 to the case of an arbitrary polyvector of variance r; Vahlen's relation may then be obtained from (31.25). 11. Show that a simple polyvector of variance r is determined by specification of

(n-r)r+ I numbers. 12. Prove that by alternating the product Wj1i2 l,wj1,2 j, with respect to any (r+l) or (r+2) of its indices and equating the result to zero, we obtain a relation which is the necessary and sufficient condition for the r-vector wi1i2...j, to be simple. 13. Prove that, if 1

2

1

2

Wabc K', l1:"µ', rk Wabc = 0, 1

2

one of the tri-vectors w, w is equal to zero.

(31.26)

363

GEOMETRIC INTERPRETATION. WEITZENBUCK'S NOTATION

32.1

14. Prove that, if I

2

2

I

(31.27)

W[ab[iwik"cl+ WiGbij ii'ik;c 1 ='= 0, 2

1

one of the tri-vectors w, w is equal to zero.

§ 32. Geometric interpretation of polyvectors. Weitzenbock's complex symbolic notation

32.1 We restrict ourselves in the text to the interpretation of polyvectors in projective, multi-dimensional geometry *); their affine significance (for the simplest cases) will be stated in Exercises 20-26. We will first consider a simple contravariant r-vector specified by

w = [xx

... x],

12

r < n,

(32.1)

r

where X. x, ..., x are linearly independent contravariant vectors. In 1

r

2

projective geometry of order it, these vectors correspond to r independent points determining an (r-1)-dimensional plane E,_ 1. If in the same plane we select r other independent points z, z, ..., z, then 1

2

r

fl = 1, 2, ..., r,

z'

(32.2)

a

where Aa are scalars satisfying the condition that the determinant (32.3)

d = 1, 1.11 # 0;

the formula (32.2) involves summation with respect to ft. The alternated product of the vectors z, z, ..., z is an r-vector whose components are determined by 1 2 r

ZZ'2 1

r! zl''z'2 . 1

2

. .

z'']

Z 2

r ZiI r

. . .

1

.

. Z'

2

2

. . .

4 x'' 2 x'2 ... A; 1

1

z'2 .

Z'2 r

zI'

Zir

r

1xlr

1

a

;,ft x'' P

AOx1' a

a

A x'2

p j . . . A.Z Xir

a

Afx'2 r e

D

..

.taxi' r e

by virtue of the theorem on products of determinants, the component written down is equal to the product of the determinant (32.3) and the *) Before becoming acquainted with the contents of this section the reader will find it useful to review the results of § 10, Example 4 and 8.2.

CHAP. VII

POLYVECTORS

364

corresponding component of the polyvector (32.1), so that [zz

...

12

z] = d[xx ... x]. 12

r

r

Thus, replacement of the points x, x, . . ., x by any other r independent 2

1

r

points of the same (r-1)-dimensional plane E,- 1 changes the polyvector w only by a scalar factor. Hence, one has Theorem 32.1: The simple contravariant r-vector (32.1) (defined apart from a scalar factor) determines an (r-1)-dimensional plane E, through the points x, x, ... , X. 1

2

r

Obviously, this plane E,_ 1 consists of those and only those points which correspond to vectors belonging to the space of the polyvector vi, (cf. 31.1). The components of the polyvector w are called the contravariant Plucker coordinates of the (r-1)-dimensional plane E,_ 1. By virtue of the duality principle, one has Theorem 32.2:

The (n - r -1)-dimensional plane E. 1

representing

2

the intersection of the hyperplanes p, p, ... , p corresponds to the simple covariant r-vector 12

r

w=[PP...p]#0,

r 1) determining the (2s- I)-dimensional plane E2,_1 and (n - 2s) points 2s+ 1, 2s + 2, ... , n no linear combination of which lies in Eli _ 1. Let E. _ 3 (i = 1, 2, ... , s) denote the (n - 3)-dimensional plane ) We will say that the point x is a linear combination of the points y, z, ..., if such a relationship connects the corresponding vectors of the points.

34.3-34.4

387

THE 81-VECTOR

passing through all the straight lines g except g; and through all the points 2s+ 1, 2s+2,.. ., n. The linear complex of straight lines K1 will be given uniquely, if we demand that the straight lines g1i g2, . . ., g, are conjugate in pairs with respect to the complex K1 and do not belong to it, that the

points 2s+1, 2s+2, ..., n are singular for K1 and that some point x, lying in none of the E. - 3, is conjugate to a given hyperplane p passing through these points 2s + I, 2s+2 ..., n and through the s points each of which is the intersection of the straight line g; with the hyperplane containing the corresponding E _ 3 and the point x. Under the conditions of Theorem 34.6, again denote by 1, 2, 3,..., 2s points the first two of which lie on the straight line g,, the next two on 92, etc. Any point n of the straight line [12] will be conjugate to the hyperplane n], by the construction of 34.1; joining the point x to all points n [t34 by straight lines and applying the same construction, we find a hyperplane conjugate to all points y of the plane [x12]. The construction of 34.1 will

...

not be applicable to the points r of the straight line [xe] lying in the hyperplane p and, consequently, belonging to K1. For such a point T, we can proceed in the following manner: draw through this point an arbitrary straight line in the plane [.x12]; since for all other points of this straight line the conjugate hyperplanes are known, we can again fall back on the construction of 34.1. Now, starting from an arbitrary point y of the plane [x12], we can apply the same reasoning also to the points of the plane [y34]; this means that we can construct conjugate hyperplanes for all points of the four-dimensional plane [x1234], etc. If Az+jtt is a singular point, then the points z and t are readily seen to correspond to one and the same conjugate hyperplane; taking this fact into consideration, we see that we can construct by this process the conjugate hyper-

planes for all points of the space, and thereby all straight lines of the complex K1 will be determined. We have arrived at the confirmation of Theorem 34.6 by geometric means. 34.4 In the following sections, we will present another, simpler rule for the determination of the rank of a bi-vector and establish a complete system of its invariants. For this purpose, consider concomitants of the bi-vector v+1 which are themselves polyvectors of the form (2k-1)!!vti,J&vt:1:

...

vikJkl,

(34.14)

where, as usual, (2k- I)!! denotes the product I - 3 5 ... (2k-1). Each of the components of the polyvector (34.14) is called a Pfaf

CHAP. V11

POLYVECTORS

388

aggregate of order k. If we expand the alternation in (34.14), we obtain a sum of (2k)! terms which fall into groups of identical terms; such identical terms are obtained from each other either by permutation of the multipliers v together with their indices or by transposition of indices in one or several factors [cf. § 10, Example 4, also (10.12)]. Hence it is clear that in each group there will be 2k - k! equal terms, as a consequence of which the number of different terms in a Pfaff aggregate of order k is (2k)! = 1 - 3 - 5 ... (2k-1) = (2k-1)!!.

2k-k!

Note that if we expand (34.14) with respect to i1 following the instructions of 31.3 [cf. (31.7)], we obtain k terms all of which are equal to each other. This is readily verified by transposing the indices in these terms in a suitable manner. Consequently, the polyvector (34.14) can be written in

the form (2k-1)!!Vi1[1,V212V131,

..

. Viklo

(34.15)

Examples: 1]

3v[12v341

2]

15v112v34v561 = 1Sv1[2v34v561 = V12V34V56-v12V35V46

=

3v1[2v341

= v12v34-v13v24+v14v23

+012 036 045 -V 13V24v56 +v13 v25 v46 - v13 v26v45 +v14 v23 V56 -014 02 5 036 + VX4 V26 V35 - V 15 V23 V46 +v15 V24 V36 - V15 V26 V34

+V16 V23045 - V16 V24V35 +V16V25 V34

We will now prove Theorem 34.7: If the polyvector (34.14) is equal to zero for k = s+ 1, it will be simple for k = s. By assumption [cf. (34.15)], (34.16)

v[1,1, V1212 ... Vi.l.vk]1 = 0.

If the polyvector vanishes for k = s, nothing has to be proved; however, if it is non-zero for k = s, construct for the polyvector (2k-1)!!v[i,1,v1212

... vi.J.]

_ (2k -1)! lv11[J1 v12J2

... vi. J.]

"'i111hh...i.J. _

the expression w[i,1,12J2...1.J. wk1]11k212 ... k,1,

= {(2k-1)11j'v[i111 v12J2 . . . vr.J. Vk1 x11 Vk212 ... vk,l.];

(34.17)

389

THE BI-VECTOR

34.4-34.5

by (34.16), this is equal to zero and, consequently (cf. Theorem 31.9), the polyvector (34.17) is simple.

The induction from (s-1) to s is now readily accomplished by the following rule for the determination of the rank of a bi-vector: Theorem 34.8: If the polyvector (34.14) is equal to zero for k = s+ I and non-zero for k = s, the rank of the bi-vector is 2s (for s = 1, Theorem 34.8 is true by Theorem 31.12). By assumption, the equality (34.16) is true. By Theorem 31.7, the polyvector (34.17) is simple: 1

2

1

2

s

s

V[i1J1 VI2J2 ... visJ,) = p(j.gj.piegJ2 ... pi,gj.), 1

2

1

2

s

s

where p, q, p, q, ... , p, q are 2s linearly independent vectors, as a consequence of which (34.16) assumes the form 1

2

1

2

pti,gJ.pi2gJ2 .

s .

.

s

pi,gJ,vk)1 = 0.

(34.18)

The relation (34.18) shows that all the vectors vklxl of the space of the 1

1

2

2

s

bi-vector v can be expressed in terms of the vectors p, q, p, q, ...,

s

p, q,

whence it follows that the rank of the bi-vector v1 j is equal to 2s (it cannot be less than 2s by the inductive assumption).

Next, we will seek a complete system of invariants of the bivector vi j and, as always, denote its rank by p. For p < n, the vector v does not have non-zero invariants: in accordance with the fundamental 34.5

Theorem 17.4, one can restrict consideration to invariants which are derived from products of the form vi, j, vi2J2 . . . vi,,j,,, 2k = ng, with the aid of g

total alternations. Place the bi-vector into its space; then we find that all these invariants vanish identically (since every index enters under one of the alternations over n indices, and n > p).

For p = n = 2s (where s is an integer), we have the invariant of weight I P = (n-1)!!v[12 v34 ... ve-1..,] ,

(34.19)

which is non-zero (cf. Theorem 34.8). It is not difficult to show that every invariant I of the bi-vector v can differ from a power of P only by a numerical factor. In fact, reduce the bi-vector v to the canonical form (34.9) 1

1

2

2

s

and take p, q, p, q, ..., p, q as coordinate vectors; then v12 = V34 = v56 = ... = vA -1, R = 1, while the remaining components of the bi-vector v

POLY VECTORS

390

CHAP. VII

vanish and (34.19) gives P = 1. If the weight of the invariant I is equal to g and its value in the above system of coordinates is c, one has in this coordinate system

I = c Pe; since both sides of this equality represent invariants of weight g, it is true for any coordinate system. Thus, one has

Theorem 34.9: Bi-vectors whose rank p is less than the order n of the space, do not have non-zero invariants; however, if n is an even integer

and the rank p = n, a complete system of invariants of the bi-vector v consists of the single invariant P defined by (34.19). Exercises

1. Let an (r-1)-dimensional plane E,_1 be given by the simple, contravariant rvector ui1i2 ... it and the complex K1 by the covariant bi-vector v1J. Find a covariant

r-vector w1112 ... i, corresponding to the (n-r-1)-dimensional plane conjugate to ',_1 with respect to K1. 2. Establish the geometric meaning of the invariant equality Vtl it Vi2j2 ... V ,j,

W1111... Jr 1y>J1 J2... J, = 0, 1

2

where v;5 is the covariant bi-vector to which the complex K, corresponds and w and w

are simple contravariant r-vectors which determine (r-1)-dimensional planes E1_1 and E; 1. 3. Let there be given a covariant bi-vector of rank p and two linearly independent covariant vectors p and q; what is the rank p' of the bi-vector v = v+ [pql? 4. Reduce to the canonical form (34.9) the bi-vectors with components: -2; n= 4, a) vas=2, v1s -1, vu=0, as'=4, vs,= 3, vu= b) vis = 1, v16 = 2. V23 = -3, V34 = 1, V13 = V14 = Vsi, = v,1 = V33 = V{s = 0;

n=5.

S. Prove Cayley's Theorem: A skew-symmetric determinant iv;,j of even order n is the square of the Pfaff aggregate (34.19). 6. Suppose there has been constructed for the concomitant 2v[,ls v,,],] of the bi-vector v,, a determinant D of order n(n- 1)/2 in which the rows are determined by the pair of indices i, j (I > j) and the columns by the pair of indices x, y (x > y). Prove that D is an invariant of the bi-vector v,l and for even n express D in terms of the invariant P [cf. (34.19)].

§ 35.

The tri-vector

As in the case of symmetric tensors, the two fundamental problems (cf. 21.5) become considerably more complicated as a consequence of a transition from variance r = 2 to higher values of r for polyvectors. 35.1

At this stage, the theory of the tri-vector is still far from complete;

34.5-35.2

THE TRI-VECTOR

391

the available results have been derived by very involved methods. In the case of non-simple r-vectors for r z 4, only trivial cases have been treated [when the r-vector is correlative (cf. 32.1) to a bi-vector or tri-vector of a type already studied]. Due to shortage of space, we will restrict the study

here to results obtained in connection with the classification of trivectors, omitting all proofs (tri-vectors are always assumed to be covariant; by the duality principle, all results are readily extended to contravariant tri-vectors). If the order of the space is n _< 2, all tri-vectors are equal to zero. For n = 3 and n = 4, every tri-vector w is simple (cf. Theorems 31.1-31.3); thus, one has here only two types of tri-vectors with the canonical forms I:

w = 0,

II: w = [qrs],

where q, r, s are linearly independent covariant vectors. Since a tri-vector can always be placed in its space, the above two types correspond to the values of the rank p = 0 and p = 3; for n = 5, a third type has to be added comprising tri-vectors of rank p = 5 (the value p = 4 being impossible; cf. Theorem 31.8). In a fifth order space, every tri-vector w is divisible by some vector p (§ 31, Exercise 4, § 33, Exercise 2): w = [vp], where v is a bi-vector; the rank of w will be equal to 5, if the rank of v is equal to 4. Taking into consideration Theorem 34.4, we arrive at the result that a

tri-vector of rank 5 can always be reduced to the .canonical form III: w = [aqp] + [brp], where the covariant vectors a, b, p, q, r are linearly independent. 35.2 Next, consider tri-vectors of a space of order n = 6. If the rank of the tri-vector p < 6, it belongs to one of the types studied in 35.1. For p = 6, the tri-vector w may be reduced to one of the two canonical

types IV:

w = [aqr] + [brp] + [cpq],

V:

w = [abc] + [pqr],

where a, b, c, p, q, r are linearly independent vectors; this was first shown by Reichel *). When n 5 5, every type of tri-vector is completely characterized by the corresponding value of the rank p; for n = 6, this is no longer the case: the value p = 6 corresponds to both types IV and V. In order to give *) Reichel, W., Uber die trilinearen alternierenden Formen in 6 and 7 VerAnderlichen (Dissertation, Greifswald), 1907.

CHAP. VII

POLYVECTORS

392

arithmetic characteristics for all types, new arithmetic invariants will be introduced. Construct for a given tri-vector w the concomitant (35.1)

Wtijk Waib, lc, l Wa2b21C21 ... Wasbrlca;

the tensor (35.1) is readily seen to be symmetric (cf. § 31, Exercise /) with respect to the indices ct , c2 , . . ., c,. Its rank p, with respect to any of these indices is (cf. Theorem 14.11) an arithmetic invariant of the tri-vector wiJk. Giving s the values 1, 2, 3,

.

.

. we accomplish the objective

of finding a series of such invariants; it is not difficult to see that for

n= 6, p,= 0,ifs _ 2. The numbers (pp 1) also give arithmetic characteristics of the five types,

stated above; their values are I: (00),

II: (30),

III: (51),

IV: (63),

V: (66).

(35.2)

The characteristics (35.2) have the following two remarkable properties: 1) The sum p+p1 is always divisible by 3; 2) The law of complement applies (cf. 22.6, 23.1): there corresponds to every type another type related to it in the following manner: if we write underneath the characteristics of the first type the characteristics of the second type in reverse

order and add, we obtain as the sum (66). Such types are said to be complementary. The pairs of complementary types are (1, V), (II, IV), (III, III). Next, we will give the algebraic characteristics of the types I-V; the concomitants of the tri-vector w W[iJk Wab]c ,

W[ijk Wab[c Wxy]z] ,

W[ijkWab[cWdef Wx7]=] ,

to be denoted by C, C, C, respectively, participate in their formation. 1

2

3

If CO 0, the tri-vector belongs to Type V, if C = 0, C # 0, to Type IV, 3

3

2

if C = 0, C # 0, to Type III, if C = 0, w # 0, to Type II (cf. Theorem 2

1

1

31.12) and, finally, if w = 0, the tri-vector belongs to Type I. 35.3 The classification of tri-vectors in a seventh order space is due to Schouten "). The types enumerated in 35.1, 35.2 must now be augmented ) J. A. Schouten, Klassifizierung der alternierenden GrOssen dritten Grades in 7 Dimcnsionen. Rend. circ. mat. Palermo, 55, 1931. Reichel (cf, above) gave an incomplete classification with omission of Types V1 and VIII.

393

THE TRI-VECTOR

35.2-35.3

by another five: their canonical forms can be unified in the following formula:

w = a[abc] +fl[grs] + y[agp] + b[brp] + e[csp],

(35.3)

where a, b, c, p, q, r, s are linearly independent vectors and at, P, y, 6, e are numbers which are equal to 0 or 1. These five types are obtained for the

following values of a, fl, y, 3, a [the arithmetic characteristics (pp, p2) are given in brackets; for n = 7, p., = 0, if s z 31:

VI: (711) a=f=0, y6=e= 1, VII: (741) a=0, fl=y=b=e= 1,

VIII: (762) 6=e=0, a=1=y= 1, IX: (774) a=0, a=fl=y=b= 1, a=J3=y=6=e= 1. X: (777)

(35.4)

It is not difficult to see that the canonical forms of the Types I-V are also obtained from (35.3) with suitable values of a, fl, y, 6, e (for

Type IV: a=c=O, j9= y= 6= 1, for Type V: y=6=e=0, a = f = 1); their arithmetic characteristics are the same as in (35.2), but, in addition, one has a zero in each bracket (since p2 = 0). The properties stated in 35.2 for the characteristics (35.2) are also retained for n = 7: the sum p + p l + P2 is always divisible by 3; the ten

types form five pairs which are complementary with respect to each other: (), X), (11, IX), (III, VIII), (IV, VII), (V, VI). For each pair, if we write the characteristics of the one type in reverse order underneath the characteristics of the other, we obtain as sum (777). Next, consider the algebraic characteristics of the types I-X. Introduce the concomitants C, C, C, C, C of the tri-vector walk representing the 4

tensors W4[bcWdefWij]k,

3

6

8

7

W[o[bcWdef

Wa[6cWdef Wi1HkWpgr],

Wpgre"gkpqr

Wa[bc Wdef Wil][k WxylzI Wi]yk

Wa[bcWdef Wii][k Wi,l,k, WxylzI Wick Wpgre

zokpg

.

If C # 0, the tri-vector belongs to Type X, if C = 0, C :A 0, to Type IX, 8

8

7

if C=0,C#0, to Type VIII, if C = 0, C# 0, to Type VII, ifC=0, 7

6

6

5

5

C :O 0, to Type VI, if C = 0, CO 0 (cf. 35.2) to Type V. For the other 4

4

3

four types, the algebraic characteristics remain the same as in 35.2.

394

CHAP. VI!

POLYVECTORS

Note the interesting link which exists between the algebraic and arithmetic characteristics. The algebraic characteristics of each type have the

form: C = 0, C # 0; if for the concomitant C we arrange the numbers i+t

I

I

indicating the number of indices in each of the alternations in nonincreasing order, we obtain the arithmetic characteristics of the type (in this context, an index which does not occur under any alternation sign is regarded as an alternation of one index). 35.4

For the case n = 8, a classification of tri-vectors was first

established by the author of this book *). For p < 8, we have the 10 types of 35.2 and 35.3; tri-vectors of rank 8 fall into 13 types whose canonical

forms can be obtained from the formula w = a[abc] + /3[grs] + y[aqp] + 6[brp] + e[csp] + 2[bst] + p[crt] (35.5)

by giving the scalars a, /3, ..., p the values 0 or 1. For n = 8, p, = 0, when s > 3; the values of the arithmetic invariants p, P I , P2 turn out to be insufficient for the distinction of all types. One has to augment them by three arithmetic invariants 03, az , a3 Construct the following three concomitants of the tri-vectors w: WtijkWablel Wde[fWxy]z],

(35.6)

W[i jk Wablcl Wde[ f W9hI Wxy]z] ,

(35.7)

W(ijk Wablcl Wde[f W9hi Wpglrl Wsy]z]'

(35.8)

The invariant aI is the rank of the tensor (35.6) with respect to the index c and the invariants a2 and a3 are equal to the ranks of the tensors (35.7), (35.8), respectively, with respect to the same index c. The following table gives the values of the scalars a, , ..., µ which must be introduced into (35.5) in order to obtain the canonical forms of the 13 types mentioned above and also their arithmetic characteristics (PPIP2; QIQ2a3):

XI: (863; 100) a= f3=, =0, y=6=E=µ= 1, XII: (874;200)a=2=0, f3=Y=6=e=µ= 1, XIII: (884; 400) 6=a=,i=0, a=B=y=p= 1, *) The arithmetic characteristics are stated in 35.2, 35.3, 35.4 and the algebraic charac-

teristics in 35.3, 35.4. Cf. G. B. Gurevich, On tri-vectors in a space of 7 dimensions. Dokl. A. N. SSSR, I[1, No. 8-9, 1934; Classification of tri-vectors of rank 8, Dokl. A. N. SSSR, 11, No. 5-6, 1935.

395

THE TRI-VECTOR

35.3-35.4

XIV: (886; 410) c=A=0,

f =y=b=µ= 1,

XVIII: (888; 741) a=E=0,

1,

XV: (887; 520) A=0, =fl=y=8=E=µ= 1, XVI: (888; 411) a=08=e=0, y=2=u= I, XVII: (888; 621) at =$=e=0,y=S=µ=1, XIX: (888; 822)a=/3=0,y=6 =E=a=µ=1, XX: (888; 852) a=0, fl=7=b=E=A.=µ= 1,

fl=y=A=µ= 1, a=0, a=fl=y=6=A= p= 1,

XXI: (888; 873) 6=E=0, XXII: (888; 885)

XXIII: (888; 888) a=Q=y=6=E=A=p= 1. The arithmetic characteristics of Types I-X are obtained by adding three zeros in each of the brackets (35.4) and four zeros in each of the brackets (35.2) (since, for p < 8, one has Q1 = cr2 = Q3 = 0).

In each of the characteristics, the sum P+pl+p2+a1+a2+a3 is always divisible by 3. The law of complement now has exceptions: the pairs of Types (I, XXIII), (II, XXII), (III, XXI), (IV, XX), (V, XIX),

(VII, XVIII), (VIII, XVII), (IX, XVI), (XI, XV), (XII, XIV) and (XIII, XIII) are complementary to each other [if we write one set of characteristics in reverse order underneath the other and add, we obtain every time (888, 888)]; however, Types VI and X have no complements. Note further that for Types XI-XXIII also the algebraic characteristics, for which the relations to the arithmetic characteristics are stated at the end of 35.3, remain valid. The problem of the classification of tri-vectors for n ? 9 so far remains unsolved.

ANSWERS AND HINTS TO EXERCISES CHAPTER 1

§1 1. 1 °) A

-1. The segments Ms 0 and M1 M, are divided by their point of inter-

-->

---k

section in one and the same ratio A. 20) A = -1. The vector M,O = M1M,. 2. X+iY = el-(X'+ IY'); yes (the direction of the vector is isotropic, cf. 7.4). 3. The point of intersection of g and M1 M, divides M1 M, in the ratio A. 4. In (1.11), the angle q> must be measured from g1 to g,; the sign of the angle is

determined by the corresponding orientation of the plane. In (1.12), q; is the angle between the positive directions of the straight lines (the positive direction of the straight

line is obtained from the direction of the vector (AB) by rotation through +90°]

5. Hint: Reduce the equation of the curve to the form y' = 2px-(1-e')x'. Answer:

1,

t

s

e the eccentricity of the curve.

(2 p

6. ta'b', -(all ±b'), where a, b are semi-axes; the upper sign for the ellipse, the lower for the hyperbola. 7. Hint: Take the equation of the circle in its simplest form: draw the y-axis through M. Answer: The degree of the point M with respect to the circle (1.18), i.e., the product

MK1- MK,, where K1, K, are the points of intersection of the circle (1.18) with any straight line through M. 8. As the components of a vector; p, q are components of a vector perpendicular to the polar of the point M with respect to 1'. 9. The point M lies on the diameter of I', conjugate to the direction of the vector tt. 10. The square of the distance from the centre of I' to the polar of the point M with respect to r.

11. A'-C'+2B'i = e-21w(A-C+2Bi); 1' is a circle. 12. Hint: If f(u1) is continuous and f(w)f(w') - f(co +w'), then f(w) = ek-. Answer: q1 = q'(w, h, k) = ceNo. , c = 1 or 0, N an integer; if A is real, N = 0. § 2

1.w=n,h=2x0,k=2yo. 2. cu' = w; h' = h cos (w - w) - h sin (w - w) + (h - h) cos w + (k - A) sin u,, k' = h sin (w -@) + k cos (w-a,)-(h-h) sin w+(k-k) cos w, where w', h', k' are new parameters of the motion S, and (9, h, h are the parameters of the coordinate transformation.

3. (0 5 co < 2n). l °) w rA 0. Fixed point (xo, yo), where x0 = }h-}k cotan w/2,

y, = } h cotan w/2+ k. 2°) co - 0. No fixed points.

3

ANSWERS AND HINTS TO EXERCISES

397

4. Hint: For w # 0 (0 < w < 2a) transfer the origin of coordinates to the fixed point (cf. Answer 3).

6. Hint: Distinguish the cases (0 S w < 2n) : 1 °) w:0, w # n; 2°) w = 0; 3°) w =n. For w # 0, transfer the origin of coordinates to the fixed point (cf. Answer 3).

Answer: 1°) There are no invariable vectors; the directions A : B = ±i (isotropic directions); the straight lines with isotropic directions through the point which is fixed under the transformation. 7. Hint: Distinguish the cases (0 0; take the fixed point (cf. Answer 10) as origin of coordinates.

12. Hint: Transfer the origin of coordinates to the centre of transformation for e = -1; take, in addition, the axis of the transformation as x-axis (cf. 11). 14. Hint: Two conjugate hyperbolae go over into each other for an imaginary homothety.


398

§§ 4-5

§4

2. Hint: Take the point A as origin of afline coordinates, AB and AC as coordinate vectors.

3. The ratio of the parallel sides in both trapezoids must be the same. Let K and K' be the points of intersection of the diagonals of the quadrangles ABCD and A'B'C'D'; the point K' must divide the diagonals A'C' and BV in the same ratio as K divides the diagonals AC and BD. 5. Hint: Take a and b as coordinate vectors and use 4.

6. Hint: Use 4. 8. Hint: Cf. 5.

9. Hint: I is an invariant of weight -1; use 4.

11. Hint: Show that (4.43) is invariant (cf. /0); take yt and q as coordinate vectors.

12. Hint: Specify the straight line by the parametric equations: x' = xo+Z't. = 0; the centre xo is Answer: The vector has an asymptotic direction, if determined by the equations a,5xj,+a, = 0, i = 1, 2. Two diameters parallel to the vectors ;, 77 are conjugate if

13. Hint: For 6,

0.

0, reduce the equation of the curve to the form a1tx'x1 = 1

(using the origin of coordinates as centre); take coordinate vectors $, q satisfying the condition 0 (cf. Answer 12). Use (4.43).

Answer: 1: b, > 0; 8,a;1E'' > 0 for arbitrary 1:. The canonical equation: (x')'+ (x')'+ 1 = 0. Imaginary second order curve. II: 6, > 0; 63a;,l:'$' < 0 for arbitrary ¢. Canonical equation: (x')'+(x')' = 1. Ellipse. 111: b, < 0. Canonical equation: (x')'-(x2)2 = 1. Hyperbola. IV: 6, = 0. Canonical 'equation: (x')'+2x' = 0. Parabola. 14. Hint: Cf. 13.

Answer: 1: 6 # 0. Canonical equation: (x')'- (x')' = 1. Central second order curve. If: 6, = 0. Canonical equation: (x')11+2x' = 0. Parabola. § 5

1. pi, = Nrll,), pi- = -711M. 2. The double point has the coordinates x' : x' = -p=: (pi-p) = (pl, -p): -p', where p is one of the roots of the equation: p'-(pi+p',)p+p;p,*-p,p; = 0. 3. pi-t-p; = 0; it cannot. 4. {xyzt} _ -A, where A is the ratio in which z divides the segment xy. S. Hint: Cf. Answer 4. '6. If the points x, y coincide, the points z, t divide them harmonically, if at least

one of them coincides with x = y, x'y'+x'y' = 0. 7. Hint: Take three of the given points as coordinate points. '8. Hint: Take the points p, q determined by the quadratic form ail as first and second coordinate points and use 6. Answer: The points x, y divide the points p, q harmonically. 10. Hint: Write down the involution in 9. 11. Hint: Assume the vectors p, q, r to start from one point; use § 4, Exercise 12. '13. Hint: Let the discriminant of the form air be zero; take the corresponding point peas first coordinate point. Answer: At least one of the points determined by the second form coincides with

the point p. 14. The pairs of points corresponding to the forms should have at least one common point.


§§ 6-7

399

15. Hint: Take r(1-h the points corresponding to one of the forms as coordinate points.

Answer: 1= 4 1 +h/ ', where h is the cross-ratio of the pairs of points deter\ mined by the forms. § 6

2. There appear multipliers d-' and d which are not essential because of the homogeneity of the coordinates. 4. Hint: Normalize the coordinates of the four vertices x, y, z, t, so that x+y+z+ t

= 0. 6. X, y, x+y.

7. (uy)x-(ux)y. 8. Hint: Use the Answer 7. Answer: [yzt]x- [xzt]y or [xyz)t- [xy1Jz.

10. Hint: Show that the straight line joining x and s is [xys][xz]-[xzs][xyl; cf. Theorem 6.4.

Answer: I =

[xys][xzt) [Xyt] [xzs]

11. Hint: Cf. Answer 7. Answer: 1 is equal to the cross-ratio of the points x, y and of the points of intersection of the straight line [xy] with the straight lines u, v or to the cross-ratio of u, v and the straight lines joining the point [uv) to the points x, y. 12. Hint: Cf. 11.

Answer: I is equal to the cross-ratio of z, t and the points of intersection of the straight line [zt] with the straight lines [xy) and u. 13.

{[ab], [ad], (ael, [af)) {[dcl, [da], [del, [df1}

_ {[ab), [acl, [ae], [af]) {[cd], [cal, [ce], [cf]}

14. Hint: Use 11. 15. Hint: Use 12 or 11. 16. Hint: Cf. 14 and 15. 18. 1 = {xyrt}, where t is the point of intersection of the straight lines [xy] and [pql. 19. Hint: Normalize the coordinates of the vertices x, y, z, t of the complete quadrangle in such a way that (ux) = (uy) = (uz) = (ut); use the results of § 5, Exercise 10.

21. pi, = '[nCe),pi' 25.

pa. _ [ 1

0.

26. Hint: Apply 25. 27. all = aaP = itd. 29. If la"I = 0, the curve of the second class represents two points (distinct or coincident). 30. This property is not projective (Theorem 6.6); it is affine, since it may be defined

by means of the concept "between": the intersection of the straight lines tx and yz lies between y and z, etc. § 7

1. Hint: Normalize the coordinates so that (lx) = (ly) = (lz) = (it) = I. 2. For fixed coordinates 1, of an improper straight line, the covariant vector is given by the three non-homogeneous coordinates u, (i = 1, 2, 3) mod /,, i.e., two covariant

vectors u, and v, are equal, if u,-v, = ill,; the law of transformation: M. = pau,,

400


§§ 8-9

where the pi have been normalized by (7.9). The initial straight line of the covariant vector ur has the coordinates ur, the final ur+I4. 4. Hint: The general case may be reduced to the particular one when the triangles have two common vertices x. Y. Use § 6, Exercise 12. Answer (for the particular case stated): If the other vertices are z and t, the ratio of the areas is equal to the cross-ratio of the points z, t and the points of intersection of the straight line [zt] with the straight lines [xy] and 1. 6. The asymptotic directions are the points in which the improper straight line intersects T; the centre is the pole of the improper straight line with respect to F. The diameters are the polars of the improper points with respect to I'; two diameters are conjugate, if their improper points are conjugate with respect to F.

7. g,r = erel, where e,(i = 1, 2, 3) are the coordinate vectors. 8. Hint: Cf. § 4, Exercise 4. Answer: The sign (+), if the coordinate parallelogram is orientated positively, the sign (-), if it is orientated negatively (where the orientation of Cartesian systems is taken to be positive). 9. Hint: Proceed as in the case of the derivation of (7.12). Answer:

(u,x'+u,x2+h)1'g \/&Tut)'-2gitu,us gu(u:)i 10. 1 °. Let a = AD and the ellipse F. of the T-system G with centre at A intersect AB in C; A is equal to the square of the ratio in which C divides AB.

2°. The vectors a and b are parallel to two conjugate diameters of the ellipse E. 3°. Let b = AK; draw from Ka straight line parallel to the diameter of E, conjugate to the diameter AB; if D is the point of intersection of this straight line and AD, then -).

-->

µ= AD:AB. 11. Hint: Apply the conditions for asymptotic directions of § 4, Exercise 12. 13. The isotropic straight lines through the point of intersection of the straight lines u. v divide them harmonically. 14. Hint: Consider first the case of parallel segments and of segments starting from one point; cf. 7.4, Example 1. 15. The curve r passes through the cyclic point I. § 8

2. x'/x! is equal to the cross-ratio of the point x, the unit coordinate point e and the points of intersection of [xe] with the i-th and j-th coordinate hyperplanes; for the coordinate hyperplane, by the duality principle.

CHAPTER II § 9

3. c= f 1. 4. jtn(n-1)(0-n+2). 7. 6;.

rw_Pr P'n qs q# 8. cru y r t caBy' 11. ail = at, )-ra[il]-


10-14

401

13. Hint: It follows from (9.37) that u,v,x'x' = (u;x')(v,x') = 0 for an arbitrary vector x. 16. Hint: Take the vector p as coordinate vector.

17. Hint: Apply 16 to the tensor cifgpf. § 10

1. Hint: For the case when [xy] intersects the hypersurface (10.2) in two distinct in the case when this straight line y' = touches the hypersurface at the point i, let y' = Answer: In the first of the cases, I = (i+h)'/4h, where h - {xyEri}, in the second,

points , ri, let x' = I = 1.

3. Hint: Start from the condition that the three points x, y, z are to lie on a single straight line. Answer: v[" zk) _ 0. 5. The pencil of straight lines with centre at the point corresponding to the bi-vector vd,.

6. A manifold of straight lines intersecting the straight line v,1.

7. Hint: Show that for any pointy of the plane u, = c,,x'the relation c(h)e'y' = 0 is true, whence it follows that the given correlation can also be specified by the tensor C($).

8. Hint: Cf. Theorem 9.3. 9. To every point x of space corresponds a hyperplane through x and that (n - 3)dimensional plane in which the hyperplanes p and q intersect. § 11

1. Hint: Introduce vectors

such that

P = z+ax+fly = 2'+Yx*+dy',

q = Y+sx = y*+Cx'.

3. Hint: Apply the relation (11.21) to the three matrices A1i Aj, Ak of the base. § 12

2. Hint: Both sides of (12.11) are skew-symmetric with respect to all superscripts and with respect to all subscripts. 3. Hint: Contract (12.11) first with respect to one pair of indices, then with respect

to another, etc. 5. Hint: Employ the relations [abc] = ([ab]c) = (a[be]) as well as the results of 4. 6. Hint: Apply (12.13) and (12.16).

CHAPTER Ill § 14

5. Hint: Cf. 4. 6. Hint: Use 4. 7. Hint: Take u as third coordinate line and apply Theorem 5.6. Answer: a[1[lbaaua] ual = 0.

8. Hint: Introduce unit n-vectors and employ 4.

9. Hint: Reduce the equation of the curve to the form 2(x1)'+µ(x')'+v(x')' = 0 (cf. § 6. Exercise 25) and use 7.

§§ 15-17


402

10: Hint: Cf. 4. Answer:

11. Hint: Denote by b'J the minor of the element b in the determinant 1b,,I, apply 4 and 5. Answer: The polar plane of the point x with respect tho the hypersurface a,Jx'x'=O passes through the pole of the hyperplane u with respect to the hypersurface

12. Hint: Without reducing generality, one can assume that the linear form is equal to x1. 13. 1 = 2r-' a1111... [1621 .. ,,j,1, where the number of alternations is r; for even r= 2k,

f= aoa,k-C ala,k-1+Cua,a,k-,-...+(-1)k-1C;tlak-1ak+l+k(-1)kC'(ak)'. 14. Hint: Write down the law of transformation of the coefficients of the quadratic

form f in the form a, = Poao+P,'aj+P,1a,+ ..., etc. 15. 11 = }(S,'--S,), 1, _ 1(S;-3S1S2+2S3), for n = 3:

Is = JAIJ.

§ 15

1. In (15.20), the left-hand side does not change, the right-hand side assumes the

form n(g- EL1g,d,), where g, is the weight of the tensor

in Theorem 15.9, the a; word "non-absolute" must be replaced by the words: "the weight of which is not equal to the product of the weight of the tensor and the degree of the invariant". 2. Hint: The weight of the covariant is equal to the weight with respect to x1 of the

term with (x')', where p is the order of the covariant. 3. Hint: Such a form can be reduced by a linear transformation to the form (xl)r; apply Theorem 15.7. 4, S. Hint: Use 3. 7. Hint: Cf. § 14, Exercise 14. Apply Theorems 15.1 and 15.7; cf. also (15.18). Answer: r(r+ 1)/2.

8. Hint: Cf. 7 and Theorem 15.10. Answer: (N/n)(r-s). CHAPTER IV § 16

2. Hint: Apply the linear transformation (15.16) to the tensor. 3. Hint: Use 2. 5. Hint: Since the coefficients of (16.26) are integers, the solutions of (16.27) can likewise be assumed to be integers, and Theorem 16.1 remains valid. § 17

1. Hint: Cf. § 16, Exercise 3 and Theorem 15.6. 3. K'JDQ = }(Ef9g1Q+Eif EJp)

4. a) K' = 61 ,

Kfi9a = &1t9601J

b) Bt1pe = Vfi X90

c) Bsok = a,,,x'xlxk,

K,,,, = iEf19a K'ti = 6i' 6! 61,1,

d) B,Jkipgra = aj$kt a,,,r, KuhIM = je'(P e I J It a Ik It

5. B = A' A$,, K(=

tI al

neither tensor changes for transposition of the pairs

of indices i, p and j, q; B,J = c,, c,,, K" = elf Evi.

403


§§ 18-19

§ 18

1. (1 /2) [ab]'; for odd r, the interchange a" b changes the sign in the symbolic representation of the invariant; consequently, it is equal to zero. 2. Q = [abJ'[ca)(ax)+-'(bx)'-'(cx),*-'. 12

n-1

w-I

12

1

1

3. [abb...b][ubb...b1(ax), for 4.

1,

(a8) (bd)

(ab) Is

(bb)

=

(ad)

(ab)

(ac)

(ba)

(bb) (cb)

(be)

(ca)

22

w-1 n-1

b,.

.

(a')

5. Hint: Expand the determinant with respect to the elements of the third row. Answer: [abJ[ac][bc], 6. Hint: Use the Vandermonde determinant; the legality is clear, if one changes over

from the symbolic representation employed to the determinant and reasons as in 4. Answer: [baj[ca][cb)[da][db][dc] ... [la][Ib) ... ilk). 9. I, = c,,-c,,, Is = 2(c11c,s-cac,i), Is = I,-112. Cf. § 17, Exercise J. 10. Hint: Expand the symbolic determinant [abu] with respect to It, and find the coefficients of (u1)' and %u,; the remaining ones are found in an analogous manner. 14. Hint: Replace simultaneously a .- d, b i--' c.

*15. Hint: Introduce the notation (a,)' = a a,a, = a,, (a,)' = a,, and analogous notation for the parallel symbols; use 5 and (18.28) for n = 3. 17. Hint: Cf. 15. fill fills fills Answer: j = fius fins fills [cf. (14.24)].

fills fins fills 18. Hint: The expanded expression of the determinant contains the inadmissible terms aia= and -a;a,', the first of which is replaced by aiia,,, the second by -a',,,

19. -}[pq]'

20. 0 = j{[pr](gx)(sx) + [ps](gx)(rx) + [gr](px)(sx) + [gs](px)(rx));

6= j { [pr] [qsj+ [ps] [qr)} ; if 6 = 0, the cross-ratio { pgrs} _ -1 [cf. Theorem 5.61.

21. }[prJ[gs]+ 22. Hint: Use (18.47) Answer. Q = , [pgJ'(px)', D = 0. 23. [ap][aq][arj[as].

24. Hint. Apply 21. Answer: For the ternary form D = -} {[pgr] [pgs]+ [prs][grsj}, for the quadratic

D = j(pgrs]'. 25. Hint: Use 12 and 13. 26. Q = 27 I(8d9,'+R). 28. Hint: Apply result of Example 14. 29. Hint: Cf. Example 14. s

3 0 . Hint: Assume that (px) = xa, a = 1, 2,

..., k.

§ 19

1. Hint: a F-, b, arithmetic mean. 3. Hint: H can be represented in the form of a sum of three terms; in one of them a" b, arithmetic [Wean.


404

§ 19

Answer: H = 00-01. 4. Hint: Cf. 2 and 3; HQ = }[abj' [de]' [ca][gd][beJ[cgj(cx)(gx) (g is a symbol parallel to a), c *-' g, arithmetic mean.

Answer: DQ = D', QQ = -D'q2. 5. Mint: Cf. (18.52). Apply twice the third fundamental identity to the difference

H.-HI-

Answer: H, = H,+;i[xy]'. 6. Hint: Use (19.21) and 1. Answer: H = i, (3Dxgo2-D,=fq'+3D,f2). 7. Hint: In G1, a"b, arithmetic mean; in Go, a i-4 fi, arithmetic mean. 8. Hint: The same method as in 5. Answer: }D1,f-DjV. 9. Hint: Apply the results of § 18, Exercises 19 and 20. Answer: Dl,,-4D1Ds; cf. (19.19). 10. Hint: Use symmetry with respect to p, q, r and the identity (19.6).

Answer:H=-ie{[p9)'(rx)'+[prJ'(gx)'+[grJ'(px)'}=-Is aiE(Fs- s)(x- ly)'. 11. Hint: D is four times the discriminant H; use the Answer to 10.

Answer: D = - 27 L[9rp'(rpl'[pgl' _

- fi ao( - s)'(

12. Hini: Add the expression ;le {[pq][rs]-[pr][gs]+ [ps][gr]}', which is equal to zero by virtue of the identity (19.7).

Answer: i = ,,I,-{[pq]'[rs]'+[pr]'[gsl'+[psJ'[grl'}. 13. Hint: To (19.38): apply (19.6) to [ac] (bx)(dx). To (19.39): express [ob](cx) using (19.6) and take the square of both sides. 14. Hint: After the replacement shown, take the arithmetic mean and apply (19.39); we obtain two terms one of which is equal to zero; in the other, c .-+ d, arithmetic mean. 15. Hint: a r+ b, arithmetic mean and apply (19.39), after which a 4 -+ c, then b 4--, c, arithmetic mean of all three expressions.

16. Hint: In the term containing [bp]', a +- b, arithmetic mean, apply identity (19.39).

Answer: H =

0(a-1) r (r -1)

hq sp

pa

12y,tp-E.

17. Hint: Let j= (px)Pq', where p Z I and 97 is not divisible by the linear form (px) and apply § 18, Exercise 28 and § 19, Exercise 16. 19. Hint: Expand each of the symbolic determinants [abc] and (abd] with respect to the elements c,, d respectively; the coefficient of x'xt turns out to be 2Aapa.1 apt = 2DaI, (where AaP is the adjoint of a,p in the discriminant D). 20. Hint: a .-. b, then a 4-+ c, arithmetic mean. 21. Hint: a a+ d and simultaneously (x e-+ P, arithmetic mean; after application of the fundamental identity, two identical terms are obtained; further, proceed as in 20.

22. Hint: Replacing [abc)[adu] using the fundamental identity, show that the concomitant to be studied is equal to 2[abc)[abd][acuj[bdu][cduJ, after which simultaneously a r- b, c F-, d, arithmetic mean. 23. Hint: Write the right-hand side in the symbolic form, and apply the fundamental identity (19.8).

405


§ 20

§ 20

1. Hint: Cf. § 19, Exercises 2 and 15. Answer: Among the results of the single symbolic total alternation one is equal to zero, two are equal to hp; the double total symbolic alternations give zero. 2. Hint: (f, f, v)'" = 36f11[1faaq'a;a); cf. § 14, Exercise 8. Answer:

fit fit 2

fat

fit

ftt fat

Jul q'tt fia

9'ta

9'tl

fit fis

+ fat 9'a fa + 9'ai f:t fia 'Ps' fs" fn fit Ta fu

Vaa

}

[notation as in (14.24)]

3. (aob,+atbo-2a1b1)x+(aobt+asbl-2atbt)y.

4. (f f)"11 = -2(x'+y'+14xty'), (f f)") = 0, (,f)(u = 12. *5. Hint: Cf. § 19, Example 2 and § 18. Exercise 14. Answer: (H, 97)121 = (H, Q)1 a) = 0, (Q, p)(3) = -D.

6. j

}(yr, H)"', where H is the Hessian of the form +p.

7. Hint: (H, 92)"' = flab)' [bc]' [ac](ax)(cx); a .- c. 8. }lab]' [bc][del' {[ad][ce](cx)+ (ae][cd](cx)+ (cd)[ce](ax)}. 9. (V, V)111

4[abl'[ap][bpl,

(H, W)"" = li[ab)'[ac][cp]'[bp]

10. Hint: Use 9; if (px) is a linear factor of the Hessian h, then h vanishes on setting

x' = Pt, x' = _P1

-

11. Hint: Cf. 9 and 10. 12. Hint: Cf. § 19, Example 6. 13. Hint: (i, f)"' = f [abl'[bc]'[ac](ax)(cx)', a.--+ c, arithmetic mean; after application of the identiy (19.6), in the second term a -* c. 14. Hint: (f j)(21 = 1L [bc]'[ab][acl[cd)[ad)[bdl - M, M = [ac)[ab](dx)'-[del x [db][ax)' _ lab](dx) {(acl(dx)- [dc](ax)) - [dc](ax) {[ab](dx)-[db](ax)}. 15. Hint: (i, f)"' = }[ab]' [ac] [bc] (cx)-, a -- c, arithmetic mean, applying the formula for the difference of cubes and (19.39), we find that the right-hand side is

equal to -f(i,f)'V -hj. 16.

a

Ch C

L.,

h

.t=0

p+4-s

[ac]z[bc]w-x(ax)P-A(bx)e_h+d(cx)'-k.

CP+q

17. Hint: Raise both sides of the identity (19.38) to the power k. 18. Hint: Replace the expressions of the form [ab] [ac](bx)(cx) on the basis of (19.39).

19. Hint: Apply (20.44) Answer: 2H' - } iV'.

20. Hint: Use (20.44) for the proof of the second part. 21. Hint: Cf. comments to 18. 22. Hint: For D = 0, use the results of § 18, Exercise 29.

23. Hint: For proof of sufficiency apply 20 (for D # 0) and § 18, Exercise 29 (for D = 0). 25. Hint: In the expressions for D'a,, H, apply the identity (19.8) to the difference of two terms. Answer: }[ab]'(ax)'(bx)(by);

D.1,H= f[ab]'(ax)'(by)'-}i[xy]'; i = }[ab]'. Cf. § 19, Exercise 5. 26. Hint: D,,Q can be represented in the form of the sum of three terms P+2R+3S;

applying the identity (19.8), show that R-P = Answer: D,a,Q = [ub]'[ca](ay)(bx)'(cx)'- ! np[xyl.

S-R - 0.

406

27. The invariant

§§ 21-22


is

a,b,--C;a,b,_,+C'a,b,_,+ ... +(-1)ra,b, = [ab]; for

F = f, one can obtain the invariants of § 14, Exercise 13 and § 18, Exercise 1. 28. 6(f, q')"1. Cf. 3.

29. -12 D, where D is the discriminant of the form q). 30. Hint: Cf. (20.18). § 21

1. Hint: In each of the terms on the left-hand side of (21.4) replace the first factor by its complex conjugate and apply § 16, Exercise 1. 2. Hint: In (21.1), one assumes the essential components of the tensors S, T, ..., to be unknown and, in (21.4), the summation to extend likewise only over essential components.

3. Hint: Let i = 1, j = 2 in (21.22). Answer: S11k1 = (cvekl+ U,1)eit, Tt ' = (WeM- U,I)e ', where U,, is an arbitrary

covariant tensor of variance 2 and weight -1; a solution which satisfies equations of the form (21.4) is obtained for U,1 = 0.

4. S, = (w+g)&, +R;, TO = hol, -Ri, Ui = -hB,-R V, = (w-g)d", +Ri, where g and h are arbitrary numbers and R, is any affinor satisfying the condition Ra = 0. A solution which satisfies conditions (21.4) is obtained for g = h = 0, R7 = 0. 5. Hint: Contract both sides of (21.23) first with respect to k and z, then with respect

to j and y. Answer: S° = wo 6 + U, T;!' = woo - U; O, where U, is an arbitrary affinor; a solution which satisfies (21.4) is obtained for U; = 0.

6. Hint: Select in the capacity of f, the form f, = x+y. Answer: The base consists of the forms f, and fl. If we denote by Q. the quotient

of f.. by fl, then f>u-I = Qss-ifl, fu = (Qu-(x-Y)Y't-'} fi+Y't 'f, CHAPTER V § 22

1. gh invariants of the form (ux), C, invariants of the form [au and C,% invariants of the form [xx 12

... N] (for g 5 n)

... x] (for h z n). n

2. Both correspondences relate the point x to the same point y, if ai,x' :p6 0; if ajjxJ = 0, any point of the straight line corresponds by (22.5) to the point x, and by (22.6), to no point. 3. The discriminant D = }jab]' and the invariants (xy], (ax)', (ay)', (ax)(ay). 4. The forms f and 97, the discriminant D of the form f, the Jacobian of f and q', I = [ap](ax) and the resultant off and q1: R = lap]'. S. Hint: Cf. (19.39). 6. The three pairs of points determined by (22.34) are in involution (15, Exercise 10).

7. Hint: By virtue of the fundamental theorem and the results of 22.5, one has to consider only products of the form [aa][aA] ... If then in the other factors a and A occur together in [aA] or each of them occurs with x, they give the concomitants (22.37) or (22.35). There remain the two cases [aa][aA][AQ]aj ...;

[aa][aA][Ab]a,..

.

N 23-24


407

Here one must apply the fundamental identity to (t/3]a, or to («2]a,.

Answer: The forms (22.34), their discriminants D,, D,, D the Jacobians 9 _ and the invariants D,,, D,,, D,,, E1 [cf. (22.36), (f c'), 013 = (f, v'), $» _ (22.37)].

8. Hint: By virtue of the proof of the proposition of 7, the invariant of even weight

E,s must be expressible in terms of D D,, ..., D. By writing down the corresponding expression with undetermined coefficients (taking into account the weight), find these coefficients, giving the forms f, q', 'D particular values. 2D, Answer: ' = D1, 2D, D,,, . E,,, D1,

D D1i D 2D,

I

9. Hint: The result follows from the same reasoning as in 7.

Answer: The forms f f ..., f,,, their discriminants, C;, Jacobians u9 = (f,, f,), C,',+C,' invariants D,, = (f,, ft)' and E,,, = (6,,, f.)"', altogether }h(h'+3h+8) concomitants.

10. Hint: Expand with respect to the elements of the first column of the determinant in which each row contains one of the forms and three of its coefficients.

11. The pencil belongs to Type I; one possible method of reduction to the canonical form: the new base f' _ -5f+ 3q', 97' = 5V2f; the linear transformation: x1 = (3x1+2x'), x' = 5x'. § 23

1. 2. Hint: Select f and F reduced to canonical fprm. where

4.

c)

_

a).=2x+y, ri= -x+y, b) E_

(x+wy)i V3

5. f

(x

'y)i

f3

w=

x-8y

23,

X28,

3x+4y $'28

i=

where a) _ - Y}y, n = 't(x-}y); b) h = -x-2y, ri=x+y.

6. Hint: For the linear transformation (23.22), such an invariant must, on the one hand, remain unaltered, on the other hand, it must change sign.

8. (H',f) = -}HQ, (H',f)", = -}Df.

9. Hint: The canonical form remains unaltered by the transformation x = wx',

y - Oy', where w = -}+iV'4 (i' = -1), and d:94- 0. 10. Hint: Let 9, = 2H and use § 20, Exercise 5. IL Hint: Apply Theorem 23.2, taking into consideration the weight and degree of each transvectant.

Answer: (H', f')") _ W. Q11)1" = 0; (H', f')

}H'fQ

12. Hint: Compute the Hessian and discriminant of the form 2f +µQ [cf. (23.12)].

Answer: A/#= f %1(-D).

13, 14. Hint: Take the form fin its canonical form (23.3). 15. Hint: Select the forms in their canonical forms. 16. Hint: The point x' is to be selected such that in the involution a,t,y' the pointy' corresponds to x'; for the proof of the second part, reduce f to the form (23.3). § 24

1. ([ab]'(ax)(bx), (ax)').

2. ([ab]'(ax)'(bx)',

(cx)')+}[ab]' (cx)'.

408


§ 25

3. K = ([ab]'(ax)'(bx)' (cx)6)"'-}([ab]'(ax)(bx), (cx)a)(:>. 4. Hint; Such a concomitant is a term of the Jacobian of a certain covariant of the forms 97 and the transvectant (H, ip)' 11 which is equal to zero (§ 20, Exercise 5).

5. Hint: By Theorem 24.2 and the Note at the end of 24.1, the transvectant (Q, f )'h' differs from any of its terms by a linear combination of transvectants of the form

(Q, f)'o', g < h, where Q is the result of (h-g) symbolic total alternations, contained under Q (cf. § 20. Exercise 1). For h = 2, 3, 4, respectively, start from the terms [abl'[cal[ad)(cd](cx)'(dx)', [ab]'[ca][cd]'(ax)(bx)'(dx), [ab]'[ca][cd]'[ad](bx)'.

In order to find the coefficient of jip, one can set tp = 6x'y'.

(Q, p)"' = iH-lfj'p; (Q, yp)"' = 0. 6. Hint: Cf. 5; take into consideration the results of § 20, Example 3 and § 20, Exercise 1. The coefficient of iQ can be found by letting +y = x4+6x'y'; in order to compute (Q, H)"' and (Q, H)"', respectively, start from the terms It = }[abl' x (dell [cal [cdl[ad](bx)'(cx)'(ex)', v = l[ab]'[de]'[bd]'[cell fca]'(ax)(cx). Answer: (Q, H)':o = (Q, H)1O = 0. 7. Hint: Cf. 5; use § 20, Example 3; in order to compute the coefficient of if, let Answer: (Q, V)1 11 = 0;

f = x'+Y'.

8. Hint: Cf. 5 and 7; start from the term l [ab l' [ac]' [bc](ax)n`'(bx),-'(cx)'-'.

9. Hint: Applying Theorem 24.2 to (H, f)"' and (i, f)"', show that (H, f)"' _ p(i,f)"'+akf; to find p and or, let f= lp(p-1)xa-'y' and f = xa+y'. 10. Hint: R = lab] [ac)(b'x)(cx) [ab][bd)(ax)(dx) lied]'; applying the formula (19.39) to the first two factors, show that R = -lu+IJtp-liH, where u = l[ab]' x (cdJ'[ac]'(bx)'(dx)'. Further, on the basis of the theorem relating to the difference of terms of a transvectant, establish the equality u-R = 2iH. Answer: R = jrp-iH. 011. Hint: Two terms of the transvectant (H, H)"' [cf. (18.58)] have been found in 10; the third is readily found using the theorem on the difference of terms of a transvectant.

12. Hint: Apply (20.44) and (24.32). Answer: j#p(2iH-3J ).

14. Hint: Use 7, 8, 9; as is easily shown, (H, f)"I = 0. Answer: (H, j). (i, f) and (i, f)"', where H = l(f, f)"', i = l(f, f)"'. 15. Hint: Cf. § 19, Exercise 15. Answer: H. = D',H+l[xyl'i, 116 = D,',H-l[xypi, Q" = D.'Q + { [xy]i, (cf. § 20, Exercise 26), Q"' = Dx,Q-l[xy]iD:,tp-}[xy]$j. 16. Hint: Apply the Gordan-Clebsch series to the covariant Q"' of 15, regarding y' as constants. Answer: Q. = D1, D.,Q+}[yz]iD:;,yr-l[xyJlD,+,D ,y+I[yz][xy)'j+§[xz]IDs ip. f 25

1. Hint: Taking into consideration the weight and order of each of the unknown transvectants, we easily find the structure of the polynomial expressing it in terms of f, Q, H, i, J. To find the numerical coefficients, give the form f particular values; it is most convenient to take the forms x'+y', 6x'y', 6x2y'+y'.


25

409

Answer: (Q, Q)"" = 2jfH-4iH'- i'f', (Q, Q)4 =0, (Q, Qyu = }(i'-27j'),

(Q, H)"' = §jH- i'f. Cf. likewise (19.22) and § 24, Exercises 6 and 12.

2. Hint: Take two of the points determined by f as first and second coordinate points (the case when all these four points merge in one point is trivial). 3. Hint: Cf. (25.5).

4. a) f = '+ri',

_ f2(x+2y), 0 = 2i (x+3y); 1+i (2x+y), =

b) f = 6i:'ri'+rl',

0 = 1 -i (x+2y) (P _ -1).

2V'27

V'12

5. Hint: Apply Cayley's method (cf. 25.3) to the form x4+61ux'y'+y' and use all three roots of the resolvent. For the verification one may still use the absolute invariant

i'/j' (cf. 6). Answer: a' = +µ, ± I +A

13µ

, f 1+3,uµ 1

6. Hint: Reduce the form to l2xy(x-y)(x-ay). Answer:

r'

j= 108

(l-a+a')' (I+a),(2-a)'(1-20)''

where a is the cross-ratio of the four points A, B, C, D determined by the form and taken in any order. For i=0, the points A, B, C, D form an equianharmonic quadruplet, for j = 0, a harmonic quadruplet. 7. Hint: Reduce f to the form (25.8).

8. Hint: Use 7. 10. Hint: Reducing f to the form (25.8), let x' f and H [cf. (22.24)1. 11. Hint: One can apply § 19, Exercise 6.

y' = 17 and find the resultant of

.

12. Use 11, (25.19) and (25.26); consider separately the cases is-27j' > 0 and 13-27j' < 0. 13. Hint: Use 10 and 12; apply Descartes' rule to (25.27). Answer: Ia. Canonical form: x4+6µx'y'+y', /.r < -}. The form determines four real lioints. Algebraic characteristic: i$-27j' > 0; for any point of the straight line

H < 0; 12H'-if' > 0.

lb, 1C. Canonical forms: aW +61,tx'y'+y')"U > -}, a 1. The form determines four imaginary points. Algebraic characteristic: i'-27j' > 0; for any point of the

straight line of > 0 and either H Z 0 or 12H'-if < 0. Id. Canonical form: x'+6µx'y'-y4. Among the points determined by the form, two are real and two imaginary. Algebraic characteristic: i'-27j' < 0. 14. Hint: Cf. (25.26) and 11. Answer: Types IV and VI are preserved. Type II becomes four with the canonical

forms 6ax'y'+fly4, a = ±1, P = f 1, aj < 0, a8(2iH-3jf) ? 0. Type III becomes four: IIia, IIIb (for H;9 0) with the canonical forms 6ax'y', aj < 0 and IIIe, hid (for H ? 0) with the canonical forms y(x'+y')', y = f 1, yj > 0. Type V becomes two with canonical forms ax', a = ± 1.

410


§§ 26-27

§ 26

1. Each term on the right-hand side of (26.10) gives (h+ I)g terms, g of which are equal to this term, while each of the others is obtained from it by interchange of one of the indices, equal to one, with one of the twos. 2. Hint: By establishing the law of change of the coefficients of binary forms for the transformation (26.35) show that 8a;/8p = kaj_1, whence find aS/8p.

3. Hint: Cf. § 11, Example 3. 4. Hint: By virtue of the second condition, I satisfies also the equation 0! = 0, and, consequently, it remains unchanged for the transformations T. Cf. 3. 5. Hint: Each alternation (1[ ,,, is] must be replaced by [2[ ,,,]2]x4 - (1l,,,323x'.

6. Hint: By (26.27), the weight g = 2, whence I = Aaoa,+; the condition JQ! = 0 gives: A+u = 0. Answer: I= A(aoa,-a;), A arbitrary. 7. Hint: Cf. 6. Answer: I = AD, cf. (14.33). 8. Hint: Cf. 6. Answer: I = A(a0a,t-Cua,a,,._,+ ...

r=2k;1=0 forr=2k+1.

(-1)i

for

10. 1 = 9{(aoa,-ai)a,-(aoa,-a1a,)a, + (alas-ai)a,), where 0 is an arbitrary

number, ao, cc,.... are coefficients of 97, a., a ... are coefficients off. If p

f= (ax)' = (bx)', then I= }9[abl'[aa1[ba].

11. /iH', where P is a number, H is the Hessian of the form. 12. For the form (13.9), H= (aoa,-ai)x{+2(aoa,-a,a,)x'y+(aoa4+2a,a,-3a=) x x'y'+ 2(a1a,-asas)xy'+ (a.a, - a')y'. 13. S = A(aoa,-3a,a1a,+2a1).

14. If'-3Ht. 15. Hint: Eliminate a, from the expressions for I and-f; use (26.33).

18. Hint: Represent the left-hand side of (26.34) in the form (9,+9,+ ... +0,)4-

(02+08+ ... +9,)+ ... + (9,-1+91)+9,.

19. Hint: Apply the reciprocal law of Hermite (cf. 18) and Theorem 25.1. 20. Hint: Use Hermite's law (cf. 18) and Theorem 22.4. 21. Hint: Cf. 20.

CHAPTER VI § 27

1. Two sides of the coordinate triangle touch the curve, the third joins their points of contact. 3. Hint: Take the points 1, 2_3, 4 as first, second, third and unit coordinate points. 4. Hint: If four of six points (for example 1, 2, 3, 4) lie on one straight line, then (27.21) will be satisfied identically; however, if there exists among the six points four of which three are independent, one can use (27.20). Cf. also § 6, Exercise 10. 5. Hint: Apply (12.15) several times. 6. Hint: Use § 18, Exercise 24 and (12.15). Answer: D = }[y]21fy13][y]41[y131[y24][y34][1231[124[11341(234).

7. Hint: Cf. (27.21) and 6. Answer: (u3)[124)[23x1 [14x] - (u4) [1231[13x] [24x] = 0, D = }(u2) (u3) (u4) x (123]'[1241'[134]'[2341.

9. The straight lines joining x to the singular point of the curve Q and the pole divide the straight lines p and q harmonically.


§ 28

411

10. Hint: Start from (6.31) and use (12.13). Answer: The straight line u will be a secant, if F < 0, and external if F > 0; Fis the contravariant (27.7). 11. Hint: Cf. § 6, Exercise 24 and § 27, Exercise 10. Answer: The point x is internal, if Df > 0, external if Df < 0, where D is the discti-

minant of f. 12. If f belongs to Types la and Id, all vertices of any polar triangle are external points; however; if f belongs to Types lb and le, one of the three vertices of any polar triangle is an internal point, the two others are external. 13. [alu]2 = 0. 14. jalg][alh] = 0.

15. Hint: Cf. 13 and 14. Answer:

[alg] [alh]+ [Igh]p

where

p=_

[alg] [alh ] - [lgh ]p

[lbc]'(ay)' [abc]'(1y)' 20. Hint: Cf. 19.

16. -

22. D122D112-D111Dais.

23. Hint: Take the common polar triangle T of the curves f = 0 and T = 0 (cf. 20) as coordinate triangle. Answer: J = 0 is the tangential equation of the system of three vertices of T. 25. Hint: The unknown equation is [abu][cdu][ab(x][cda] = 0 or [abu]'[cda]'

+[aba]'[cdu]'-{[abu)[cda] -[abxj[rdul}' = 0. 26. The equality (27.28) denotes that the polar of the point x with respect to q' passes through the pole of the straight line u with respect to f (cf. § 14. Exercise 11); (27.29) shows that the polars of x with respect to f and q' intersect at a point on the straight line u; (27.30) applies if the polar of x with respect to f intersects that with respect to T of the pole of the straight line u with respect to f at a point on u (cf. 28). 27. The stated collineation refers x to the pole with respect to f of the polar of x with respect to p: 11 = D111, Is = D12LD111, 13 = D222Di11-

29. Hint: Cf. 23 and 26. Answer: For given it, the curve [aau](ax)(ax) = 0 [cf. (27.29)] degenerates into two straight lines, if and only if it passes through one of the vertices of the common polar

triangle of the curves f = 0 and p = 0. § 28

1. r straight lines through one point.

2. Hint: Cf. 1. 3. Hint: Cf. Theorem 5.6 and (18.10). Answer: [aau]' = 0 is the envelope of the straight lines it for which the points of

intersection with f = 0 divide harmonically the points of intersection with 9' = 0. Cf. § 14, Exercise 7.

4. Hint: Apply the duality principle to 3. Answer: [a'a`x]' = 0. 5. Hint: Cf. 3. 6. Hint: Cf. 5 and (12.13). 8. Hint: Cf. § 22, Exercise 6. Answer: The tangential equation of the unknown envelope: [aauj[ahu][- Au] = 0.


412

§ 28

9. Hint: Cf. § 25, Exercise 6 and 25.4.

Answer: The curve (K) is the envelope of the straight lines intersecting f = 0 in points forming an equianharmonic triangle; the curve (L) envelops the straight lines whose points of intersection with f = 0 form a harmonic quadruplet. If u touches both

curves, then u is a tangent at the point of inflection off = 0. 10. Hint: Find the coefficients of (u,)' and u,u,; cf. § 19, Exercise 17. Answer: The straight line u intersects the curve (28.2) at r points merging into one. 11. Hint: Cf. 10.

12. Hint: For the evaluation of the discriminant of the cubic binary form, it is more convenient to find beforehand its Hessian. Answer: a) u6+v4+w'-2(1-+-16µ')(usv3-++usw' I v3*?) 24µ'uvw(u3+v'+w') -24µ(1-4- 2p')u2VZw2 = 0;

b) 40 -9v2w = 0. 13. Hint: The binary cubic form, whose vanishing determines the coordinates of the common points y and z of the curve and straight line, belongs to Type II; therefore its linear factor, which does not enter in the second degree, is equal to the quotient of the form by its Hessian (cf. 23.1, 23.2). Answer: x = (v' w3)(vw+2µu'). 15. S = i7[abc][abd][acd][bcd], T=, 14 [def]'[abc][abd]face] [bcf]. 16. Hint: The relationship between xa and x is symmetric with respect to x0 and x; use (12.13). 17. Hint: Transform the system of coordinates so that the point of inflection P0 has the coordinates x = y = 0, the tangent at PO has the equation x = 0 and the first polar curve of PO (cf. 16) has the equation xz = 0. Answer: The point Pa (x = y = 0) will be a point of inflection of the curve f = 0 as well as of its Hessian; x = 0 is the tangent at P0 to the curve f = 0, y = 0, tangent at the same point to its Hessian; z = 0 is the common straight line of the first polar curves of Pe with respect to f = 0 and H = 0. 18. Hint: Eliminate z from f = 0, and H = 0 and let y = Ax; take into consideration the equal status of all points of inflection with the point P0(x = y = 0).

Answer: The curve f = 0 has 9 points of inflection P, (i = 0, 1, ..., 8). From the points of inflection P, start 4 straight lines on each of which lie, in addition to P,, another two points of inflection. There exist 4 triangles of points of inflection on any side of which there lie three and only three points of inflection. 19. Hint: Select a system of coordinates, so that one of the triangles of the points of inflection (cf. 18) serves as coordinate triangle and two of the points of inflection have

the coordinates (0, 1, -1) and (1, 0, -1). The coordinates of the other points of inflection can be found from the fact that they are located in threes on 12 straight lines. 20. Hint: If the tangents at a singular point are distinct, take them as straight lines

x = 0 and y = 0; however, if they coincide, give the singular point the coordinates x = y = 0 and the tangent there the equation y = 0. 21. Hint: In accordance with the results of 20, the curve f = 0 has at least one point of inflection. 22. Hint: Cf. § 27, Exercise I and § 28, Exercise 1. 23. Hint: By computing the concomitant 6 using (28.17), find the contravariant P by means of the method of 14. Answer:

I H = -f'x'±ax'y+$xy'-yz'; P - -6u11v-6flv3-12fluw'+6acvw2;

S = -fl; T = -a;

IV H = 2xyz-ez'; P = 24uvw; S = 1; T = 2;

VH

-x';P= -3v3;S=0;T=0.

§29

413


24. Hint: First show that (0,0)[2] = }[abc][acd][bcd][bduj(ax)(ux)-(0,0)121. 27. Hint: Use the expressions for H found in 23 and 26. 28. Hint: Cf. 17, 21, 23, 26.

29. Hint: The linear transformation x = p2x', y = y', z = p -1z' with determinant p leaves the form of (28.24) unchanged (it only alters the values of the coefficients

a and P); for (28.24): S = -fl, T = -a (cf. 23). 30. Hint: Assume f to have been reduced to the form (28.24); cf. also 27. 31. Hint: Writing down the usual fourth order form corresponding to the left-hand

side of (28.26), find for this form the invariants i and J and the discriminant i'-27j' (cf. Theorem 25.4). 32. The straight line u corresponds to the point x'= J[aa']'[bb'u], where the symbols a', b', respectively, are parallel to a, b. 33. Hint: Use 32.

Answer.,,H[cc']'[dd'a)b where c', d' are parallel symbols to c, d. § 29

1. Hint: Apply the identity (19.4). 3 2 2. The point of intersection of the hyperplanes u, u, ..., u lies on the hyperplane (29.7). 3. The pole of the hyperplane u lies in the hyperplane v (which is said to be conjugate

with respect to the hypersurface f . 0). 2 3 4. The point of intersection of the hyperplanes u, u, ..., u is conjugate with respect 2

3

n

to the hypersurface f = 0 to the point of intersection of the hyperplanes v, v, ..., v. 6. Hint: Take fin the canonical form (29.16). 7. For p = n, the hypersurface of the second class a"u,u1 = 0 is the second order

hypersurface [d d ... dx]t = 0, a" = did' = did! _ ...; for 2 < p < n, it is a (p-I)-

n-I 12 2 2 dimensional second order surface, for p - 2, two distinct points, for p = 1, two coincident points. 8. a) The k-th coordinate point is the pole of the k-th coordinate hyperplane with respect to the hypersurface H; in other words, the coordinate n-hedron is the polar n-hedron of the hypersurface H. 1

1

b) The (2k-1)-th and (2k)-th coordinate hyperplanes touch H in the (2k)-th and (2k-1)-th coordinate points, respectively; the 2" coordinate (m-1)-dimensional planes (2m = n) lie altogether on H, one of them is x1 = x' = x' = ... = x"-t - 0.

9. (One of the possible answers) T = (y')2+(y2)'±2y'y4, yt = x'-2x2, y' _

''2x', Y3 = 2x'+X', y' = X2-X4. 10. Hint: First of all prove that for one of the systems of unknown (k-1)-dimensional planes the equations may be assumed to have been solved for xt+', xt+2, ..., XEt.

Answer: There exist 2k systems of such planes, partially overlapping each other; one of them is determined by the equations xk+l O,txi, i, j = 1, 2, ..., k; 0 _

-0,,; for i > j, the 0 are arbitrary. 11. Instead of k < }(n-Isl-1), one must have k < n-1}(p+Js,)-1. 12. a,;$'x' = 0, where ' is the radius vector of any point of the unknown hyperplane.

14. The contravariant vector is perpendicular to the planes of the covariant vector and directed from the initial to the final plane; if d is the distance between the planes of the covariant vector, the length of the contravariant vector is 1/d.

414


§§ 30-31

§ 30

2. Hint: For the computation represent

ek by k symbols written down in

all possible orders, where the first a, symbols form al cycles with respect to one symbol, the following 2a, symbols a, cycles of two symbols, etc.

Answer: I, = Y4(S1-6S'S,+8S,S,+3S=-6S4);

I, = -120 L(S,-10S'S,+20SiS,-

30S, S4 + I 5S,Sa - 20S, S, 4 24S6).

3. Hint: Start from (30.15) and (30.17). Answer: S2 = I12-211; S, = 1;-31,1,+313; S, = 1;-4IiI,+41,13+2P-414. 4. Hint: For n = 3, 14 = I6 = 0; cf. 2. Answer: S6 = j(S,-5S;S,+5SiS,a 5S2S3). 7. Hint: If B is such an affinor, then (uBx) is an invariant of the tensors A, u, x of first degree in the components of u and x. 8. Hint: First consider the case w(A) = A9. W"().,) Answer: w(A)ek = w(A,)ek - 0)'(A1)ek+1 + -2 ek+a +

. .

.+

w'o-k'(Aj) i

eo,

k= 1,2,...,g. 9. Hint: Use 8. Answer: Sk = ).k+).':+ ... +A". 10. Hint: Take the affinors in the canonical forms of Jordan. Answer: I: p = 2, 11: p = 1, 111: p = 0. 11. Hint: Cf. 10; use results of 8. Answer: The characteristics (p,p,): 1(33), 11(32), 111(31), IV(21), V(10),.VI(00). 12. a) [(11) 11; A, 0, A, = 3; e,(l; -1; 0), e;(1; 0; -1); e;(1; 1; 1);

b) [3]; Al = Al = A, = 2; e;(l; 0; 0), e'(0, -1; 0), ee(0; 1; 1); c) [2(11)]; At = A, A4 = 1; e;(0; 1; 0; 0), e,(-1; 0; 0; -1), ea(0; 0; 1; 0), e"(0; 0; 0; 1); i; ei(1 -i; 0; 1-i; 1), a=(0; 1; - i; 0), d) [221; A, = A, == i, A3 := 2, e;(I +i; 0; 1 +i; 1), e' (0; l ; i; 0).

13. a) [n]; b) [(m, m)] for n = 2m; [(m-}- 1, m) ] for n = 2m- 1. 1 [111 ] Three invariable points.

14. For n = 3:

11 [21) Two invariable points; through those of them which correspond to a double root there pass two invariable straight lines, through the

others, one such line. 111 [(l 011 Straight line of invariable points, and outside it another invariable point (non-parabolic homology). IV [3] One invariable point. V [(21 )] A straight Iine of invariable points; one of the points of this line has the property that any straight line through it is invariable (parabolic homology).

VI [(Ill)). All points of the plane are invariable. 15. Hint: The affinor A corresponding to the transformation satisfies the equation

(A+E)(A-E) - 0; cf. S.

CHAPTER VII § 31

1. Hint: Apply Theorem 31.3. 2. The tensor w1 112 . i, w3,12 i, I li, is symmetric with respect to the indices it and j, for odd r and skew-symmetric for even r.

415


§ 32

3. Hint: Alternate the left-hand side of (31.21) with respect to the indices a, b, c. d. e, i, j and use Theorem 31.4; expand the left-hand side of (31.22) with respect to c and the sum with respect to c, f, k. Cf. 1. *4. Hint: The relation (31.8) becomes a system of n equations in the n unknowns ,p,t the matrix of which is skew-symmetric: apply the theorem on skewpit, symmetric determinants of odd order. It is useful to consider first the case r = 3.

P1,-.

5. Hint: Prove the sufficiency of (31.23) by induction with respect to s for the t

2

2

s

polyvector [wp ... p], divided by the vectors p, p, ..., p; apply Theorem 31.6. 7. Hint: Expand the right-hand side of (31.24) with respect to c and then alternate with respect to c, i, j, k; then (cf. 1) we obtain (31.13) with r= 3. In an analogous manner, one can obtain from (31.13) (for r-- 3) the formula (31.24). 8. Hint: It follows from (31.13) that the tri-vector w is divisible by the vectors p, q, r,

whence w = a[pgr]. 9. Hint: Cf. 8. Answer: (One of those possible) p, = iwi2 q, = w,,,, r; = wi13. 11. Hint: Use 10.

12. Hint: The necessity is readily proved by placing the polyvector w in its space; the remainder of the proof is analogous to the proof of Theorem 31.12. t

13. Hint: If 4125 # 0, first set in (31.26) a -- i = 1, b = j = 2, c = k = 3, then a = 1, b = 2, c = 3, leaving i, j, k arbitrary. 14. Hint: First apply to (31.27) the same transformation as that which converts (31.24) into (31.13) (cf. 7), then that which converts (31.13) into (31.24);'as a result one obtains (31.26). § 32

1. awls = H'545, (1Wl3 = W4 , aW14 = W23 , aw24 - µ91s, etc.

2. A. (r+k-1)-dimensional plane through E,_1 and the point x, x, . . ., x.

3. Hint: (pz) = 0; if v = [xy], then z = ix-=-µy.

t

k

2

Answer: z is the point of intersection of the straight line vi' and the hyperplane p.

4. Hint: Cf. 3.

r

2

3

Answer: z is the point of intersection of E,_1 and the hyperplanes p, p, ..., p. 5. Hint: By Theorem 31.11, v is a simple polyvector; use 4. Answer: To the polyvector v corresponds the (r- k-))-dimensional plane in which 1

k

2

Er_2 and the hyperplanes p, p, ..., p intersect.

6. Hint: Cf. 2 and 5. Answer: pi,i2...ir = 41,12...i,xalpz. 1

1

.,

7. Let e, e, ..., e be the first, second, ..., n-th coordinate hyperplanes, 8 the unit '

i2 6

it

coordinate point, y the point of intersection of E,.-1 and the hyperplanes e, e, ..., e; 44112...1,/µi1i2...ir is equal to the cross-ratio of the points y, i and the points of it

it

intersection of the straight line [ye] and the hyperplanes e and e (cf. § 8,Exercise 2). 9. The hyperplane pit = w, X12 , ,1r xi2 xi, ... xi, passes through E,_=; 'that and only ,

2

3

that (r-1)-dimensional plane which passes through E,_i2 and lies in the hyperplane p belongs to K,_1.

10. Hint: In 9 take x, ..., x as coordinate points and use the geometric meaning 2

r

of the coordinates of the hyperplanes (§ 8, Exercise 2).

§ 33

ANSWERS AND HINTS TO EXCERCISES

416

11. v;12...rwr+1, r+t...n] = 0;

0'

13. [vw](vY)(wY)

15. Hint: In the identity (32.25) replace e1P 8 by v[i/ wit], expand the alternation with respect to j and then contract both sides with p and x. 16. Hint: The left-hand side is equal to (4!)'v[1lwu]n(12p"] = 41v[Itwk,]7a"pa'; by expanding the alternation, we obtain the result. 17. Hint: Cf. (19.4). write 18. Hint: If the point x lies in E,_ the hyperplane ai(ax) contains with down first the unknown relation, giving E,_, with the aid of r points and the aid of r hyperplanes.

[w'J = [aa ... aJ(ao) (a0) ... (a0), where a, a,

Answer:

..., a are

ordinary

symbols, parallel to the symbol a. 19. Hint: Use 18 and 32.5, Example 3. 12

k

1

k

2

Answer: [aa ... asn-t](a0)(av) ... (a0); the (n-k-1)-dimensional plane conjugate to the (k-1)-dimensional plane [0] with respect to the hypersurface (29.7) has one point in common with the (k-1)-dimensional plane [s"-']. 20. Hint: Cf. § 11, Example 3 and § 11, Exercise 1.

Answer: The contravariant n-vector determines in a space the measure of an ndimensional orientated object. 21. Let e, e, . . ., e be contravariant coordinate vectors; the component "4112 ... In is 1

n

2

the ratio of the hyperparallelopipeds [xx ... x] and [e a ... e]. 12

i, 's

1,

22. The r-vector tb determines a system of parallel n-dimensional planes (r-dimensional direction) and the measure of r-dimensional orientated objects in .each of these planes; the component wiJ2 .. - it is equal to the ratio of the hyperparallelopipeds

[x x ... x e ... e l and [e e... e e ... e] (where the notation is the same as in the 12

r Ir+1

1"

Il 1s

1r Ir+1

in

answer to 21).

23. Hint: One can compare the bi-vector 0 with the contravariant bi-vector v related to it by v determines a system of parallel straight lines and the direction of the circuit around each of them, and likewise the measure of orientated cylindrical surfaces with generators parallel to these straight lines. 25. Hint: [Oz] = el"(pz); (pz) = 1; (px) (py) = 0. Answer: Let O = [xy]; select the vector z such that [xyz] = X. Then two planes through the starting and end points of the vector z, respectively, and parallel to the vectors x, y determine a covariant vector p.

26. If v = [pq], then select r so that [pqr] = e. The covariant vector r cuts off at the edge of the prismatic surface [pq] a contravariant vector equal to x (the directions of the contravariant and covariant vectors coincide). 33

1. Hint: On expanding the left-hand side of (33.4) with respect to k and alternating with respect to a, b, c, i, j, k, we obtain (33.22). Conversely, if we expand the left-hand side of (33.22) with respect to k and alternate with respect to x, y, z, k, then we arrive at (33.4); cf. also § 31, Exercise 1. 3. wile = v[1rpkl; p(O; 4; 9; 6; 3); v1,

Via =via =vu=vas=0.

-1; v" - 1; v" = 2; v,, = 3; vs = V14

§ 34


417

§ 34

1. Hint: Cf. § 32, Exercise 18. Answer: wtl12...1, _ "[i,11, 1°+21121...

r1,]j,UiIi2...1..

2. Hint: Use 1 and 32.5, Example 3. Answer: E, , has a point in common with the (n-r-1)-dimensional plane which is conjugate to E,'_1 with respect to K1. 3. If no linear combination of p and q belongs to the space of the bi-vector v, then

p' = p+2; if only one of their linear combinations belongs to this space, then p = p; if both p and q belong to the space of v, then p = p or (p-2). 4. (One of the possible answers) a) pi =}na;gj=na;r(0;0; -}; 0); s (0; 0; 0; 1); v [pql+[rs]; b) pi = oil; qi = on; r(0; 0; 0; -1; 6); s = (0; 0; 1; 0; 0); v = [pql + Irsl. 5. Hint: Proceed as for the proof of Theorem 34.9. 6. Hint: Denoting as an abbreviation the pair of indices if by the single symbol A, write down the law of transformation of a bi-vector in the form: vB = B VA and proceed in an analogous manner for the given concomitant. Cf. Theorem 15.1 and S. Answer: D = ±P240-1).

LITERATURE 1. SALMON, G., Lessons introductory to the modern higher algebra. Dublin, 1859. 2. KLEIN, F., Vergleichende Betrachtungen Ober neuere geometrische Forschungen. Erlangen, 1872. 3. CLEBSCH, A., Theorie der binaren algebraischen Formen. Leipzig, 1872. 4. CLEBSCH, A.-LINDEMANN, F., Vorlesungen tlber Geometric. Leipzig, 1876. 5. FAA DI BRUNO, Theorie des former binaires. Turin, 1876. 6. GORDAN, P.-KERSCHENSTEINER, G., Vorlesungen uber Invariantentheorie. Leipzig, 1887.

7. STUDY, E., Methoden zur Theorie der ternAren Formen, Leipzig, 1889. 8. DERUYTS, J., Essai d'une th6orie gbnbrale des formes algbbriques. Bruxelles, 1891. 9. MEYER, W. FR., Bericht Ober den gegenwArtigen Stand der Invariantentheorie. Jahresberichte der deutschen mathem. Vereinigung, Bd. 1, 1892. 10. MEYER, W. FR., Invariantentheorie. Enzyklopadie der mathematischen Wissenschaften, 1B2, 1892. H. ALEKSEEV, The theory of rational invariants of binary forms. Iur'ev, 1899. 12. CAPELLI, A., Lezioni sulla teoria delle forme algebriche. Napoli, 1902. 13. GRACE, J. H., AND YOUNG, A., Algebra of invariants. Cambridge, 1903. 14. DICKSON, L. E, Algebraic invariants. New York, 1914. 15. WEITZENBOCK, R., Neuere Arbeiten der algebraischen Invariantentheorie. Differen-

tialinvarianten. Enzyklopadie der mathematischen Wissenschaften. III El, 1922. 16. WEITZENBOCK, R., Invariantentheorie. Noordhoff, Groningen, 1923. 17. SCHOUTEN, J. A., Ricci-Kalktll. Berlin, 1924. 18. TURNBULL, H. W., The theory of determinants, matrices and invariants. London, 1928.

19. SHIRoKov, P. A., Tensor calculus, 1. Moscow-Leningrad, 1934. 20. WEYL, H., The classical groups. Their invariants and representations. Princeton, 1939.

INDEX A.

81

Absolute, plane in Euclidean geometry Addition of doublets Affine quantity

32, 94 35 338 339

AfFinor

, unit Aggregate, Pfaff

387

Algebra, Lie Matrix

131

Alternation

110 225 136 97 296

symbolic

total Angle between vectors in multi-dimensional Euclidean space Annihilator Area, orientated Axis

of reflection

128 122 17

21, 397

of transformation B.

Base of a linear manifold of vectors manifold of forms pencil of binary forms Basis theorem of Hilbert

346 239 253 239

Bivector

105

, simple

118, 381 C.

Cayley curve

Centre of transformation of principle group pencil of straight lines Characteristics, algebraic arithmetic Weierstrass Class of algebraic curve Classification of tensors of given order pencils of binary quadratic forms Coefficient, leading, of the covariant of a binary form Collineation of projective space Combinant of a system of mixed forms Commutator of two matrices Complex, linear, of k-dimensional planes , of straight lines symbol

324 21

58 251

252, 258 350 320 245 258 304 125

257 131

366 123 371

INDEX

420

Components, affine, of a vector , effective, of a tensor Concomitant, joint, of several tensors , mixed of a form tensor Contraction of tensors , total Contravariant of a form Contravariant tensor Convention of Einstein Coordinates, affine, of a second order curve of a straight line of a point of a geometric object Coordinates, axial, Pliicker, of a straight line Coordinates, Cartesian, of a second order curve , Cartesian, of a straight line , homogeneous, Car- sian, of a straight line , of a point in a plane of a hyperplane of a second order hypersurface Plticker, contravariant, of a k-dimensional plane , of a straight line covariant, of a k-dimensional plane projective, of a point of a plane straight line second order curve straight line on a plane radial, Plucker, of a straight line Coordinate points of a multidimensional affine space projective space

Correlation of projective space Covariant of a form Covariant tensor Curve, first polar, of a curve of order r

24 103, 105 140 142 141

140

112, 113 113

142

106, 107 25 33

29 23

10, 37, 100 122 8

6

40 51

88, 93 116

364 122 364 52

42 62 54 122

90 87 123

142 101

324 65

of second class D.

Degree of a covariant Determinant, bordered of affine transformation of linear transformation of space of n covariant vectors of n contravariant vectors of projective transformation Direction, isotropic orientated plane r-dimensional

142 158

27, 90 99 146 145

43, 53, 86 397 129 129

416

INDEX

Discriminant of binary cubic forms of binary forms of order r of binary quadratic forms of n-ary quadratic form of symmetric contravariant second order tensor of symmetric covariant second order tensor of ternary cubic form of ternary quadratic form Divisibility of a polyvector by alternated product of vectors of a polyvector by a vector Doublet , coordinate

421 154

320

49, 249 147 157 147

325 62 356 356

31, 94 38

E.

Equality of contravariant vectors in multidimensional

92

affine space

doublets of tensors Equation, characteristic, of two ternary quadratic forms of Hamilton-Cayley tangential Equivalence of figures tensors Expansion of an alternation with respect to an index

31, 94 108

316 342

65, 150 20, 36 245 342

F.

Factor, symbolic, of first kind of second kind of third kind Finishing line of a doublet Form, bilinear binary , quadratic characteristic, of pencil of quadratic forms cubic multilinear, of contravariant vectors of covariant vectors of covariant and contravariant vectors of order r of n variables symmetric

nary

190

191, 192 192 31

116 141

104, 115 256 104

102 106 107 104 114 141

of a set of variables of order r of contravariant vector covariant vector

104 104

relative

134

, ternary quadratic trilinear Formula of Laguerre Newton Waring

141

141

308 102 83 351 351

422

INDEX

G.

Geometry, affine

in real domain in complex domain mufti-dimensional Euclidean, multi-dimensional Lobachevskian Minkowskian projective

in complex domain in real domain multi-dimensional of the plane of the straight line Riemannian Group, affine

35, 91 37, 91 37, 91 91

95 83

83

65 88 88 88 Sl

40 83

22, 98 18

principal

41, 54, 97

projective

property of linear transformations of space of motions of motions and reflections of matrices of order n , unimodular, of linear transformations

14

99

14, 98 17, 96 131

113

H.

Harmonic property 47, 58 Hermite's reciprocal law 307 Hessian of a form 153 321 an r-th order curve Homology, non-parabolic 414 414 parabolic 413 Hyperplanes, conjugate with respect to a second order hypersurface with point, with respect to linear complex of straight lines

381

89 coordinate 338 diametral, of second order hypersurface 88 independent 144 of a finite doublet 93 of multidimensional affine space projective space 88 a regular linear complex 368 parallel 93 polar, of points with respect to second order hypersurface 124, 329 tangent to a second order hypersurface 150, 329

Hypersurface

95

algebraic of class r

of order r

117 117 117

423

INDEX

117

second class second order Hypersurface of, central

116 126

0

Identity, first fundamental second fundamental third fundamental Invariant absolute algebraic arithmetic integral, rational , rational, homogeneous joint , of several tensors metric relative of group of motions of a form of a tensor Involution , elliptic , hyperbolic , parabolic Involutions, pencil of

211 211

212 1

19, 139 159 146 159 164

8

140 17

19, 139 16 141

39 0 252 252 253 260

J.

Jacobian of a system of forms Jordan canonical form of an affinor

152 349

K.

Kronecker delta

53

L.

Law of compliments inertia of quadratic forms Length of a vector in tisnlti-dimensional Euclidean space Lie matrix algebra

-

258, 392 336 96 131

M.

Matrix algebra of Lie Method, symbolic, of Aronhold Minor, reduced Motion Multiplication of affinors

131

189

27 13, 98 339

424

INDEX

Multiplication of doublet by number tensors tensors with contraction N.

n-hedron, coordinate , polar, of second order hyperplane n-vector n-vector, unit, covariant , contravariant Null-system Number of dimensions of a linear set of vectors of a space

87 413 105 135 135

124 346

86, 90

0.

Object, geometric, afiine Cartesian projective with respect to group of motions with respect to linear transformation of space Operation Sl

0

Order of a covariant group of matrices

37, 91 10

65, 87 16

100 295 296 142 131 106, 107

general tensor

matrix algebra of Lie

131 85

space

101, 106

a tensor

24, 92

Origin of affine coordinates P.

Pairs of hyperplanes of a pencil not dividing each other dividing each other straight lines of a pencil dividing each other harmonically not dividing each other of a pencil dividing each other points of a straight line dividing each other harmonically not dividing each other dividing each other Parabola, isotropic Parallelogram, coordinate Pencil of binary quadratic forms hyperplanes straight lines Pencils, projective, of straight lines Pfaff aggregate Plane, k-dimensional, of affine space tangential to second order hypersurface

90 90 58 58 58

47

46, 89 46, 89 397 38 253 89 58 315 387 94 329

425

INDEX

of multidimensional affine space projective space , projective Planes, k-dimensional, conjugate with respect to second order hypersurface parallel

94 89 51

374 95

multidimensional, conjugate with respect to linear complex of straight lines Point at infinity of straight line circular, at infinity

conjugate with respect to second order hypersurface linear complex of straight lines second order curve coordinate

diagonal, of perfect quadrangle double, of curve of order r , of projective transformation of straight line external with respect to second order curve fixed, of transformation independent

internal with respect to second order curve of inflection of curve of order r of multiplicity p of curve of order r of multidimensional affine space projective space of view of group theory regular, of plane of straight line singular, of curve of order r of linear complex of second ordersurface Polar invariant line of points with respect to second order curve Polarity Pole of hyperplane with respect to second order hypersurface straight line with respect to second order curve Polynomial, characteristic, of an affinor Polyvector

381

40 81

116 381

63 43, 61, 87

66 319 50 315 16 57, 89

315 321

323 89 86 20 51

41

318 367 148 230 64 124 337 64 342 105

, simple Polyvectors, correlative

Power of an affinor , oblique, of an affinor of points with respect to a circle

Principle of Clebsch

duality Process, invariant Process, polar Product of an isation affinor and a number affinors

alternated, of vectors

354 366

339 341 396 319

60, 89, 106 223 230 339 339 112

426

INDEX

of a contravariant vector and a number in multidimensional afl5ne space

92

of a doublet by a number scalar, of two vectors in multidimensional Euclidean space of a covariant and a contravariant vector of tensors Projections of points from E, on to Et Projective quantity Property, affine, of figures group projective, of figures Propositions, correlative , dual

31, 94 96 30, 144 112 369 65 35 14 65

60 60

Q. Quadrangle, perfect Quadri-vector Quantity, affine , projective

66 105 35 65 R.

r-vector

105

Radius vector

24, 92

Range of points Rank of an affinor a bivector covariant r-vector

linear complex of straight lines non-symmetric second order tensor pencil of quadratic forms symmetric covariant tensor second order covariant tensor system of linear equations of points

of vectors

56 157 156 157, 353

385 157 256 156 149 .176 66

146

tensor with respect to one of its indices

157

trivector

156

quadratic form Ratio, cross-, of four hyperplanes of a pencil of four straight lines of a pencil of four points of a straight line of two bivectors

Ray Reflection with respect to a straight line Renormalisation of coordinates Resolvent, cubic , quadratic Resultant of two binary quadratic forms Root, characteristic, of an affinor

148

90 58

45, 89 127

122 17

57

289 264

221, 256 343

INDEX

427

S.

Scalar

absolute relative Segment in multidimensional affine space Semi-invariant of binary form Series of Jordan-Clebsch Set of four points, harmonic equianharmonic Shear Signature of a quadratic form Space, affine of a polyvector projective Square, scalar, of a vector in multidimensional Euclidean space Starting line of a doublet Straight line, external with respect to second order curve improper of plane isotropic of multidimensional affine space projective space

108 134 133

95 303 278

47, 50 48

127, 374 334, 335 90

- 353, 354 86 96 31

315 52 81

95 89

40

projective

secant of second order curve Straight lines, conjugate with respect to second order lines coordinate dependent, in plane independent, of space of n-dimensions Sum of affinors contravariant vectors in multidimensional space doublets tensors Surface, k-dimensional Symbol, ordinary , parallel Symbolic of Aronhold Symmetrization of a tensor System, complete of invariants Systems of affine coordinates projective coordinates Syzygy

of Cayley

315 68 61

58 385 339 92

32, 94 Ill

95 371 193 189 108

235

90 86 248 268

T.

T-system of curves Tangent to a second order curve

Tangent to a line of order r at its double point Tensor, absolute

,contravariant

77

63 318 318 134 106

INDEX

428

covariant equal to zero metric mixed relative skew symmetric

101

108 78, 96 107

134 105

with respect to two indices several indices

105

structural, of Lie algebra

105 133

symmetric

103

with respect to two indices several indices with constant components Tensors, equal , equivalent Term of a transvectant Terms, adjacent, of a transvectant Theorem of Boole Cayley Ceva Desargues

Hilbert Menelaus Pascal Steiner basis, of Hilbert first fundamental, of the symbolic method fundamental, of invariant theory second fundamental, of the symbolic method Three pairs of points of involution Trace of an affinor Transformation, affine , homothetic linear of space projective elliptic hyperbolic involutionary parabolic Transformations, linear, contragredient Translation, parallel Transvectant of order k

, index of Triangle, coordinate , of points of inflection of third order curve , polar, of second order curve Trivector Type of a tensor Types, complimentary

103 103

170 108

245

273 273 231

390 84 67, 138

239 84 315 315 239 206 188 213 50

340 21, 90

18

99

43, 54, 87 50 50 50, 352

50

59, 100 31

225, 226 225, 226 61

412

68, 311 105

245

258, 263, 392, 393, 395

INDEX

429

V.

Vector belonging to the space of a polyvector conjugate with respect to a second order hypersurface coordinate contravariant covariant

function, linear, of the first kind imaginary , parallel of invariable direction of an affinor of null direction of an affinor real

354 338 24, 92 26, 91, 99 29, 94, 100 130 189 193 343 341 189

W.

Weight of an invariant component of an absolute tensor of a relative tensor

Weight of a relative tensor of a scalar

of a term of an integral rational invariant

139 165

168 134 134

165, 168

Foundations of the Theory of Algebraic Invariants

Foundations of the theory of algebraic invariants

Theory of algebraic invariants

Theory of algebraic invariants

Algebraic invariants of links

Theory of Algebraic Invariants (Cambridge Mathematical Library)

Algebraic invariants

Foundations of Algebraic Geometry

Introduction to the Algebraic Theory of Invariants of Differential Equations (Nonlinear Science : Theory and Applications)

Treatise On The Theory Of Invariants

Algebraic theory of molecules

Algebraic theory of molecules

The algebraic theory of semigroups

Theory of algebraic integers

Algebraic Theory of Numbers

Algebraic Theory of Automata

Algebraic theory of numbers

Algebraic theory of numbers

A treatise on the theory of invariants

The foundations of quantum theory

Foundations of Electromagnetic Theory

The algebra of invariants

Foundations of set theory

Foundations of Galois theory

Foundations of Potential Theory

Foundations of Garside Theory

Foundations of potential theory

Foundations of Set Theory

The Invariants of Nature

Elements of the Theory of Algebraic Curves

Foundations of the Theory of Probability

Foundations of the Theory of Algebraic Invariants