00
fc CD
160510
>m
OSMANIA UNIVERSITY LIBRARY Call No. 5/3 # % 3 Accession No^, 33 .
Author
This book should he r...
35 downloads
1170 Views
6MB Size
Report
This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form
00
fc CD
160510
>m
OSMANIA UNIVERSITY LIBRARY Call No. 5/3 # % 3 Accession No^, 33 .
Author
This book should he returned on or before the date
last
marked below,
THE UNIVERSITY OF CHICAGO PRESS
.
CHICAGO
THE BAKER & TAYLOR COMPANY, NEW YORKJ THE CAMBRIDGE UNIVERSITY PRESS, LONDON; THE MARUZEN-KABUSHIKI-KAISHA, TOKYO, OSAKA, KYOTO, FUKUOKA, SENDAIJ THE COMMERCIAL PRESS, LIMITED, SHANGHAI
INTRODUCTION TO ALGEBRAIC THEORIES By A.
ADRIAN ALBERT THE UNIVERSITY OF CHICAGO
THE UNIVERSITY OF CHICAGO PRESS CHICAGO ILLINOIS
1941 BY THE UNIVERSITY OF CHICAGO. ALL RIGHTS RESERVED. PUBLISHED JANUARY 1941. SECOND IMPRESSION APRIL 1942. COMPOSED AND PRINTED BY THE UNIVERSITY OF CHICAGO PRESS, CHICAGO, ILLINOIS, U.S.A.
COPYRIGHT
PREFACE During recent years there has been an ever increasing interest in modern algebra not only of students in mathematics but also of those in physics, chemistry, psychology, economics, and statistics. My Modern Higher Algebra was intended, of course, to serve primarily the first of these groups, and its rather widespread use has assured me of the propriety of both its contents and its abstract mode of presentation. This assurance has been conits successful use as a text, the sole prerequisite being the subject matter of L. E. Dickson's First Course in the Theory of Equations. However, I am fully aware of the serious gap in mode of thought between the intuitive treatment of algebraic theory of the First Course and the rigorous abstract treatment of the Modern Higher Algebra, as well as the pedagogical difficulty which is a consequence.
firmed by
The
publication recently of more abstract presentations of the theory of equations gives evidence of attempts to diminish this gap. Another such at-
tempt has resulted in a supposedly less abstract treatise on modern algebra which is about to appear as these pages are being written. However, I have the feeling that neither of these compromises is desirable and that it would be far better to make the transition from the intuitive to the abstract by the addition of a
mathematics
new
course in algebra to the undergraduate curriculum in a curriculum which contains at most two courses in algebra
and these only partly algebraic in content. This book is a text for such a course. In
fact, its only prerequisite maa knowledge of that part of the theory of equations given as a chapter of the ordinary text in college algebra as well as a reasonably complete knowledge of the theory of determinants. Thus, it would actually be pos-
terial is
a student with adequate mathematical maturity, whose only training in algebra is a course in college algebra, to grasp the contents. I have used the text in manuscript form in a class composed of third- and fourth-year sible for
undergraduate and beginning graduate students, and they all seemed to find the material easy to understand. I trust that it will find such use elsewhere and that it will serve also to satisfy the great interest in the theory of matrices which has been shown me repeatedly by students of the social sciences. I wish to express my deep appreciation of the fine critical assistance of Dr. Sam Perils during the course of publication of this book. A. A.
UNIVERSITY OF CHICAGO September
9,
1940
v
ALBERT
TABLE OF CONTENTS CHAPTER I.
III.
1
1.
Polynomials in x
1
2.
The
4
3.
Polynomial
4. 5.
Polynomials in several variables Rational functions
6.
A greatest common divisor
7.
Forms
13
8.
Linear forms Equivalence of forms
15
9.
II.
PAQB
POLYNOMIALS division algorithm
5
divisibility
8 9
process
17
RECTANGULAR MATRICES AND ELEMENTARY TRANSFORMATIONS 1. The matrix of a system of linear equations
.
.
19 19
2.
Submatrices
3.
4.
Transposition Elementary transformations
5.
Determinants
6.
Special matrices
26 29
7.
Rational equivalence of rectangular matrices
32
21
22 24
*
EQUIVALENCE OF MATRICES AND OF FORMS 1.
Multiplication of matrices
2.
The
3. 4.
Products by diagonal and scalar matrices Elementary transformation matrices
5.
The determinant
6.
Nonsingular matrices
7.
Equivalence of rectangular matrices forms
associative law
of a product
8. Bilinear
-
.
36 36 38 39 42 44 45 47 48
Congruence of square matrices
51
10.
Skew matrices and skew
11.
Symmetric matrices and quadratic forms Nonmodular fields
52 53 56
13.
Summary
58
14.
Addition of matrices
15.
Real quadratic forms
9.
12.
IV.
6
of results
LINEAR SPACES 1.
2.
bilinear forms
.
.
.
Linear spaces over a Linear subspaces
59 62 .
66
:
66
field
66 vii
TABLE OF CONTENTS
viii
CHAPTBR
PAQBJ
3.
Linear independence
4.
The row and column spaces The concept of equivalence
5. 6.
7. 8. 9.
10.
11.
67 69 73 75 77 79 82 86 87
of a matrix
Linear spaces of finite order Addition of linear subspaces
Systems of linear equations Linear mappings and linear transformations Orthogonal linear transformations Orthogonal spaces
V, POLYNOMIALS
WITH MATRIC COEFFICIENTS
1.
Matrices with polynomial elements
2.
3.
Elementary divisors Matric polynomials
4.
The
characteristic matrix
and function
Similarity of square matrices 6. Characteristic matrices with prescribed invariant factors 5.
7.
VI.
Additional topics
FUNDAMENTAL CONCEPTS 1.
Groups
2.
Additive groups Rings Abstract fields
3. 4. 5. 6. 7.
8. 9.
Integral domains Ideals and residue class rings
The The
ring of ordinary integer^ ideals of the ring of integers
Quadratic extensions of a field quadratic fields
10. Integers of 11.
12.
INDEX
Gauss numbers An integral domain with nonprincipal
ideals
.
.
.
89 89 94 97 100 103 105 107 109 109 Ill 112 113 114 116 117 119 121 124 127 130
133
CHAPTER
I
POLYNOMIALS 1. Polynomials in x. There are certain simple algebraic concepts with which the reader is probably well acquainted but not perhaps in the terminology and form desirable for the study of algebraic theories. We shall thus begin our exposition with a discussion of these concepts. We shall speak of the familiar operations of addition, subtraction, and multiplication as the integral operations. A positive integral power is then
best regarded as the result of a finite repetition of the operation of multiplication.
A polynomial f(x) plication of a finite
in x is
any expression obtained number of integral operations
as the result of the apto x and constants. If
is a second such expression and it is possible to carry out the operations indicated in the given formal expressions for/(z) and g(x) so as to obtain two identical expressions, then we shall regard f(x) and g(x) as being the
g(x)
same polynomial. This concept is frequently indicated by saying that f(x) and g(x) are identically equal and by writing f(x) = g(x). However, we shall usually say merely that /(a;) and g(x) are equal polynomials and write the polynomial which is the constant f(x) = g(x). We shall designate by zero and shall call this polynomial the ze^o polynomial. Thus, in a discus-
= will mean that f(x) is the zero polynomial. confusion will arise from this usage for it will always be clear from the where context that, in the consideration of a conditional equation f(x) =
sion of polynomials f(x)
No
we seek a constant solution c such that /(c) = 0, the polynomial f(x) is not the zero polynomial. We observe that the zero polynomial has the properties g(x)
=
+ g(x)
,
=
g(x)
for every polynomial g(x). Our definition of a polynomial includes the use of the familiar
term conterm we shall mean any complex number or function independent of x. Later on in our algebraic study we shall be much more explicit about the meaning of this term. For the present, however, we shall merely make the unprecise assumption that our constants have the usual properties postulated in elementary algebra. In particular, we shall assume then either the properties that if a and 6 are constants such that ab = stant.
By
this
1
INTRODUCTION TO ALGEBRAIC THEORIES
2
a or 6 is zero; and 1 1 a"" such that aar
a
if
we
the label
If f(x) is
is
a nonzero constant then a has a constant inverse
1.
assign to a particular formal expression of a polyit occurs in f(x) by a constant c, we ob-
nomial and we replace x wherever tain a corresponding expression in
c which is the constant we designate by is that /(c). any different formal expression of a polyg(x) = nomial in x and that f(x) g(x) in the sense defined above. Then it is
Suppose now
evident that /(c) = g(c). Thus, in particular, if /&(x), q(x), T(X) are poly= h(c)q(c) nomials in x such that }(x) = h(x)q(x) r(x) then f(c) r(c)
+
for
any
+
2z 2 + 3z, h(x) = re For example, we have f(x) = a; 3 = 2z, and are stating that for any c we have x, r(x)
c.
= x2 g(x) 2 2c + 3c= (c-
l)(c
2
-
c)
+
1,
c8
2c.
the indicated integral operations in any given expression of a polynomial f(x) be carried out, we may express f(x) as a sum of a finite numIf
ber of terms of the form ax k Here k
a non-negative integer and a is a The terms with the same exponent fc
.
constant called the
coefficient of
x
k .
is
may be combined into a single term whose coefficients, and we may then write f(x)
(1)
The constants a unless f(x)
is
=
a xn
+ aixn ~ + l
.
.
.
+
a n -\x
are called the coefficients of /(x)
the zero polynomial,
sum
coefficient is the
+ an
.
and may be
we may always take
ao
^
is most important since, if g(x) pression (1) of f(x) with do 5^ polynomial and we write g(x) in the corresponding form
g(x)
(2)
=
6
zm
of all their
0. is
zero,
The
but ex-
a second
+ fciz"- +...+& 1
= n, a = 6* for then/(x) and g(x) are equal if and only if m n. In other words, we may say that the expression (1) of a i = 0, is polynomial unique, that is, two polynomials are equal if and only if their with 60
.
5^ 0, .
.
,
expressions (1) are identical.
The integer n of any expression (1) of f(x) is called the virtual degree of the expression (1). If a 7* we call n the degree* of f(x). Thus, either any = a n is a constant and will be f(x) has a positive integral degree, or /(#)
^
called a constant polynomial in x. If, then, a n we say that the constant polynomial f(x) has degree zero. But if a n = 0, so that f(x) is the zero polynomial, we shall assign to it the degree minus infinity. This will *
Clearly
any polynomial of degree no may be written as an expression of the form (1) any integer n ^ no. We may thus speak of any such n as a virtual de-
of virtual degree gree of /(a;).
POLYNOMIALS
3
be done so as to imply that certain simple theorems on polynomials shall hold without exception. The coefficient ao in (1) will be called the virtual leading coefficient of this expression of f(x) and will be called the leading coefficient oif(x) if and only if it is not zero. We shall call/(x) a monic polynomial if ao = 1. We then have the elementary results referred to above, whose almost trivial verifica-
we
tion
leave to the reader.
LEMMA
The degree of a product of two polynomials f (x) and g(x) is the and g(x). The leading coefficient of f(x) g(x) is the leading coefficients of f (x) and g(x), and thus, if f (x) and product of
sum the
1.
of the degrees of f(x)
g(x) are monic, so is f(x)
LEMMA stant if
2.
=
3.
if both factors are constants.
of
=
Let f(x) be nonzero and such that f(x)g(x)
f(x)h(x).
Then
h(x).
LEMMA 4. The f (x)
g(x).
product of two nonzero polynomials is nonzero and is a con-
and only
LEMMA g(x)
A
and
+ g(x)
degree of f (x)
is at
most the larger of the two degrees
g(x).
EXERCISES* State the condition that the degree of /(x)
1.
+ g(x)
be
+ g(x)
if
than the degree of
less
either /(x) or g(x).
What
2.
can one say about the degree of f(x)
and
/(x)
g(x)
have posi-
tive leading coefficients? 2 3 3. What can one say about the degree of/ of/ of/* for/ k a positive integer? ,
=
f(x) a polynomial,
State a result about the degree and leading coefficient of any polynomial
4.
= /i
s(x)
+.+/?
Make
5.
and
,
for
t
>
1,
/
= /(x)
a polynomial in x with real coefficients.
a corresponding statement about g(x)s(x) where g(x) has odd degree Ex. 4.
real coefficients, s(x) as in
6.
State the relation between the term of least degree in f(x)g(x) and those of
least degree in /(x) 7.
State
why
and
it is
g(x).
true that
if
x
is
not a factor of f(x) or g(x) then x
is
not a fac-
tor of f(x)g(x). 8. Use Ex. 7 to prove that if k and only if x is a factor of /(x). 9.
is
a positive integer then x
is
a factor of
[/(x)]*
if
Let / and g be polynomials in x such that the following equations are satisfied Show, then, that both /and g are zero. Hint: Verify first that other-
(identically).
* The early exercises in our sets should normally be taken up orally. choice of oral exercises will be indicated by the language employed.
The
author's
INTRODUCTION TO ALGEBRAIC THEORIES
4
=
wise bothf and g are not zero. Express each equation in the form a(x) apply Ex. 3. In parts (c) and (d) complete the squares. a) 6)
/ /
2
+ xg* = - zy =
+ 2xfY + (* + 2zfa - a# =
4
2
/ 2 /
c)
d)
*)0
4
b(x)
and
=
2
^
10. Use Ex. 8 to give another proof of (a), (6), and (5) of Ex. 9. Hint: Show that / and g are nonzero polynomial solutions of these equations of least possible de= xfi as well as g = xgi. But then/i and g\ are also solutions grees, then x divides/
if
a contradiction. "*
to show that if/, g, and h are polynomials in x with real coefficients the following equations (identically), then they are all zero: satisfying
Use Ex. 4
11.
b) c)
^
-
2
/ 2 / 2 /
a)
xg*
=
xh*
= xg* + h* + g + (x + 2)fc 2
2
=
Find solutions of the equations of Ex. 11 for polynomials/, coefficients and not all zero. 12.
g,
h with complex
2. The division algorithm. The result of the application of the process ordinarily called long division to polynomials is a theorem which we shall call the Division Algorithm for polynomials and shall state as
Theorem g(x)
7* 0.
1.
Let f (x) and g(x) be polynomials of respective degrees n and m, there exist unique polynomials q(x) and r(x) such that r(x)
Then
has virtual degree
m
f (x)
(3)
For
let /(a;)
Then, either
n
>
m.
n
and
n
q(x) is either zero or has degree
1,
=
q(x)g(x)
+ r(x)
g(x) be defined respectively
< m and we have
(3)
with q(x}
If Ck is the virtual leading coefficient of
m, and
.
by
=
(1)
and
(2)
=
0, r(x)
with 6
f(x),
ora
^
0.
j* 0,
a polynomial h(x) of virtual k l k 1. b^ CkX g(x) is
m+ m, a virtual degree of h(x) n'^n l 1, and a finite repetition b^ a Q x g(x) is n n~ l of this process yields a polynomial r(x) = f(x) + .)g(x) of b^ (a x m and leading for of virtual degree m hence n and (3) degree 1, q(x) degree
m+k>
Thus a virtual degree of f(x)
.
coefficient
degree
m
Lemma
a^ ^ 1
1,
1
states that
t(x)g(x) is the t(x)
=
0. If also f(x)
0, g(z)
=
q Q (x)g(x)
then a virtual degree of s(x)
sum
=
of
if
m
t(x)
and
?o(z), r(x)
The Remainder Theorem
=
=
+r
Q (x)
r(x) is
m
1.
But
the degree of s(x)
q Q (x) 7* the degree of t(x). This
is
=
impossible; and
r Q (x).
of Algebra states that
=
for r (z) of virtual
r Q (x)
q(x)
Algorithm to write /Or)
=
.
q(x)(x
-
c)
+r(x)
,
if
we
use the Division
POLYNOMIALS = x = /(c). The
has degree one and r = r(x) is necessarily a constant, obvious proof of this result is the use of the remark in = r c) paragraph of Section 1 to obtain /(c) = q(c)(c r, /(c)
so that g(x)
then r the
fifth
5
c
+
as desired. It
is
for this application that
we made
the remark.
The Division Algorithm and Remainder Theorem imply the Factor Theorem a result obtained and used frequently in the study of polynomial equations. We shall leave the statements of that theorem, and the subsequent and theorems on the roots and corresponding factorizations of polynomials* with real or complex coefficients, to the reader. * If f(x) is a polynomial in x and c is a constant such that /(c) = then we shall
definitions
call c
a root not only of the equation /(x)
=
but also of the polynomial /(x).
EXERCISES 1.
Show by formal
differentiation that
if
m
m of f(x) =
a root of multiplicity
c is
m
1 of the derivative /'(x) of /(x). (x c) q(x) then c is a root of multiplicity What then is a necessary and sufficient condition that /(x) have multiple roots? 2.
Let
cients.
be a root of a polynomial /(x) of degree n and ordinary integral coeffito show that any polynomial h(c) with rational
c
Use the Division Algorithm
may
coefficients
numbers 3.
6
,
.
.
.
l
c.
,
=
Let /(x)
+ + b n-ic n~ for rational + r(x) and replace x by
+
be expressed in the form 6 &ic 6 n-i. Hint: Write h(x) = q(x)f(x)
x3
+
3x 2
+
4 in Ex.
Compute
2.
the corresponding &< for each of
the polynomials 6 a) c
6)
3.
c
4
Polynomial
+ 10c + 25c + 4c + 6c + 4c + 2
4
3
c)
2
divisibility.
1
d)
c
6
(2c
Let f(x) and g(x)
by the statement that g(x) divides
f (x)
-
2c 4
+c
2
+ 3)(c + 3c)
2
3
^
be polynomials. Then there exists a poly-
we mean that
divides /(x) if and nomial q(x) such that/(x) = q(x)g(x). Thus, g(x) T only if the polynomial r(x) of (3) is the zero polynomial, and we shall say in this case that f(x) has g(x) as a factor, g(x) is a factor o/f(x). We shall call two nonzero polynomials/(x) and g(x) associated polynomials
= q(x)g(x)> g(x) = f(x) divides g(x) and g(x) divides /(#). Then f(x) = q(x)h(x)f(x). Applying Lemmas 3 and 2, we have h(x)f(x), so that/(x)
if
q(x)h(x)
=
1,
q(x)
and
h(x) are nonzero constants.
Thus
f(x)
andg(x)are
associated if and only if each is a nonzero constant multiple of the other. It is clear that every nonzero polynomial is associated with a monic polynomial. Observe thus that the familiar process of dividing out the leading coefficient in
a conditional equation f(x)
equation by the equation g(x) associated with/(x).
=0
=
is
where g(x)
that used to replace this is the monic polynomial
INTRODUCTION TO ALGEBRAIC THEORIES
6
We
Two associated monic polynomials are equal. see from this that if g(x) divides f(x) every polynomial associated with g(x) divides f(x) and that one possible way to distinguish a member of the set of all associates
We
shall use this property of g(x) is to assume the associate to be monic. ater when we discuss the existence of a unique greatest common divisor (abbreviated, g.c.d.) of polynomials in x.
In our discussion of the
g.c.d. of
polynomials
we
shall obtain
a property
which may best be described in terms of the concept of rational function. It will thus be desirable to arrange our exposition so as to precede the study
common divisors by a discussion of the elements of the theory of polynomials and rational functions of several variables, and we shall do so.
of greatest
EXERCISES be a polynomial in x and define m(f) = x mf(l/x) for every positive integer m. Show that m(f) is a polynomial in x of virtual degree m if and only if m is a virtual degree of /(x). 1.
Let/ =
2.
Show
3.
that m(0)
Define/
Show
f(x)
=
0,
=
m[m(f)}
=
and/ =
n(f)
/ 0, "n j for every
if
=
that m(f)
=
xm
/.
m>
Let g be a factor of/. Prove that $
4.
if /is
n and
any nonzero polynomial of degree n. that, if / 7* 0, x is not a factor of/.
a factor of m(f) for every
is
m which is at
least the degree of /.
Polynomials in several variables. Some of our results on polynomials extended easily to polynomials in several variables. We define a polynomial/ = f(x\, x q ) in xi, x q to be any expression obtained x q and as the result of a finite number of integral operations on #1, 4.
in x
may be
.
.
.
.
,
.
.
,
.
As in Section 1 we may express f(xi, number of terms of the form
constants. finite
azji
(4)
We
x$
.
.
x qk
.
.
.
.
,
.
x q ) as the
.
,
sum
of a
.
of the term (4) and define the virtual degree in k q the virtual degree of a parXi, jX q of such a term to be k\ ticular expression of / as a sum of terms of the form (4) to be the largest of call
a the
coefficient
+
.
the virtual degrees of kq exponents k\,
its
terms
(4).
.
If
.
.
+
,
two terms of / have the same
set of
we may combine them by adding their coefficients and thus write /as the unique sum, that is, the sum with unique coefficients, .
(5)
/
.
.
,
,
= f(xi, ...,*)= k
- 0,
1,
POLYNOMIALS coefficients a^ k q are constants and n/ is the degree of x q ) considered as a polynomial in Xj alone. Also / is the zero polynomial if and only if all its coefficients are zero. If / is a nonzero poly-
Here the .
.
.
.
.
.
,
nomial, then some a kl
maximum sum
+
fci
.
.
.
.
.
.
kq
T* 0,
+k
q
and the
for a kl
.
.
degree of
^
* fl
.
is
/
defined to be the
As before we
0.
assign the
and have the property that nonzero constant polynomials have degree zero. Note now that a polynomial may have several different terms of the same degree and that consequently the usual definition of leading term and coefficient do not apply. However, some of the most important simple properties of polynomials in x hold also for polynomials in several x and we shall proceed to their degree minus
infinity to the zero polynomial
v-,
derivation.
We
observe that a polynomial / in x\, polynomial (1) of degree n = n^in x = x q with .
.
.
,
xq
be regarded as a a a n all
may
its coefficients
,
.
.
.
,
x q -\ and a not zero. If, similarly, g be given by , polynomials in x\, with 6 then a virtual degree in x q oifg is ra not n, and a virtual (2) zero, = a 6 are then and nonzero polyof a If coefficient is &o. 2, leading q fg .
.
.
+
nomials in Xi and ao6 ^ by Lemma 2. Then we have proved that the product fg of two nonzero polynomials / and g in x\, x 2 is not zero. If we x q-i prove similarly that the product of two nonzero polynomials in x\ .
.
.
,
}
and hence have not zero, we apply the proof above to obtain ao&o ^ x q is not proved that the product fg of two nonzero polynomials in #1,
is
.
We have
zero.
Theorem
.
thus completed the proof of The product of any two nonzero polynomials in
2.
.
,
Xi,
.
.
.
,
XQ
is not zero.
We have
the immediate consequence 3. Let f, g, h be polynomials in
Theorem fg
=
Then g h. continue our discussion
fh.
To
we
shall
.
. ,
Xq
and
f
be nonzero,
need to consider an important special x q ) a homogeneous poly(5) have the same degree
type of polynomial. Thus we shall call /(xi, xq if all terms of nomial or a form in Xi, .
.
=
.
Xi,
=
+
+
.
.
,
.
.
,
Then, if / is given by (5) and we replace x> in (5) by + x is replaced by y k ^~ *Xj> power product x k x q ) identixq and thus that the polynomial/(i/xi, yx q ) = y f(xij x q ) is a form of degree k x q if and only if /(xi, cally in y Xi, in xi, xq The product of two forms / and g of respective degrees n and m in the same Xi, x q is clearly a form of degree m + n and, by Theorem 2, is nonzero if and only if / and g are nonzero. We now use this result to obtain the second of the properties we desire. It is a generalization of Lemma 1. k
ki
yx^ we .
.
.
.
.
kq
.
see that each
.
.
.
.
.
.
t
.
.
.
.
.
,
.
.
,
.
.
.
.
,
.
.
.
,
.
,
.
.
,
INTRODUCTION TO ALGEBRAIC THEORIES
8
all the terms of the same degree in a nonzero polybe grouped together into a form of this degree and then may express (5) uniquely as the sum
Observe nomial (5)
we may
first
that
/
(6)
=
/(*!,...,*,)
=/o+-..+/n,
where /o is a nonzero form of the same degree n as the polynomial/ and /< a form of degree n i. If also g
(7)
=
for forms g> of degree
g(xi,
m
.
fg
.
,
=
x q)
+
g
.
.
=
h
+
.
.
.
+
.
^
and such that g Q
i
(8)
.
h m+n
m+n
+ gm 0,
is
,
then clearly
,
=
By Theorem 2 /o(7o. Thus if we call/o the leading form of/, we clearly have Theorem 4. Let f and g be polynomials in xi, Xq. Then the degree of f g is the sum of the degrees of f and g and the leading form of f g is the prodwhere the
forms of degree
hi are
i
and
Ao
Ao T* 0.
.
.
.
,
forms of f and g. above is evidently fundamental for the study of polynomials in several variables a study which we shall discuss only briefly in these uct of the leading
The
result
pages. 5.
The integral operations together with the operaby a nonzero quantity form a set of what are called the
Rational functions.
tion of division
A
rational operations.
rational function of xi,
.
.
.
,
xq
is
now
defined to be
any function obtained as the result of a finite number of rational operations x q and constants. The postulates of elementary algebra were on xi f
seen
.
.
.
,
by the reader in his earliest algebraic study to imply that every rational
function of
x\>
.
.
.
,
xq
may
be expressed as a quotient
x q ) 7* 0. The coefficients of x q ) and 6(xi, x q ) and 6(xi, x q ) are then called coefficients of/. Let us obx q with complex serve then that the set of all rational functions in xi, coefficients has a property which we describe by saying that the set is
for polynomials a(xi, a(xi,
.
.
.
,
.
.
.
.
.
,
.
.
.
.
,
,
.
.
.
,
closed with respect to rational operations. By this we mean that every rational function of the elements in this set is in the set. This may be seen to be
due to the definitions a/6 + c/d Here b and d are necessarily not
=
(ad
zero,
+ 6c)/6d, (a/6)
(c/d)
=
and we may use Theorem 2
(ac)/(6d).
to obtain
POLYNOMIALS bd
7* 0.
erties if
/
9
Observe, then, that the set of rational functions satisfies the propin Section 1 for our constants, that is, fg = if and only
we assumed
=
or g
=
0,
while
if
/
then /- 1 exists such that //~ 1
j
=
1.
6. A greatest common divisor process. The existence of a g.c.d. of two polynomials and the method of its computation are essential in the study of what are called Sturm's functions and so are well known to the reader who has studied the Theory of Equations. We shall repeat this material here
because of
importance for algebraic theories. of polynomials f\(x), /,(x) not all zero to be any monic polynomial d(x) which divides all the fi(x), and is such that if g(x) divides every fj(x) then g(x) divides d(x). If do(x) is a second such polynomial, then d(x) and d Q (x) divide each other, d(x) and d (x) are associated monic its
We define the g.c.d.
.
.
.
,
polynomials and are equal. Hence, according to our definition, the g.c.d. f (x) is a unique polynomial. If g(x) divides all the/i(x), then g(x) divides d(x), and hence the degree
of /i(x),
.
.
.
,
t
of d(x) is at least that of g(x). Thus the g.c.d. d(x) is a common divisor of the /i(x) of largest possible degree and is clearly the unique monic com-
mon
divisor of this degree.
If dj(x) is the g.c.d. of /i(x),
.
.
.
,
/,<x)
and d
(x) is
the g.c.d. of
d,-(x)
and
the g.c.d. of /i(x), ,//+i(x). For every common divisor h(x) of /i(x), ,/,-+i(x) divides /i(x), ,/,-(x), and hence both
//+i(aO,
then d
(x) is
.
.
dj(x)
and
d,-(x)
.
.
.
//+i(x), h(x) divides
the divisor
The
.
.
of /i(x),
.
.
.
d
(z).
,/,-(x),
.
.
Moreover, d
(x) divides /,-+i(x)
d Q (x) divides /i(x),
.
.
.
and
,fs+i(x).
above evidently reduces the problems of the existence and construction of a g.c.d. of any number of polynomials in x not all zero to the case of two nonzero polynomials. We shall now study this latter problem and state the result we shall prove as Theorem 5. Let f (x) and g(x) be polynomials not both zero. Then there exist polynomials a(x) and b(x) such that (10)
result
d(x)
=
+ b(x)g(x)
a(x)f(x)
a monic common divisor o/f(x) and g(x). Moreover, d(x) is then the unique 0/f(x) and g(x). For if /(x) = 0, then d(x) is associated with g(x),a(x) = Iand6(x) = b^ 1 is a solution of ( 10) if g(x) is given by (2) Hence, there is no loss of generality if we assume that both f(x) and g(x) are nonzero and that the degree of is not the of than For of notation g(x) greater degree /(x). consistency
is
g.c.d.
.
we put (11)
*.(*)-/(*),
h l (x)
=
g(x).
INTRODUCTION TO ALGEBRAIC THEORIES
10
By Theorem
1
h
(12)
(x)
= q^h^x)
+ h (x) 2
,
where the degree of h z (x) is less than the degree of may apply Theorem 1 to obtain hi(x)
(13)
=
+ W*)
fc(x)M*)
hi(x). If ht(x)
^
we
0,
,
where the degree of hz(x) is less than that of h^(x). Thus our division process yields a sequence of equations of the form (14)
hi- while n< > conclude that our sequence must terminate with .
.
.
,
unless
We
WX)
(15)
-
+ hr(x)
g r_i(oO/l r_i(z)
and (16) for r
*,(*)
>
^
A r_i(z)
,
[qr-i(x)qr (x)
+
be replaced by h r^(x) = divides both h r ~i(x) and h r-i(x). If we
that
implies
(16)
Thus h r (x) hi(x) and
l]h r (x).
hi-.$(x).
=
both ho(x)
Equation b*(x) bi(x)
f(x)
and
hi(x)
=
Ai- 2 ()
6i_i(o:)gr(a:)
Thus we obtain
/i r (o;)
=
divisor of /(#)
=
by
ar (x)f(x) gf(x)
= =
a\(x)j(x)
+ 6 j or all i <j, then |A| = |An| |A|. Evidently Theorem 1 is the case where the A, are one-rowed square matrices and the result con.
.
.
sidered in (24) the case where t = 2. In connection with the discussion just completed we shall define a notation which is quite useful. Let be a square matrix partitioned as in (6)
A
and suppose that
s
=
t}
the
An are
all
square matrices, and every
A,-/
=
RECTANGULAR MATRICES
31
composed of zero matrices and matrices Ai = An which are what we may call its diagonal blocks, and we shall indicate this
for i 7*
Then
j.
A
is
by writing
A =
(25)
diag{Ai,...,A,}.
of A is the product of the determinants of its submatrices Ai, At. In closing we note the following result which we referred to at the close of Section 4.
As above, the determinant .
LEMMA
9.
.
.
,
Every nonzero
m
by n matrix
A
is rationally equivalent to
a
matrix
(26)
/I
0\
\o
or
I is an identity matrix. For by elementary transformations of type
where
a pq
7*
1
we may carry any element
A into the element 6u of a matrix B which is rationally equivalent
of
1 elementary transformation of type 3 defined for a = frji we = 1. We then apply replace B by a rationally equivalent matrix C with c\\ = of 2 with row transformations c c r \ to replace C by elementary type for the rationally equivalent matrix D = (d< ) such that du = 1, d r \ =
to A.
By an
?
r c
> =
1,
and then use elementary column transformations of type 2 with dir
to replace
an
m
1
row. Clearly,
A
and
Now Ai
D
by the matrix
by n
1 matrix, and I\ is the identity matrix of one are rationally equivalent. Moreover, either AI = and we have (26) for / = /i, or our proof shows that AI is rationally equivalent to a matrix is
A
(
But t
(29)
then,
by Lemma
1,
A
is
rationally equivalent to a matrix
INTRODUCTION TO ALGEBRAIC THEORIES
32
After finitely
We
shall
such steps we obtain
many
show
later that the
number
(26).
of rows in the matrix I of (26)
is
uniquely determined by A.
EXERCISES Carry the following matrices into rationally equivalent matrices of the
1.
form
(26)
3
/
a)
10
5
\-l -1 -
2
1
2\
i
iy
/I
1
2\
(2
3 2
5] 3/
4\
211
1
I
6)
O/
\1
2023 0000 o
(1i
Apply elementary row transformations only
2.
arid carry
each of the following
matrices into a matrix of the form (26)
10
10^ 22-50 -21-52
00
ly
3. Apply elementary column transformations only and carry each of the following matrices into a matrix of the form (26).
a)
-x
2
3
/
3-2\
2
1
,
-27
4
\5
/! 6) \
J
/
.
c)
Show
4.
that
if
3
1
V-l
2
4
-1\ 5/
d)
/O
1
VO
5
2\ 1
3J
the determinant of a square matrix A is not zero then A can be by elementary row* transformations alone. Hint:
carried into the identity matrix
carry
A
^
preserved by elementary transformations. Some element must not be zero, and by row transformations we may a matrix with ones on the diagonal and zeros below and then into /.
The property \A\ in the first
column
into
of
is
A
The largest order of any nonvanishing minor of a rectangular matrix A is called the rank of A. The result of (20) states that every (t + l)-rowed minor of A is a sum of nu7.
* e.g.,
Rational equivalence of rectangular matrices.
Not every matrix may be take
A =
(11).
carried into the
form
(26)
by row transformations
only,
RECTANGULAR MATRICES merical multiples of its Crowed minors, and the former. We thus clearly have
LEMMA most
is at
minors of
the latter are
A vanish.
l)-rowed minors of
all
Then
zero so are
the
A
rank of
r.
We may also LEMMA
+
10. Let all (r
if
33
state this result as
A have a nonzero r-rowed minor A vanish. Then r is the rank of A.
+
l)-rowed
of the zero matrix
and that
11. Let
Note that we are assigning zero as the rank
and
let all (r
the rank of any nonzero matrix is at least 1. The problem of computing the rank of a matrix
A would seem from our and lemmas to involve as a minimum requirement the computation of at least one r-rowed minor of A and all (r + l)-rowed minors. The number of determinants to be computed would then normally be rather large, and the computations themselves generally quite complicated. Howdefinition
problem may be tremendously simplified by the application of elementary transformations. We are thus led to study the effect of such transformations on the rank of a matrix. Let then A result from the application of an elementary row transformation of either type 1 or type 3 to A. By Lemmas 3 and 5 every Crowed minor of A is the product by a nonzero scalar of a uniquely corresponding Crowed minor of A, and it follows that A and Ao have the same rank. If Ao results when we add to the ith row of A the product by c 7^ of its 0th row and B is a Crowed square submatrix of A, the correspondingly placed submatrix B of A is equal to B if no row of B is a part of the ith row of A. If, however, a row erf B is in the ith row of A and a row of B is in the 0th row of B, then by Lemma 2.4 we have B = B If, finally, a row of B is in the ith row of A but no row is in the 0th row of A, then by Lemma 8 Bo = B + c C where C is a Crowed square matrix all but one of whose rows coincide with those of B, and this remaining row is obtained by ever, the
.
|
|
I
1
|
|
|
I
1
\
,
B in the ith row of A by the correspondingly columned elements in its 0th row. But then it is easy to see that C is a minor of A as well as of AO. If A has rank r we put t = r + 1 and see of A that |B = |C| =0, Bo = Of or every (r + 1) -rowed minor |B in A, and our proof shows the Also there exists an r-rowed minor B 5* = B or B = B + c C in existence of a corresponding minor B =0 implies that |C| = -cr \B\ ^ 0, and A has a Ao. But then |B nonzero r-rowed minor |C|, A and A have the same rank. We observe, finally, that if an elementary row transformation be applied to the transpose A' of A to obtain Ai and the corresponding column transformation be applied to A to obtain Ao, then AJ = A\. By the above
replacing the elements of
|
\
.
|
|
|
|
|
|
|
1
|
|
|
l
|
1
|
|
\
\
INTRODUCTION TO ALGEBRAIC THEORIES
34
proof A' and AI have the same rank, by Lemma 2 A and A! have the same minors and hence the same rank, so that A\ and A( = AQ have the same rank. Hence, A and A have the same rank. Thus, any two rationally equivalent matrices have the same rank. Conversely, let the rank r of two by n matrices
m
A
and
B
be the same
9 to carry A into a rationally equivalent matrix (26). It is clear that the rank of the matrix in (26) is the number of rows in the matrix /. By the above proof A and this matrix have the same rank, I = I r
and use
Lemma
,
A
is
rationally equivalent to
2
EQUIVALENCE OF MATRICES AND OF FORMS A
Let
5.
-1
by
47
(d) above and B be the matrix of (e). Compute C and verify that (AE)C = I by direct multiplication.
be the matrix of
(28)
=
7. Equivalence of rectangular matrices. Our considerations thus far have been devised as relatively simple steps toward a goal which we may now
We
attain.
first
DEFINITION. exist
make the Two m by n
nonsingular matrices
matrices
P and Q
A and B
PAQ = B
(30)
Observe that
P
are called equivalent if there
such that
.
and Q are necessarily square matrices
of
m and n rows,
respectively. By Lemma 2 both P and Q are products of elementary transformation matrices and therefore A and B are equivalent if and only if A and B are rationally equivalent. The reader should notice that the definition of
rational equivalence
to be regarded here as simply another
form of the above and, while the previous definition is more useful for proofs, that above is the one which has always been given in previous expositions of matrix theory. We may now apply Lemma 1 and have the principal result of the present chapter. is
definition of equivalence given
Theorem the
9.
Two
m
by n matrices are equivalent
if
and only
if they
have
same rank.
We
emphasize in closing that, if A and B are equivalent, the proof of 2.2 shows that the elements of P and Q in (30) may be taken to be rational functions, with rational coefficients, of the elements of A and B.
Theorem
EXERCISES 1.
P and Q for each of the matrices A of Ex. 1 of Section 2.6 PAQ has the form (2.30). Hint: If A is m by n, we may obtain P by ap-
Compute matrices
such that
plying those elementary row transformations used in that exercise to I m and simi-
(The details of an instance of this method are given in the example at the end of Section 8.)
larly for Q.
2.
that the product AB of any three-rowed square matrices A and B of not zero. Hint: There exist matrices P and Q such that A = PAQ has the
Show
rank 2
form
is
(2.30) for r
the same rank as 3.
illustrative
= 2. Then, if AB = 0, we have A*B* = where B Q B and may be shown to have two rows with elements
Compute the ranks
into a simpler matrix
of
A, B,
AB
QrlB has all zero.
Carry A B into B Q = BQ by
for the following matrices. Hint:
A Q = PA by row transformations
alone,
INTRODUCTION TO ALGEBRAIC THEORIES
48
column transformations alone, and thus compute the rank stead of that of
a)
of AoB
= P(AE)Q
in-
AB.
A=
2
3
4
B
,
B
4^
1-213
1-1 0-2
5-83 (3-52 8. Bilinear
forms. As
5/
we indicated at the close of Chapter I, the problem two forms of the same
of determining the conditions for the equivalence of
restricted type customarily modified by the imposition of corresponding restrictions on the linear mappings which are allowed. We now precede is
the introduction of those restrictions which are linear forms
made
for the case of bi-
by the presentation of certain notations which will
simplify our
discussion.
The one-rowed matrices x'
(31)
=
(xi,
.
.
.
,
x m)
y'
,
=
(yi,
.
.
.
,
y n)
have one-columned matrices x and y as their respective transposes. A be the m by n matrix of the system of equations (1) and see that system may be expressed as either of the matrix equations
let
x
(32)
= Ay
x'
,
=
y'A'
We
have called (1) a nonsingular linear mapping singular. But then the solution of (1) for yi, .
xi,
.
.
.
,
xn
is
.
if .
,
m=
= A-
1
x
,
y'
=
z'(A')-
1
We
A
tation (34)
.
A
is
non-
= x'(A~y
of (32) for y in terms of x (or y' in terms of x'). and variables Xi and y, and shall by n matrices
m
n and
y n as linear forms in
the solution
y
(33)
.
We this
x
=
P'u
shall again consider
now
introduce the no-
EQUIVALENCE OF MATRICES AND OF FORMS for a nonsingular linear
=
mapping carrying the
Xi to
new
49
variables Uk, for
i,
r m, so that the transpose P of P is a nonsingular m-rowed 1 = Um). Similarly, we write (u\, square matrix and u
fc
1,
.
.
.
,
.
.
.
,
= Q
y
(35)
for a nonsingular n-rowed square matrix return to the study of bilinear forms.
A
bilinear
scalar which
form /
=
=
Zziauj//, for i
may be regarded
Q and
v'
1,
,
.
.
.
=
(v\ t
.
.
m and j ~
as a one-rowed square matrix
.
,
1,
We now
v n ).
.
and
.
is
.
,
n, is
a
then the
matrix product
/
(36)
=
x'Ay
.
(31), and we call the m by n matrix A the matrix rankle rank of/. Also let g = x'By be a bilinear form in #1, xm m n B. Then with matrix we shall that and ?/i, and are yn / by say g
Here x and y are given by
of/, its
.
.
.
.
.
.
,
,
there exist nonsingular linear mappings (34) and (35) such um and Vi, v n into which / is that the matrix of the form in ui,
equivalent
if
.
carried
by these mappings /
B = PAQ
=
is
B.
.
But
.
.
,
=
(u'P)A(Qv)
.
.
,
(34) holds, then x'
if
u'(PAQ)v
=
u'P and
,
A
and are equivalent. Thus, two bilinear forms f and g are equivalent if and only if their matrices are equivalent. By Theorem 9 we see that two bilinear forms are equivalent if and only if they have the same rank. It follows also that every bilinear form of rank r is equivalent to so that
the form (37)
zit/i
+
.
.
+xy r
.
r
.
These results complete the study of the equivalence of bilinear forms.
ILLUSTRATIVE EXAMPLE
We shall find nonsingular linear mappings
into a form of the type (37).
The matrix f
of
/
2-3 -1
W6
(34)
is
1^
5
3
and
W
(35)
which carry the form
INTRODUCTION TO ALGEBRAIC THEORIES
50
We interchange the first and second new second row, add
6 times the
which evidently has rank the
first
row by
1,
2.
rows of A, add twice the new
new
first row to the row to the third row, and obtain
first
We then add the second row to the third row, multiply
the second row by
0-5\
/I
PA=
and obtain
J,
o
i
\o
o
-y o/
The matrix P is obtained by performing the transformations above on the threerowed identity matrix, and hence /
P-f-* \
We
continue and carry
times the
first
PA
column and
into
V
PAQ
1
-1 -I -4
of the
0\
o
.
I/
form
(2.30) f or r
times the second column of
PA
=
2
by adding
five
to its third column.
Then
5\
/I
Q =
1
f
I I
We
verify
by
xi
Z2 Z8
(34)
= - Jtt2 + Uz = -Ui - ftt2 ~ = ^3
-
17
\0
The corresponding linear mappings
V
and
(35) are given respectively (
4W3
Vi 2/2
,
I i 2/3
= = =
+
5^3
30t;8
+
vi
by
t>2
^3
direct substitution that
(-6vi
-
3i; 2
as desired.
EXERCISES Use the method above to
find nonsingular linear following bilinear forms into forms of the type (37). 1.
o) 2x
1
3
3
2
1
2
21-1
1
1
\-l -1
-
1
O/
Write the symmetric bilinear forms whose matrices are those of (a), (6), (c), Ex. 2 and use the cogredient linear mappings obtained from that exercise to obtain equivalent forms with diagonal matrices. 3.
and
(d) of
4.
Apply the process
5.
Which
of the matrices of
bers as elements of 6.
linear
of
of Ex. 3 to (e), (/), (g),
and
(h) of
Ex. 2 are congruent
if
Ex. 2 for quadratic forms.
we
allow any complex
num-
Pf
= xf x\ are not equivalent under that the forms / = xf x\ and g the possible signs of values with Hint: real coefficients. Consider mappings
Show
/ and
of g.
+
INTRODUCTION TO ALGEBRAIC THEORIES
56
Nonmodular fields. In our discussion of the congruence and the equivalence of two matrices A and B the elements of the transformation matrices P and Q have thus far always been rational functions, with rational coefficients, of the elements of A and B. While we have mentioned this fact before, it has not, until now, been necessary to emphasize it. But the reader will observe that we have not, as yet, given conditions that two symmetric matrices be congruent, and our reason is that it is not possible to do so without some statement as to the nature of the quantities which we allow 12.
as elements of the transformation matrices P. algebraic concept which is one of the subject the concept of a field.
A
We
shall thus introduce
most fundamental concepts
an
of our
complex numbers is a set F of at least two distinct complex and a/c are in F for every such that a + &, ab, a fc, fields in of such F. c 7* are, Examples then, the set of all real numa, 6, of all the set of all rational functions and the set complex numbers, bers, with rational coefficients of any fixed complex number c. field of
numbers
If
F
is
a,
.
fe,
any
.
.
,
field of
K
= F(x) of all rational complex numbers, the set F is a mathematical system having prop-
functions in x with coefficients in
with respect to rational operations, just like those of F. Now it is true that even if one were interested only in the study of matrices whose
erties,
elements are ordinary complex numbers there would be a stage of this study where one would be forced to consider also matrices whose elements are rational functions of x.
Thus we
shall find it desirable to define the concept
K
of a field in such a general way as to include systems like the field defined shall do so and shall assume henceforth that what we called con-
above.
We
stants in Chapter
The
I and scalars
thereafter are elements of
we have already mentioned
fields
all
a fixed field F.
contain the complex number
unity and are closed with respect to rational operations. But it is clearly possible to obtain every rational number by the application to unity of a finite number of rational operations. Thus all our fields contain the field of all rational called
called nonmodular fields. The fields be defined in Chapter VI. We now make the fol-
numbers and are what are
modular fields
lowing brief DEFINITION.
will
A
set of elements F is said to form a nonmodular field if F contains the set of all rational numbers and is closed with respect to rational operations such that the following properties hold: I.
II.
III.
+ b) + c = a + (b + c) a(b + c) = ab + ac ab = ba a + b = b + a (a
for every a, b, c of F.
,
;
,
(ab)c
=
a(bc)
;
EQUIVALENCE OF MATRICES AND OF FORMS The difference a
F
tion x in
57
a solu-
6 is always defined in elementary algebra to be
of the equation
+b=
x
(46)
and the quotient a/6 to be a solution y yb
(47)
a
F of the equation*
in
=
a
,
.
Thus our hypothesis that F is closed with respect to rational operations should be interpreted to mean that any two elements a and b of F determine b and a unique product ab in F such that (46) has a a unique sum a in F and solution (47) has a solution in F if b j 0. In the author's Modern Higher Algebra it is shown that the solutions of (46) and (47) are unique.
+
In fact,
be concluded that the rational numbers
may
it
and
have the
1
properties
a
(48)
=
+
=
al
a
aO
,
=
0,
F and that there exists a unique solution x = aoix -}- a ~ and a unique solution y = 6" of yb = 1 for 6^0. Then the solutions = ab~ of (46) and (47) are uniquely determined by z = a + ( b), y
for every a of
1
l
.
We
also see that the rational
1=0, and a
1
a,
(
= a)
thus
1 (
6)
(
1
+
number
=
l)a
a
=
=
0,
a
+
=
ab for every a and b of a
a.
Hence
a
1
defined so that
1 is
l)a = also true that
whereas a.
It is
field
(
1
+
+ +
1)
(
1
a
(a) =
F.
EXERCISES 1. Let a, 6, and c range over the set of all rational numbers and F consist of all matrices of the following types. Prove that F is a field. (Use the definition of addition of matrices in (52).)
a
2.
Show
N
/a
^(b
that
-2b\
a)
if a, 6, c,
'~
'
r
"-
/a
x
b
.,
b
and d range over all rational numbers and i 2 form a quasi-field which is not a
set of all matrices of the following kind
/a
+ bi
\c
di
3(c
=
l,
the
field.
-)
a
Note that in view of III the property II implies that (b c)a = ba ca, and the existence of a solution of (46) is equivalent to that of b x = a, of (47) to that of by = a. But there are mathematical systems called quasi-fMs in which the law ab = ba *
+
+
+
does not hold, and for these systems the properties just mentioned must be additional assumptions.
made
as
INTRODUCTION TO ALGEBRAIC THEORIES
58
13. Summary of results. The theory completed thus far on polynomials with constant coefficients and matrices with scalar elements may now be clarified by restating our principal results in terms of the concept of a field.
We
observe
first
that
if
and
f(x)
g(x) are nonzero polynomials in x with
F then they have a greatest common divisor d(x) with and d(x) = a(x)f(x) + b(x)g(x) for polynomials a(x), b(x)
coefficients in a field coefficients in F,
with coefficients in F.
We next assume that A and B are two m by n matrices with elements in F and say that A and B are equivalent in F if there exist nonsingular matrices P and Q with elements in F such that PAQ = B. Then A and B a
field
are equivalent in
F if and only if they have the same rank.
Moreover, corre-
spondingly, two bilinear forms with coefficients in F are equivalent in F if and only if they have the same rank. Since the rank of a matrix (and of a
corresponding bilinear form) is defined without reference to the nature of the field F containing its elements, the particular F chosen to contain the
elements of the matrices
is
relatively
unimportant for the theory.
A and B are square matrices with elements in a field F, then we call A and B congruent in F if there exists a nonsingular matrix P with elements in F such that PAP = B. Similarly, we say that the bilinear forms x'Ay and x'By are equivalent in F under cogredient transformations* if A and B are congruent in F. When A = -A the matrix A is skew, and every matrix B congruent in F to A is skew, two skew matrices with elements in F are congruent in F if and only if they have the same rank. Hence, the precise nature of F is again unimportant. Let A = A' be a symmetric matrix with elements in F so that any matrix B congruent in F to A also is a symmetric matrix with elements in F. If
f
1
Then two corresponding quadratic forms x'Ax and x Bx f
are equivalent in
and only if A and B are congruent in F. Moreover, we have shown that every symmetric matrix of rank r and elements in F is congruent in F to a diagonal matrix diag (a a a r 0, in F and 0} with a* ^ that correspondingly every quadratic form x'Ax is equivalent in F to
F
if
,
.
.
.
,
,
.
.
.
,
The problem
of finding necessary and sufficient conditions for two quadforms with coefficients in a field F to be equivalent in F is one involving the nature of F in a fundamental way, and no simple solution of this problem exists for F an arbitrary field. In fact, we can obtain results only ratic
after rather complete specialization of F,
vary as *
F
we change our assumptions on
We leave
and these
results
may
be seen to
F.
to the reader the explicit formulation of the definitions of equivalence in linear
two forms, of two bilinear forms, and of two bilinear forms under cogredient mappings, where all the forms considered have elements in F. of
EQUIVALENCE OF MATRICES AND OF FORMS
59
The simplest Theorem 12.
conditions are those given in Let F be a field with the property that for every a of F there exists a quantity b such that b 2 = a. Then two symmetric matrices with eleare congruent in F if and only if they have the same rank. = diag ^{ai, = A' of rank r is congruent to ar , 0, For every , = a 6? for in F. for and a, Then if = diag {&rS 7* 6< , 0} >
ments in
F
A
A
.
.
67
.
^
.
PA
1 ,
0,
.
.
.
,
0},
P'
F
also congruent in
=
.
-
B=
B' has rank r, then 5 is 0} and hence to A. The converse follows
diag [I r , 0}. If also
to diag {/ r
.
P
,
from Theorem 2.2. We then have the obvious consequences.
COROLLARY
I.
Two symmetric
num-
matrices whose elements are complex
bers are congruent in the field of all complex the same rank.
numbers
if
and only
if they
have
COROLLARY II. Let F be the field of either Theorem 12 or CoroUary I. Then two quadratic forms with coefficients in F are equivalent in F if and only if they have the same rank. Hence every such form of rank r is equivalent in
F
to
(49)
x?
+
.
.
.
+ x?
.
There is one other result on symmetric matrices which will be seen to have evident interest when we proof involves the computation of the product of two matrices
14. Addition of matrices.
over an arbitrary state
it.
Its
field
which have been partitioned into two-rowed square matrices whose elements are rectangular matrices. If these matrices were one-rowed square matrices, that is to say in F, we should have the formula
But
it is
also true that
the partitioning of any
if
A
B is
and
carried out so
that the products in (51) have meaning and if we define the sum of two matrices appropriately, then (51) will still hold. Thus (51) will have major
importance as a formula for representing matrix computations. m and j We now let A = (ay) and B = (6
>
t
1
1
.
}
and
1
(61)
r
t
+
(x\
.
.
elements
+
.
-
*?)
1.
Thus, /
is
.
.
.
,
.
.
,
F
equivalent in
to
+ ...+*).
(* +1
it is equivalent in F to (61) and = n and index s so that g is equiva-
the same rank and index as/,
Now, if 0has
hence to /. Conversely,
F
\
,
such that p* = diy and if P is the nonsingular matrix diag then PAP is the matrix diag {1, 1, -1, -1}
elements
lent in
.
.
.
t,
,
^
numbers p
have rank r
let g
to
(*?+...+
(62)
- (J +l +
*2)
.
.
.
+3
.
propose to show that s = t. There is clearly no loss of generality if we assume that s ^ t and show that if s > t we arrive at a contradiction.
We
Hence, let s > t. Our hypothesis that the form / defined by (61) and the form g defined by (62) are equivalent in the real field implies that there exist real numbers d^ such that if we substitute the linear forms
in .
.
/ .
= f(x
.
l9
+ yj).
.
.
Put
,
dnyi
=
=
x2
resulting
.
t
.
.
These are
linear
t
numbers
The remaining n numbers
certain 0,
(ttf +1
.
.
+
.
,
.
0,
.
+d
.
.
in
=
=
xt
and
+
.
.
.
.
,
y, +1
+ =
.
.
.
.
.
.
=
1,
.
.
+ yj) =
n)
,
(yj +1
=
yn
.
,
+
in (63)
v9
not
all
zero
(t
in s
and
>
t
=
1,
.
.
.
,
*)
.
unknowns, and there
satisfying these equations.
equations of (63) then determine the values of x, as = t = 1, n, and we have the result h
t
+
Uj for j
w l+1
(y\
(i
+ cky. =
.
.
homogeneous equations v\,
yn
equations
AMI
(64)
exist real
+
x n), we obtain as a result
Xi
and consider the
/(O,
=
Xi
(63)
,
.
+ ul) ^
.
.
0,
,
tO =
.
t>f
.
.
+
,
.
.
.
+
0J
>
0.
But
clearly
ft
=
a contradiction.
We have now shown that two quadratic forms with real coefficients and the same rank are equivalent in the field of all complex numbers but that, their indices are distinct, they are inequivalent in the field of all real
if
numbers. We shall next study in some detail the important special case t = r of our discussion.
INTRODUCTION TO ALGEBRAIC THEORIES
64
A
A
symmetric matrix
are called semidefinite of rank r
A
if
is
.
.
Thus /
is
semidefinite
+ xj). We call A and/ definite
.
x'Ax
a matrix
\
in F.
F to
congruent in
air
for a 5^
=
and the corresponding quadratic form /
;
=
if r
+
equivalent to a form a(x\
if it is
n, that
A
is,
is
both semidefinite
and nonsingular.
F is
If
the field of
positive or take a tive
if
and only
numbers, we may take a = 1 and call A and / and call A and / negative. Then A and / are nega-
all real
=
1
A
if
and
/ are
Thus we may and
positive.
shall re-
our attention to positive symmetric matrices and positive quadratic forms without loss of generality. If f(xi, x n ) is any real quadratic form, we have seen that there exists a nonsingular transformation (63) with real d^ such that / = y\ strict
.
.
.
,
+
+
we put
if
y t
= d
and only if the
=
t
iff
is
.
+
,
Ci,
.
.
.
are
cn
,
numbers and
real
any
l,/(ci,
all zero.
.
.
.
,
cn )
t
=
Now if < r, we have/ < < 0. Conversely, if /(ci,
y\
for y\
t
.
+
.
.
.
+
/(ci,
j/J,
.
.
.
,
.
.
,
=
, for all real ci not all zero. if f (ci, As a consequence of this result we shall prove
if
and only .
15.
if f (ci,
real quadratic form f(xi,
.
.
.
cn )
,
.
.
.
,
^
.
Theorem
16.
Every principal submatrix of a positive semidefinite matrix submatrix of a positive definite matrix
is positive semidefinite, every principal is positive definite.
For a principal submatrix B of a symmetric matrix A is defined as any im th rows m-rowed symmetric submatrix whose rows are in the tith, of A and whose corresponding columns are in the corresponding columns of A. Put XQ = (x^, z m ) so that g = x Q Bx Q is the quadratic form with B as matrix, and we obtain g from / by putting Xj = in / for j 9^ i*. for all values of the x ik and for all z = a, then g ^ Clearly, if / .
.
.
,
f
.
.
.
,
,
EQUIVALENCE OF MATRICES AND OF FORMS hence
B
A
is positive definite positive semidefinite by Theorem 15. If = for x not all and hence for the then singular, zero, / = g ik
is
and B is Xj above
65
all
zero
and
for the x ik not all zero, a contradiction.
The converse of Theorem 16 is also true, and we refer the reader to the author's Modern Higher Algebra for its proof. We shall use the result just obtained to prove
Theorem Then AA' For we
is
m
A be an by n matrix of rank r and with real elements. a positive semidefinite real symmetric matrix of rank r.
17. Let
may
write
A-- pP (I' A (o
where
P
o)
>
00' QQ
Q are nonsingular matrices of m and n rows, respectively. a positive definite symmetric matrix, and we partition QQ' so an r-rowed principal submatrix of QQ'. By Theorem 16 the ma-
and
Then QQ' that Qi
\o Q
is
is
trix Qi is positive definite,
is
congruent to the positive semidefinite matrix
C
of rank r and hence has
the property of our theorem.
EXERCISE What tion 11?
are the ranks
and
indices of the real
symmetric matrices of Ex.
2,
Sec-
CHAPTER
IV
LINEAR SPACES 1.
Linear spaces over a
The
field.
U =
(1)
(d,
Vn
set .
.
.
of all sequences
Cn)
,
may be thought of as a geometric n-dimensional space. We assume the laws of combination of such sequences of Section 1.8 and call u a point or vector, of Vn We suppose also that the quantities Ci are in a fixed field F and call c the ith coordinate of u, the quantities Ci, c n the coordinates of u. The entire set V n will then be called the n-dimensional linear space .
.
.
.
,
over F.
The
(u
(2)
a(bu)
(3)
=
for all a, 6 in
by
O.is
+
+w=
v)
(o6)w
(a
,
+
u
be easily seen to imply that
(v
+ w)
+ V)u =
+ bu
aw
+v=v+u
u
,
+
a(w
,
= aw
v)
+ aw
F and all vectors w, v, w of Fn The vector which we designate .
that vector
all
u =
(
Ci,
.
.
+ = .
w =
(5)
Note that the
first
whose coordinates are
of
w
(4)
where
F may
properties of a field
1
,
of (5)
is
u+(u)
u,
cn ).
,
zero,
=
and we have
Q,
Then u = u
u
,
the quantity
of F,
=
1
u
.
and the second zero
is
the
We
shall use the properties just noted somewhat later in an abstract definition of the mathematical concept of linear space, and we leave
zero vector.
the verification of the properties
(2), (3), (4),
and
(5) of
Vn to the reader.
Linear subspaces. A subset L of V n is called a linear subspace of V n if bv is in L for every a and b of F, every u and v of L. Then it is clear that L contains all linear combinations 2.
au
+
u =
(6)
for
a in F, and w^ in L.
aiUi
+
.
.
.
+ amum
,
LINEAR SPACES
67
We observe that the set of all linear combinations
m
ber
of given vectors, ui, to this definition. If now
.
.
um
,
any
finite
num-
a linear subspace L of Vn according we shall say that u\, Um
is
so defined, shall write
L and
span the space
.
L
(6) of
is
.
.
.
,
L-lm,...,!!*}.
(7)
It is clear that, if e,- is the vector whose jth coordinate is unity and whose other coordinates are zero, then ei, e n span Vn The space spanned by the zero vector consists of the zero vector alone .
and may be
.
.
.
,
L =
and designated by
called the zero space
{0}.
In what
fol-
our attention to the nonzero subspaces L of Vn and, for the time being, we shall indicate, when we call L a linear space over F that L is a linear subspace over F of some V n over F. lows
we
shall restrict
,
}
Linear independence. Our definition of a linear space L which is a V n implies that every subspace L contains the zero vector. If Oum. Hence the zero vector of {MI, u^, then = Oui
3.
subspace of
L = L is expressible .
ui,
.
.
.
,
.
.
+
,
(6) in at least this one way. u m are linearly independent in F or that ui, .
Vn over F), if
linearly independent vectors (of (6).
.
.
.
ai
shall say that
shall say that
um are a
,
set of
no other such expres-
is
and only
For
is unique.
cial case
= .
.
aiUi
+
if the
u =
+
(am
.
.
.
+
.
,
.
.
.
,
u
expression of every
L =
of
{ui,
this property clearly implies linear 0.
.
.
if ui,
Conversely, am um = biui
+
+
b m )um a
iUi + = ar = 0, a contradiction. But then + OrUr = 0, from which a\ = l l u r }. It follows that, u = ( a Oi)ui + + ( a^ an )ur is in [u^ if ui, ... Mm are any m distinct nonzero vectors, we may choose some largest number r of vectors in this set which are linearly independent in F and we will then have the property that all remaining vectors in the set are pendence in case
Then
let ui,
.
.
.
that au
1 is
.
.
a
,
.
}
.
if
only
.
.
,
.
.
.
.
.
.
.
.
.
.
.
.
,
|
F of these r. We state this result as
linear combinations with coefficients in
m
1. Let L be spanned by distinct nonzero vectors Ui, has a basis consisting of certain r of these vectors, 1 < r < m.
Theorem
L
Then
.
.
.
,
um
.
EXERCISES Determine which of the following
1.
subspace they span. Hint: It u\
and
=
=
6)
Ul
=
c)
tii
=
*
d) u,
=
ui
=
tii
=
)
/)
2.
of the
To
see
if ui,
dependent we
HZ are linearly
u^
(1, 3,
-3)
(3, 1, 2)
,
(-1, -1, (1,
1)
,
-2, -1,
(1, 1, 1,
(1,
,
-1,
3)
-1) 1,
,
,
-1)
w2
=
u,
=
u,
=
u,
=
u,
=
u2 =
,
(2, 5,
-2)
,
u*
= =
(4, 1, 3)
,
m
(3, 2, 1)
,
7/3
(2,
-1,
(2, 2, 2,
=
6)
,
u =
-2)
,
u3 =
1,
(1, 1, 1, 0)
,
3
=
u,
(1, 1, 6)
(-1, -1,
0)
(7, 3, 9)
-1, -3,
(0,
(1, 2, 3, 4)
-1,
(5,
5,
1)
-3)
that the space L = {MI, w2 , ^3} spanned by the following sets of vectors Hint: In every case one of the vectors, say ui, has first coordinate not zero, x 2 ui = (0, 6 2 c 2 ), u 3 contains u* XsUi = (0, 63, c*). Some linear combina-
Show
is 7s.
and
form a basis
+ yui and solve for x and y.
xu\
a) MI
sets of three vectors
easy to see whether some two of the vectors, say
Ui are linearly independent.
write uz
>
is
L
,
two quantities has the form (0, 0, d) and L contains e 3 = (0, 0, 1). = Fs. easy to show then that L contains e 2 = (0, 1, 0) and e\ = (1, 0, 0), L
tion of these is
a) ui
3.
=
6)
m-
c)
I*
-
(1,
-2,
3)
,
u2 =
(0, 1, 8)
t*2
-
,
(0, 3, 2)
u,
=
,
(2, 3, 1)
(1,
-3,
(0, 2, 1)
u3 = (-1,
,
6) ,
,
u*
=
11,
(1,
3, 2)
-1, 23)
(1, 5, 4)
Determine whether or not the spaces spanned by the following
wi, Ut coincide with those
spanned by
the corresponding
It
vi,
v2
.
sets of vectors
Hint:
If
Li
=
LINEAR SPACES =
Lu
{tfi,
+ #2^2, a)
t>2
02}
=
there
,
must
exist solutions xi,
+ #4^2 such
^3^1
m=(3, -1,2,1),
U!
=
(1,2,
=
u,
(1, 4, 6, 1)
c)
in
=
(1,
-1,
2,
v,
d)
2,
-3)
=
(3,
-3,
Vl
/)
6,
=
Wl
(1, 0, 1, in
(3,
-2)
=
1, 1)
,
-2,
4,
=
0.
4,
(6, 0, 4, 1)
-6)
n=
,
-9)
(4,
-4,
8,
(2,
-2,
0, 6)
,
-1,1,0),
(2,
t*=(-l, -1, -1,1)
,
=
11,
^
#2X3
(7, 2, 10, 3)
(2, 1, 0, 1)
=
u,
-3,
,
,
=
*=
-2, 1,2),
,
(2,
-
,
-2,
(2,
-9)
ti,
=(1, -2,0,1),
=
x3 ou of the equations
(1,2,3,4),
=
u,
i-(l, -1,0,3),
in
t> 2
-13, -4),
,
i(-l, e)
=
u2
-1,2), in=(l,
a? 2 ,
that the determinant x\x
tn=(7, -11, -6, 1), 6)
69
(0,1,
-8)
-1,2),
*=
,
(1, 1, 0, 0)
4. The row and column spaces of a matrix* We shall obtain the principal theorems on the elementary properties of linear spaces by connecting this theory with certain properties of matrices which we have already derived.
Let us consider a set of
m vectors,
of
=
Ui
(9)
(aa,
.
.
.
,
a in )
(i
=
.
1,
.
.
,
rn)
,
V n over F. Then we may regard w as being the tth row of the correspondum as being m by n matrix A = (ag'=f(g)
(15)
(read g corresponds to g'} such that every element g of G determines a unique corresponding element g' of (?'. In elementary algebra and analysis the sys-
tems G and G' are usually taken to be the field of all real or all complex numbers and (15) is then given by a formula y = /Or). But the basic idea there is that given above of a correspondence (15) on G to (?'. This concept may be seen to be sufficiently general as to permit its extension in many directions.
Suppose now that
f
a correspondence such that every element g of G is the corresponding element f(g) of one and only one g of G. Then we call (15) a one-to-one correspondence on G to G'. It is clear that (15) then (15) is
r
defines a second one-to-one correspondence
M=
(16)
which
is
now on G
ence between
G
to G, and we may thus call (15) a one-to-one correspondand G' and indicate this by writing f
g+
(17)
Note, however, that,
and
0'-
