Multiparameter Eigenvalue Problems, Volume 1: Matrices and Compact Operators (Mathematics in Science & Engineering V. 82-I)

MULTIPARAMETER EIGENVALUE PROBLEMS

VOLUME I Matrices and Compact Operators

This is Volume 82 in MATHEMATICS IN SCIENCE AND ENGINEERING A series of monographs and textbooks Edited by RICHARD BELLMAN, University of Southern California The complete listing of the books in this series is available from the Publisher upon request.

MULTIPARAMETER EIGENVALUE PROBLEMS VO L U M E I Matrices and Compact Operators

F. V. Atkinson Department of Mathematics University of Toronto Toronto, Canada

I972

@

ACADEMIC PRESS

New York and London

COPYRIGHT 0 1972, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED NO PART OF mis BOOK MAY BE REPRODUCED IN ANY FORM, BY PHOTOSTAT, MICROFILM, RETRIEVAL SYSTEM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS.

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United Kingdom Edition published by ACADEMIC PRESS. INC. ( L O N D O N ) LTD. 24/28 Oval Road, LondonNWl7DD

LIBRARY OF CONGRESS CATALOG CARDNUMBER: 78-182608 AMS (MOS) 1970 Subject Classifications: 15A18, 15A69, 34B99, 35P10 PRINTED IN THE UNITED STATES OF AMERICA

CONTENTS

ix xi

Preface Contents of Volume I1

Part I. PRELIMINARIES FROM LINEAR ALGEBRA 1. Linear Spaces 1.1 1.2 13 1.4 1.5 1.6 1.7 1.8 1.9 1.10

1

Introduction Linear Maps Composite and Induced Maps Direct Sums Linear Dependence and Dimension Dimensions of Kernel and Image Further Dimensional Results Topologies Connectedness Semilinear Maps Notes for Chapter 1

4 6 8 10 13 14 16 18 19 19

2. Bilinear and Multilinear Functions 20 23

2.1 Multilinear Functions 2.2 Bilinear Functions V

vi

2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10

CONTENTS

Bilinear Functions on a Single Space Bilinear Forms Sesquilinear Functions Sesquilinear Forms and Endoniorphisms The Zeros of Hermitian Forms Pairs of Hermitian Forms Three Hermitian Forms General Remarks on the Range of a Set of Forms Notes for Chapter 2

24 26 21 30 31 33 35 37 39

3. The Decomposition of Finite-Dimensional Endomorphisms 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10

Ascent and Descent The Case of Equal Ascent and Descent Eigensubspaces and Root Subspaces The Splitting Off of Root Subspaces The Finite-Dimensional Case Several Commuting Operators The Hermitian Case Orthogonality Some Modifications Reduction of Pairs of Hermitian Forms Notes for Chapter 3

4.

The Tensor Product o f Linear Spaces

4. I 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10

Introduction The Definition by Means of Functionals Rases and Dimension The Real and Complex Cases Subspaces Induced Homoniorphisms Exactness Properties Universal Property Bilinear Forms and Tensor Products Products of Sesquilinear Forms Notes for Chapter 4

40 43 45 46 41 50 51 53 55 57 59

60 61 63 66 61 69 71 73 74 76 78

5. Tensor Products and Endomorphisms 5.1 5.2 5.3 5.4 5.5 5.6 5.7

Introduction The Hermitian Case Eigenvalues and Ranks Decomposition The Kronecker Sum and Product Kronecker Sums and Eigenvalues Separation of Variables

79 80 82 83 84 86 88

vii

CONTENTS

89 91 92

5.8 The Tensor Product of Identical Factors 5.9 Induced Maps of Symmetry Subspaces Notes for Chapter 5

Part 2 MULTIPARAMETER PROBLEMS FOR MATRICES 6. Simultaneous Equations in Linear Spaces 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9

Introduction Determinantal Maps Singular Determinantal Maps in the Case k Rectangular Arrays Definiteness Requirements Solutions for Rectangular Arrays Nonformal Determinantal Properties Eigenvalues for a Rectangular Array Decomposition Notes for Chapter 6

=

2

97 101 103 105 107 108 110 111

112 113

7. Simultaneous Eigenvalue Problems for Hermitian Matrices 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9

Introduction The First Definiteness Condition and Its Consequences Orthogonality of Eigenvectors Stronger Definiteness Conditions Splitting of Multiple Eigenvalues Decomposable Orthogonal Eigenvectors A Connectedness Property The Main Result on Positive Definiteness The Eigenvector Expansion Notes for Chapter 7

115 117 119 121 123 127 129 132 133 135

8. The Singularity of Square Arrays 8.1 8.2 8.3 8.4 8.5 8.6 8.7

Introduction Equivalent Singularity Conditions An Algebraic Lemma The Inductive Argument Singularity and Decomposable Tensors The Positive Definiteness Theorem Reproved Eigenvalues and Singularity Notes for Chapter 8

136 137 138 140 143 144 145 146

9. Arrays of Hermitian Forms 9.1 Introduction 9.2 Two by Two Arrays

147 148

...

CONTENTS

Vlll

9.3 9.4 9.5 9.6 9.7 9.8 9.9

Two by Three Arrays General Square Arrays A Property of Convex Cones The Case of Several Cones Square Arrays of Hermitian Forms, Continued Rectangular Arrays of Hermitian Forms Relation between Definiteness Conditions I and I1 Notes for Chapter 9

150 151 153 159 161

163 168 171

10. Completeness of Eigenvectors in the Locally Definite Case 10.1 Introduction 10.2 Eigenvalues and Eigensubspaces 10.3 Eigenprojectors 10.4 Existence of a Nonsingular Determinantal Map 10.5 Completeness of the Eigenvectors 10.6 The Eigenvector Expansion

172 173 175 177 179 180

11. Arrays of Compact Operators 11.1 Introduction 11.2 Notions from Hilbert Space Theory I I .3 Discreteness of the Spectrum 11.4 Truncated Problems 11.5 Sequences of Truncations 11.6 Convergence of the Eigenvalues 1 I .7 Convergence of the Eigenvectors 1 I .8 Introduction of Tensor Products 1 1.9 Discussion of the Expansion 1 I . 10 A Special Case Notes for Chapter I 1

181 183 184 187 188 189 190 192 197 199 20 1

References

203

Index

207

The idea of studying simultaneous eigenvalue problems involving several parameters arises naturally in the treatment of the wave equation by the technique of the separation of variables. Depending on the coordinate system concerned, it may, or may not, be possible to separate the spectral parameters, or “separation constants,” when going from the partial differential equation to a set of ordinary differential equations. [See Morse and Fesbach (1953.)] The former case is the source of Sturm-Liouville theory and the spectral theory of ordinary differential operators. The latter leads to a nontrivial generalization, involving several differential operators and several parameters in an essential manner. The existence of this area of multiparameter eigenvalue problems for differential operators did not escape the attention of early investigators, notably Bocher, Klein, and Hilbert, to mention but a few. Of recent years, the theory has seemed to suffer, if not absolute, at least relative neglect in contrast to the spectral theory of a single differential operator, in spite of a continuing interest on the side of the literature of special functions and mathematical physics. Accordingly, it is part of the aim of this work, and specifically that of Volume 11, to set out the main facts concerning the extension of Sturm-Liouville theory to the multiparameter setting. When the latter is attempted, one is soon struck by another hiatus in the literature, namely that concerning corresponding multi parameter eigenix

Y

PREFACE

value problems for matrices. The ordinary theory is concerned with the singularity of linear combinations of two matrices, one of which usually has to be the identity. In the extension, one associates eigenvalues, sets of scalars, with arrays of matrices by considering the singularity of linear combinations of the matrices in the various rows, involving the same coefficients i n each case. Attention to this area was called in the early 1920’s by R. D. Carmichael, who pointed out in addition the enormous variety of mixed eigenvalue problems with several parameters. Matrix theory serves in a variety of ways as an analog and as a foundation for more difficult problems concerning linear differential operators, or linear operators generally. It has thus seemed to me desirable to give first a treatment of the matrix case; it turns out that this raises problems which have no analog in the eigenvalue theory of a single matrix. Volume I is, accordingly, devoted to the theory of the completeness of eigenvectors associated with arrays of matrices. To keep the length of this volume within convenient bounds, and to minimize the requisite algebraic machinery, the treatment has been largely restricted to the most important case, roughly speaking that of “self-adjoint” situations concerning hermitian matrices. While the eigenvalue theory of arrays of matrices has, I believe, its own interest, I have .presented it here with one particular application in mind. One of the simplest of the many proofs of the completeness of the eigenfunctions of a Sturm-Liouville problem proceeds by way of finite-dimensional approximations, resting on the completeness of the eigenvectors of a symmetric matrix. In the multiparameter Sturm-Liouville case, avenues for proving the completeness of the eigenfunctions appear more limited. However the method of finite-dimensional approximation is still available, and rests naturally on the corresponding property for eigenvectors of arrays of matrices, which is dealt with here. In addition, in Chapter I 1 of Volume I, I have gone through in a linear space setting one version of the procedure of finite-dimensional approximations, to give a completeness result associated with arrays of compact operators. One may treat the differential equation case by means of this result, or again directly. Volume I falls roughly into two halves. In Chapters 1-5, I have given a presentation of some more or less standard material in linear and multilinear algebra; it would otherwise have been necessary to refer the reader to many sources, in order to have the notions and results in the appropriate setting. The multiparameter theory proper is developed in Chapters 6-1 1.

CONTENTS OF VOLUME II

Differential Equations

1. 2. 3. 4. 5. 6. 7. 8.

Introduction, Separation of Variables Properties of Stackel Determinants Reality and Discreteness of the Spectrum Oscillation Theorems Distribution of Eigenvalues Completeness of Eigenfunctions Finite-Dimensional Approximations and Completeness Singular Cases

xi

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PART I ~

PRELIMINARIES FROM LINEAR ALGEBRA

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CHAPTER 1

LINEAR SPACES

1.1 Introduction

The theory of linear spaces draws its inspiration from sources ranging from operations on vectors in euclidean space, the theory of linear equations in a finite number of unknowns, through Fourier series and the theory of equations in an infinity of unknowns and specific function-spaces such as the LP-spaces, to abstract spaces such as those associated with the names of Hilbert and Banach. In the most general notion of all, we assume nothing more than that we have a collection of unspecified objects that admit linear operations, subject to certain formal axioms such as the commutative and distributive laws. This constitutes the basis of the abstract or “algebraic” theory of linear spaces. Although its meager foundations leave little room for analytical subtleties, certain formal constructions can be made, and some formal results proved with the advantages of generality and compact formulation. It is the latter advantage that is significant in what follows. We shall consider mainly finite-dimensional spaces, and so will really be dealing with matrix algebra, but have much need to consider constructions involving two or more spaces. In a linear space, we are able to add the elements, and to multiply them by “scalars,” or numbers from some field K. In what follows, we have in 1

2

1 LINEAR SPACES

mind mainly the complex field, but will often leave the field unspecified. It will always be assumed that K has “characteristic zero”; by this we mean that if c E K is not zero, then 2c, 3c, ... , are also not zero. Frequently, but not invariably, we assume that K is algebraically closed, so that an algebraic equation n

1c,xr = 0, 0

with c, E K , c, # 0, n 2 1,

will have at least one root x in K . This of course holds for the complex field. A linear space X over the field K is then, in brief, an additive abelian group whose elements admit multiplication by elements of -K. In more detail, the following properties hold. For X , ~ EX , there is defined x + y~ X , addition being commutative and associative. (ii) There is a zero or neutral element 0 for addition in X , so that 0 x = x, and for each x an additive inverse (-x), so that (-x) x = 0; we write x - y in place of x + ( - y ) . (iii) For C E K, X E X , there is defined cx E X . This product is distributive in ‘both factors, and is associative in the sense that (clc2)x = (c,(c2x)), for any c l , c2 E K . In addition, we have lx = x, Ox = 0, if 1, 0 denote the identity and the zero elements in K on the left, and 0 denotes the zero element in X on the right; multiple use of the symbol “0” will be unavoidable.

(9

+

+

A trivial example of a linear space over K will be the zero space, formed either by the zero element of K only, or simply by a single abstract zero element, possibly the zero element in some nontrivial space. The simplest nontrivial linear space over K will be K itself. A wide class of examples is given by 1.1.1 If K is aJield, and S any set, the collection of functions THEOREM from S to K forms a linear space.

The import of this actually incomplete statement is that it is possible to define the linear operations and the zero element in such a way that this collection of functions is a linear space over K in a nontrivial manner. We shall use this manner of speech occasionally in what follows. For the proof, we define the sum of two functionsf(s), g(s) from S and K as the function that associates with s E S the value f ( s ) + g(s) E K . The zero element will be the zero function, associating 0 E K with every s E S. The negative of a function f ( s ) will be -f(s). Finally, for c E K and any

1.1

3

INTRODUCTION

function f ( s ) : S + K, (cf)(s) will be the function associating c(f(s)) with s E S. It is a trivial matter to check that the linear space axioms hold. In the most important single case, S is formed by the integers 1, ... ,n. A function from S to K is then given by an ordered set or n-tuple (a,, . .. ,a,) of elements of K. The operations are the usual vectorial or matrix ones, namely (Ul,

... ,a,)

+ (q‘,... ,a,‘)

= (a,

+ a,’,

... ,a,

+ a,’),

c(a,, ... ,a,) = (ca,, ... , ca,),

(1.1.1) ( I. 1.2)

while the zero element is the zero n-tuple (0, ... , 0). If K is the real field, this is denoted by R”,and in the complex case by C“. In Eqs. (1.1.1) and (1.1.2) we have exhibited the elements of such spaces as row matrices, though we shall more often consider them as column matrices. It may happen that the set S in the construction we have just given is finite, but does not have any particular appropriate linear ordering. In connection with tensor products we meet the situation that S is a collection of k-tuples of positive integers (t,,

... , tk),

1 I t, I n,, r = 1, ... ,k.

(1.1.3)

In such a case, functions from S to K are given by collections of n, elements of K , say a

,,...

fk,

l s r r s n r , r = l , ..., k .

nk

(1.1.4)

Whereas it is possible to arrange the elements (1.1.4) linearly in a systematic way-for example, lexicographically-it is perhaps more appropriate to consider them as forming a k-dimensional matrix. Of the many ways in which linear spaces give rise to additional such spaces, we note here two simple cases. A “subspace” of a linear space Xis a collection of elements of X that forms a linear space itself; naturally, the field is the same in the two cases, and the linear operations in the subspace are those inherited from the space X . In order that a collection Y of elements of X should form a linear space it is sufficient that Y be closed under addition, and under multiplication by scalars. For a linear space X , particular examples of subspaces are given by the zero subspace, and the whole space X . We use the term “proper” subspace for other subspaces than these. If Yis a subspace of X , we denote by X / Y, the “quotient-space,’’ elements of which are formed by “cosets” of Y. In other words, we divide the elements of X into equivalence classes, any two members of the same class differing by an element of Y . Any element of such a class can serve a “representative” of the class. The linear operations on X / Y can then be defined in terms of

4

1 LINEAR SPACES

the corresponding operations on representatives. The zero element in X / Y is the class containing the zero element of X ; that is, the class formed by Y itself. 1.2 Linear Maps

Let X , Y be two linear spaces, not necessarily distinct, over the same field K . By a “linear map,” or “linear operator,” or “homomorphism” from X to Y, we mean a function f:X - t Y that preserves sums and scalar x’) =f(x) +f(x’), and multiples. Thus, for x, x’ E X we must havef(x if c E K, ~ ( c x )= cf(x). The space X is the domain o f f ; we do not consider maps defined only on subspaces. The set of values off(x), as x varies in X , will be termed the “range” or “image” o f f ; Y may be termed the “range space,” whether or not f(x) assumes every value in Y. Various special cases have their own names. We termfan

+

( i ) endomorphism if Y is the same as X , so thatf maps X into itself, (ii) monomorphism iff is “injective,” or if f(x) = 0 implies that x = 0, (iii) epimorphism’ifit is “surjective” or onto, so that to each y E Y there is at least one x E X withf(x) = y , (iv) isomorphism if both (ii) and (iii) hold, so that f effects a one-to-one map of Xonto Y, (u) automorphlsm if all of (i)-(iu) hold, so that f gives a one-to-one map of X onto itself.

The set of values off(x) as x varies in X,the image off, will be denoted by Imf; thus epimorphisms f:X -t Yare characterized by Imf = Y . The set of zeros off, that is, the set of x E X such that f(x) = 0, is termed the kernel off, and written Kerf; monomorphisms are thus characterized by Kerf = 0, where the latter denotes the zero subspace of X . We note

THEOREM 1.2.1. The kernel o f f , Kerf, and the image o f f , I m f , are subspaces of X , Y, respectively. As remarked in the last section, in order to show that these collections of elements are subspaces, we need only show that they are closed under addition and under scalar multiplication. In the case of Kerf, we have that x, x’ E Kerf, or f(x) = 0, ,f(x’) = 0, implies that f(x) f(x’) = 0, and so f(x x’) = 0, so that x x’ E Kerf; likewise, if f(x) = 0, c E K, then f(cx) = cf(x) = 0, and so cx E Kerf. Passing to the case of ImS, suppose that y , y ‘ E Imf, or that y =f(x), y ’ =f(x’); we then have y y’ =

+

+

+

+

1.2

+

5

LINEAR MAPS

+

f ( x x’), and s o y y’ E Imf; again, ify = f ( x ) , c E K, we have cy = f ( c x ) , and so cy E Imf. In view of the last theorem, we may define certain quotient-spaces. The most important of these is Cokerf= Y/Imf, the cokernel o f f ; an epimorphism is characterized by the vanishing of the cokernel. The parallel definition is that of Coimf = X/Kerf, the coimage off. We pass now from the consideration of a single linear map from a linear space X to a linear space Y, over a field K, to the collection of all such maps from X to Y. This is denoted by Hom(X, Y ) ;one may also write Hom,(X, Y), but we shall be concerned only with homomorphisms over one and the same field, and so will not find this necessary. Much as in Theorem 1.1.1, we assert 1.2.2. The set of homomorphisms Hom(X, Y ) is a linear space. THEOREM

As before, if f, g E Hom(X, Y ) , we define f + g by (f+ g ) ( x ) = f ( x ) g(x), and cf by (cf)(x) = c ( f ( x ) ) , for all x E X . The zero element is to associate 0 E Y with every x E X . One must verify first that these prescriptions yield maps that are linear, and so in Hom(X, y ) . We do this for the case off g . We first check that this map is additive, or that

+

+

(f+ g ) ( x + x’)

=

(f+ g)(x> + (f+ g)(x‘>.

These are, respectively, equal to

f(x

+ x? + g(x + X I ) ,

by the definition off same as

(f(x)

+

+ g ; sincef, g E Hom(X, fW

+ (f(x‘) + g b ’ ) )

Y ) , the first of (1.2.1) is the

+f(x’> + g ( x ) + g(x’),

which is equal to the right of (1.2.1). We must also show that f homogeneous, or that

or

(1.2.1)

+g

is

(f+,g>(cx) = c ( f + g>(x),

f(cx> + g(cx> = c ( f ( x ) + g(x>), and this again follows from the fact thatf, g E Hom(X, Y). In a similar way, we show that cf E Hom(X, Y), while it is immediate that the zero map is in Hom(X, Y ) . Thus the linear operations of addition and scalar multiplication are defined from Hom(X, Y) into itself. Once more, we pass over the verification that these operations satisfy the requirements for a linear space.

1

6

LINEAR SPACES

Two special cases deserve special notation. If Y = X , Hom(X, Y ) becomes the space of endomorphisms of X , which we denote by End X . If Y = K , Hom(X, Y ) becomes Hom(X, K ) , the set of linear maps from X to the coefficient field K ; these maps are usually termed “linear functionals,” and the linear space that they form “the dual space” denoted by X * . Since we have imposed no continuity requirements on these functionals, the space in question is sometimes termed the “algebraic dual.” 1.3 Composite and Induced Maps

We start with some formalities concerning arbitrary maps between arbitrary sets. Let S, T be any sets, and f any function from S to T. If S’, T’ are further sets, and we have further functions g: S‘ + S, h : T + T‘, we can form the “composite” functions f’o

g: S’ + T,

(1.3.1)

: S + T’.

(1.3.2)

It0 f

For example, f g takes s’E S‘ first into g(s’) E S, and then into f ( g ( 5 ’ ) ) E T, as in the diagram 0

S‘ZSLT.

(1.3.3)

We now add a further formality. The transformation in which we start with a map f : S + T and end with a map f 0 g as in (1.3.1) can be considered as a transformation of the varying mapfor class of maps f by a fixed map g. In other words, g “induces” a map, which we may denote by gt, with action gt: Map(& T ) 4 Map(S’, T),

(1.3.4)

where Map(S, T) denotes the set of maps or functions from S to T. Similarly, (1.3.2) defines an induced map

At: Map(& T ) + Map(S, T‘).

(1.3.5)

Extending this notation, we have constructed maps M a d s ’ , T) + Map(Map(S, TI, Map(S’, TI),

(1.3.6)

Map(T, T’) + Map(Map(S, T ) , Map(& T‘)).

(1.3.7)

Here (1.3.6) and (1.3.7) correspond, respectively, to (1.3.4) and (1.3.5). Our point in this section is that these formalities can be specialized to the linear context, the sets being linear spaces over some fixed field, and Map being restricted to Hom. One principal property is that the composite of two

1.3

7

COMPOSITE AND INDUCED MAPS

linear maps is again linear, provided that the composite is well defined, the domain of the second map including the range of the first THEOREM 1.3.1 Let X , Y, Z be linear spaces, and let A E Hom(X, Y ) , B E Hom( Y, Z ) . Then B A E Hom(X, Z ) . For any X E X , we now write A x for the image of x under A . We then have, as a matter of definition, B(Ax) = (BA)x, and so may write BAx unambiguously for them. For any x , x’ E X , we then have BA(x

+ x ’ ) = B ( A ( x + x ’ ) ) = B(Ax + A X ’ ) = BAx + BAx’,

and likewise, for a scalar c, BA(cx) = B(A(cx)) = B(cAx) = c(BAx),

so that indeed B A E Hom(X, Z ) . There is a similar result for induced maps. THEOREM 1.3.2 If A E Hom(X, Y ) , B E Hom( Y, Z ) , the transformation that takes B into BA, for fixed A and varying B, defines an induced map (1.3.8)

A t : Hom( Y, Z ) + Hom(X, Z ) ,

which is linear. The transformation taking A into BA, with fixed B and varying A, defines an induced map

Bt : Hom(X, Y )

also linear.

--f

(1.3.9)

Hom(X, Z ) ,

In each case, one needs to show that the induced map preserves sums, and that it commutes with multiplication by scalars. In the case of (1.3.8) the required properties are that, with B, B I , B, E Hom( Y, Z ) , A E Hom(X, Y ) , (B1

+ B J A = B I A + BZA,

(cB)A

=

c(BA).

(1.3.10)

The meaning of the first of these is that for any x E X we have { ( B , f B2)A)x = B1Ax

+ B2Ax.

In fact, the left is the same as ( B , + B2)(Ax),and this, by the definition of addition in Hom( Y, Z ) , is the same as B,(Ax) B,(Ax), which coincides with the right. Similarly, for the second of (1.3.10), we must show that

+

(cB)Ax = c(BAx), which follows from the definition of scalar multiplication in Hom( Y, Z). The discussion of (1.3.9) is similar.

8

1

LINEAR SPACES

An important case is given by taking, in (1.3.8), Z = K , the field. We have then that a map A E Hom(X, Y ) induces a map A t E Hom( Y*, X * ) of the dual spaces, proceeding in the reverse direction. Specifically, let f E Y*, so that if y - Y , f y denotes a scalar. The action of A t o n f i s to produce the functional that takes x E X into the scalar f A x ; it is natural to denote this functional by f A . In finite-dimensional terms, one may view f A x as the product, in that order, of a row, a rectangular, and a column matrix. 1.4

Direct Sums

Among the many additional ways of generating new spaces out of linear spaces already given are two which bear a close resemblance to addition and multiplication. These operations are the “direct sum” and “tensor product.” The latter is a comparatively recondite matter, which will be taken up in Chapter 4; the direct sum construction is quite elementary. There are actually two constructions which bear this name. The “external” direct sum of a number of linear spaces may be formed whether or not there is any relation between the spaces. The “internal” direct sum can be formed only for subspaces of one and the same space, and then only for restricted sets of subspaces. In the external direct sum, we suppose given k linear spaces G, , ... ,Gk, over the same field K . In one extreme, the spaces might be all the same, and in the order there might be no relation between the spaces. In any case, the external direct sum, written (1.4.1) G1 @ ’ * . @ Gk is specified as consisting of k-tuples of elements grEGr, r = 1,...,k,

(gi,...,gk)9

(1.4.2)

taken one from each of the spaces. Addition and scalar multiplication take place componentwise, as in (gl

9

...

Y

gk)

+ ( g l ’ , ...

9

gk’) = (81

+ gl’,

9

gk

+ gk’),

, cgk)Finally, the zero element is specified as (0, ... , 0), where turn the zero elements in G,, ... ,Gk . c(gl Y

**’

Y

gk) = (cgl Y

(1.4.3) (1.4.4)

“0” denotes in

This external direct sum coincides with the Cartesian product of the G1 , ... , Gk , so far as the elements of the two spaces are concerned. However the Cartesian product lacks the linear structure. In the special case that G,, ... , G k all coincide with some linear space G, we obtain the direct sum of k copies of G. In particular, if G is the field K,

1.4

9

DIRECT SUMS

we obtain effectively the space of row or column matrices, with entries in K . In standard cases this space is written as a power rather than a sum, as in Ckfor k-dimensional complex space. Suppose now that C, , ... ,G, are all subspaces of a single linear space X . In this case, we can form a sum, written

+ Gk,

GI -l

(1.4.5)

which is the set of elements of the form gl

+ + gk,

g,

E

r = 1, ... k.

G,,

9

(1.4.6)

This is a subset of the elements of X , which is easily seen to be closed under the linear operations. It is therefore a subspace of X . We can still form the external direct sum (1.4.1) of elements (1.4.2); the latter are not in X , if k > 1, but are rather elements of the direct sum of k copies of X . If we form the map that takes (1.4.2) into (1.4.6), we obtain a map GI @ ... @ Gk

-+

GI 4-

+ Gk,

1..

(1.4.7)

which is linear and onto, and so an epimorphism. However, it need not be monomorphic. For example, if k = 2, and G I = G 2 , elements of the kernal of (1.4.7) are given by pairs (8, -g), g E G I , and so the kernel is nonzero unless G, = G 2 = 0. The appellation “direct” is reserved for sums of subspaces in which (1.4.7) is monomorphic, and so an isomorphism. More explicitly, if GI , ... ,Gk are subspaces of X , the collection (1.4.5) of elements (1.4.6) is termed the (internal) direct sum of the G, if g,

implies that g, (1.4.5)

+ ..- + g, =

0, r

=

=0,

r

g,EG,,

= 1,

... , k ,

(1.4.8)

1, ... , k . We indicate this by writing, in place of GI

4

.**

4 Gk.

(1.4.9)

Sometimes we use a notation for such sums. It may happen that for some set of subspaces G I , ... ,G, of X we have GI

4

* * a

4 G,

=

X.

(1.4.10)

In this case we speak of a “direct sum decomposition” of X . In such a case every g E X has a unique expression g

=

g,

+ ... + gk,

g , E G,, r = 1, ... , k.

It is a prime task of eigenvalue theory to achieve direct sum decompositions which satisfy additional conditions. We need later a result on the extension of such decompositions.

10

1 LINEAR

SPACES

THEOREM 1.4.1. Let (1.4.10) be a direct sum decomposition of the linear space, and let the last summand have the decomposition ck = Hk 4 J , . We then have a direct sum decomposition

We must show that for any x E X , a representation

where the summands belong respectively, to the spaces in (1.4.11), both exists and is unique. For existence, we note that there is, by (1.4.10), an expression x = g , + ... + g , , gr E G,, r = 1, ... ,k,and, by hypothesis, a decomposition g, = h, + j , , which yields (1.4.12). For the uniqueness, we suppose that (1.4.12) holds with x = 0, and must show that all terms on the right are zero. We have

and here all k terms on the right must be zero, by (1.4.10). In particular, we have h, j , = 0, and by the directness of the decomposition of Gk we deduce that h, = j k = 0. This completes the proof.

+

1.5 Linear Dependence and Dimension A finite number of elements g , , ... ,g, of a linear space G are said to be linearly dependent if there holds a relation of the form

in which the scalars c, are not all zero. Equivalently, they are linearly dependent if some one of them can be expressed as a linear combination of the others. They are said to be linearly independent if no such relation holds. The linear space G is said to be finite dimensional, or of finite dimension, if there is an upper bound to the number of elements in a linearly independent set. The dimension of G, which we denote by dim G, will then be the number of elements in a maximal linearly independent set. If linearly independent sets of any size can be found, G is said to be infinite dimensional. The field, considered as a space over itself, is one dimensional. The dimension of the zero space is zero. We collect some basic properties.

1.5 LINEAR

DEPENDENCE AND DIMENSION

11

THEOREM1.5.1. Let dim G = n, 0 < n < co, and let the elements ... ,Gn of G be linearly independent. Then any element g E C admits a represen tation g,,

(1.5.2) If the representation were not unique, then on taking two distinct such representations of g and subtracting we should obtain a nontrivial relation of the form (1.5.1), which is impossible. To see that such a representation of g exists at all, we note that, by hypothesis, the elements g , , ... , g,, g must be linearly dependent, so that there holds a relation n

C1 drgr + dg

=

0,

where the scalars dr , d are not all zero. Here we cannot have d = 0, for then we should have a nontrivial relation of the form (1.5.1). We thus have n

g = C(-dr/d)gr, 1

which has the form (1.5.2). A maximal linearly independent set will be termed a “basis” of the space. According to our definition, the dimension of the space, if finite, is the greatest number of elements in any basis. Actually, all bases contain the same number of elements, this number being the dimension. This will follow at once from THEOREM1.5.2. Let dim G = n, 0 < n < co. Then any linearly independent set j , , ... , j , of elements of G, when m < n, can be extended by the addition of.further elements to.form a basis. It will be sufficient to show that we can add a further element j , + l without destroying the linear independence. Suppose that this cannot be done, and that the set j , , ... ,j,, is a maximal linearly independent set. The argument of the proof of Theorem 1.5.1 then shows that every element of G has a unique expression as a linear combination of j , , ... ,j,. Let g , , ... ,gn be a linearly independent set; the hypothesis that dim G = n implies that one such set must exist. We can express the gr in terms of the.j, in the form m

gr

=

C1cr,j,,

s=

r

=

1,

... , n.

(1.5.3)

12

1 LINEAR SPACES

We now claim that it follows from (1.5.3), in which m < n, that the gr are linearly dependent. Indeed, we claim that this is true of the gr,

r=1,

..., m + l .

We prove this by induction over m. If m = 1, the proposition is that two multiples of one element are linearly dependent, which is certainly true. Passing to the general case, with m 2 2, suppose first that

Then g, , ... ,gm are linear combinations of j , , ... , j m - , and so, by the inductive hypothesis, g, , ... , g, are linearly dependent. Suppose next that at least one of the crm,r = 1, ... , m + 1 is not zero. To simplify the notation we suppose the g, numbered so that c l m # 0. Then the m elements

are linear combinations o f j , , ... , j m - land so, by the inductive hypothesis, are linearly dependent. Thus we have a relation of the form m+ 1

1 dAgr 2

Crmgl/Clm)

=

0,

where d,, ... ,dm+ are not all zero. This constitutes a nontrivial linear relation connecting g , , ... ,gm+,, and so the result is proved. We thus have a contradiction with the hypothesis that g , , ... , g, are linearly independent, and so have completed the proof of the theorem. We deduce a monotonic property of the dimension-function.

T ~ O R E1.5.3. M Let H be a subspace of the finite-dimensional space G. Then,for H # G it is necessary and suficient that dim H < dim G .

(1.5.4)

It is obvious that if H is contained in G, then dim H I dim G, and that equality holds if H = G. The condition is thus sufficient. Conversely, suppose that H is strictly contained in G, that g E G, g f! H , and t h at j , , ... , j mare linearly independent elements of H forming a basis of H , so that m = dim H . Since g is not a linear combination of j , , ... ,j,, , the set j , , ... ,j m, g is linearly independent, and so dim G 2 m 1, which proves (1.5.4).

+

1.6

DIMENSIONS OF KERNEL AND IMAGE

13

1.6 Dimensions of Kernel and Image We sometimes need to argue that a set of homogeneous linear equations possesses solutions, not by exhibiting them, but by considering the dimensions of the spaces involved. The same procedure may serve also to identify the solution-space, if we can indicate a space of the correct dimension which contains this solution space. We need

THEOREM 1.6.1. Let A E Hom(X, Y ) , where X , Y are linear spaces and X is finite-dimensional. Then dim X

=

dim Ker A

+ dim Im A .

(1.6.1)

We dispose first of the case that Ker A = 0, so that A is monomorphic; the result then asserts that dim X = dim Im A . Let x , , ... ,x,, be linearly independent elements forming a basis of X.We then claim that the elements A x , , ... , A x , are linearly independent, and form a basis of Im A ; this will prove the result. To show that the A x , are linearly independent, suppose that c,Ax, = 0. We have then A x ; c,x, = 0, and so c,x, = 0, since A is monomorphic. Since the x, are linearly independent, we have that all the c, = 0. To show that the A x , generate Im A , we take an arbitrary element A x in Im A . Then x admits a representation in the form c,x,, and so Ax = crAx,. This completes the proof. For the event that A is not monomorphic, we write dim Ker A = m, and choose a basis x , , ... ,x, of Ker A . Let dim X = n, so that m 5 n. If m = n, then Ker A = X,A is the zero map on X into Y, and (1.6.1) takes the form n = n + 0. Suppose then that m < n . We extend the set x,,... , x,, to a basis x1, ... ,x, of X . We then claim that then - m elements x,,,+,, .. . , x,, are such that A x , + , , ... ,Ax,, are linearly independent, and form a basis of Im A . It will then follow that dim Im = A = n - in, so that we have (1.6.1). For an arbitrary X E X , we may write x = C; c,x,, and have then C,X, = c,Ax,. This shows that the A x , , r = In + 1, ... ,n Ax = generate Im A . It remains to prove their linear independence. We assume that c,Ax, = 0, so that Tm+, crxr E Ker A. It then follows that c,x, is a linear combination of x , , ... ,x,, and this is only possible if the c, are all zero, since x, , ... ,x,, are linearly independent. This completes the proof. In ptdicular, we have

1;

1;

1;

1;

rm+, rm+ ,

EL+

14

1

LINEAR SPACES

THEOREM 1.6.2. If A E Hom(X, Y ) , dim Y < dim X
0, dim H = n > 0. The proof proceeds, in effect, by representing elements of Hom(G, H ) by matrices with mn elements. Let bases of

1.7

FURTHER DIMENSIONAL RESULTS

15

G, H be, respectively, g , , ... , gm and h , , ... , hn. We define a collection of maps Ars

by

Arsgs =

r

1,

=

... , n,

s = 1, ... i n ,

Arsgt = 0 ( t Z s)*

hr,

These are extended to the whole of G by linearity; for a general g = we define m

(1.7.5) (1.7.6)

1:c,g, (1.7.7)

The elements (1.7.5) are mn in number, so that to prove the theorem we must show that they are linearly independent, and generate Hom(G, H ) . On the first point, suppose that for some scalars b,, we have (1.7.8)

On applying the left to g , we obtain n

C brthr r=l

=

0,

and since the h, are linearly independent, we have b,, = 0, as was to be proved. We claim next that for any A E Hom(G, H ) there is a representation A = CCbrSArs.We determine the coefficients b,, so that the two sides agree in their action on the basis elements g,. Since Ag, is a linear combination of the hr, we can write

if the b,, are determined in this way we shall have

by (1.7.6), as required. The operators A , CCbrSArsmust then coincide on the whole of G. In the special case Y = K , the field, we have

THEOREM 1.7.3. r f the linear space G isJinite dimensional, and G* is its dual, then (1.7.9) dim G = dim G*. This follows from Theorem 1.7.2. since dim K = 1. The details of the argument deserve an indication for the present specialization. If dim G = m,

16

1 LINEAR SPACES

and g , , ... ,g , are a basis of G, we can form a basis of G * , say h , , ... , h,, which are, in a sense, biorthogonal to the g, . The functionals h, in question are defined by hrgs

=

1

1

( r = s),

0

( r # s);

( 1.7.10)

this actually defines their action on the basis elements of G, and we are then to extend their action to the whole of G by linearity. Bases of a space and its dual, with the biorthogonal property (1.7.10), are termed “dual bases.”

1.8 Topologies Although our considerations are mainly algebraic, we shall at certain crucial points appeal to principles such as the following:

(i) a continuous real-valued function, whose domain is connected, and which does not vanish, has fixed sign, (ii) a continuous real-valued function, whose domain is compact, attains its upper and lower bounds, (iii) a bounded subset of a finite-dimensional linear space is sequentially compact, this being the Bolzano-Weierstrass property. For this purpose we must introduce notions of convergence in a linear space, and of the continuity of functions with domain or range in such a space. One must start by choosing a topology in the field K , so that the notion of the convergence of a sequence of elements of K has sense. The notion may then be extended immediately to the direct sum of k copies of K ; the convergence of a series of k-tuples (Cln,

... , Ckn)

+

(c19

... , C k ) ,

n

+

m,

(1.8.1)

means precisely that

crn+cr,

n+m,

r=l,

..., k.

(1 3.2)

The procedure can then be extended to an abstract linear space G of finite dimension k . We choose a basis g , , ... ,g, of G, and can then associate with a general g E G a k-tuple (cl , ... ,c,) of scalars by means of the representation g= c,g, . A sequence of elements of G then converges to a limit if the

1:

1.8

TOPOLOGIES

17

same is true of the corresponding k-tuples of scalars. Of course it is desirable to show that the definition does not depend on the choice of basis. Similar remarks apply to the notion of boundedness. In the real and complex cases of relevance here, we accommodate both convergence and boundedness by treating K as a metric space endowed with a norm. We treat one method of norming finite-dimensional linear spaces in 1.8.1. Let G be an n-dimensional linear space, over the real or THEOREM the complex field. Let g , , ... ,gn and h , , ... , hn be two bases of G. For any g E G, we define two norms IJgII,llgll, as follows:

if

(1. 8.3)

then

(1.8.4) These norms are equivalent, in the sense that there are positive constants a, /3 such that (1.8.5) allgll 5 llgllt 5 Pllgll.

It will be sufficient to prove one of (1.8..5), and we choose the second. We have to show that llgll'/llgll is bounded above when g # 0, or that Jlgllt is bounded above when g # 0, or that llgllt is bounded above when llgll = 1. In fact, if the bases are related by n

gr =

then

C arshs, s=

r = 1, ... , n,

1

n

n

n

and on comparison with (1.8.3) we obtain n

d, =

Hence

C crarS, r= 1 n

C1 Ids1 5

s = 1, ... , n. n

n

r=l

IcrI

1 IarsI,

s=l

and so the second of (1.8.5) holds with

P This completes the proof.

n

=

max C IarsI. r

s=l

18

1

LINEAR SPACES

A bounded subset S of G will be one for which there exists an M > 0 such that llgll < M for all g E S; the definition is independent of the choice of basis, by means of which the norm is defined. The same is true for a number of other definitions using such a norm. A sequence g(’), g(’), ... will converge, with limit g , if IIg(m)- gll

+

0

as m + 03. A closed set will be one that includes the limits of all convergent sequences of elements of the set. A function g(t) from a real interval [a, b ] to G will be continuous at t‘ if Ilg(t) - g(t’))I + 0 as t + t’. The image in G of a function continuous in [ a , b ] may be termed an “arc” or “curve” in G. A real- or complex-valued function f defined on G is continuous if, for every g’ E G, we have that g + g’ implies that f ( g ) +f’(g’). If f ( g ) is continuous and g(t), a 5 t _< b , is a (continuous) curve, thenf(g(t)) is continuous. 1.9 Connectedness

We use this term in the sense of “arcwise” connectedness. A subset S of a finite-dimensional real or complex linear space G is then connected if for every pair g , , g , S~there is an arc g(t), a It 5 by such that g(a) = g , , g(b) = g , , which is contained entirely in S. In particular, it might happen that g , , g , can be connected within S by the segment g,(l

- t ) + g,t,

0 I t I1,

(1.9.1)

or that they can both be connected within S to a third point of S by segments. Generally, if g , , g , can be connected by a finite sequence of such segments, we say that they are polygonally connected, and that S is so connected if this is the case for all pairs of points of S. We use later

THEOREM 1.9.1. The set of nonzero elements of a jinite-dimensional complex linear space is connected. Let g , , g , be two nonzero elements of the space. They are certainly connected by the segment (1.9.1). Furthermore, if g , , g , are linearly independent, no point of (1.9.1) will be zero, and so we shall have connected g,, g, within the set of nonzero elements. It remains to deal with the case that g , , g , are linearly dependent, or that g , = cg, , for some scalar c # 0. The segment (1.9.1) will serve in this case also, except when c is real and negative. In the latter event we can join g , ,g , to ig, by segments of the form (1.9.1), of which no point will be zero.

1.10

19

SEMILINEAR MAPS

1.10 Semilinear Maps The complex field has the feature of admitting a nontrivial involution, namely the map which takes a complex number c into its complex conjugate C. This map preserves sums and products, is not the identity, while its square is the identity, in that C = c. For this situation we have, in addition to the ordinary linear maps, a second class of maps. For two complex linear spaces X , Y, we term a map A : X + Y “semilinear” if it is additive, so that A(x

+ x’)

= Ax

+ Ax‘,

x, x’ E

x,

(1.10.1)

whereas in respect of homogeneity it behaves according to A(cx) = C(Ax),

x EX.

(1.10.2)

Other terms for this type of map are “antilinear” or “conjugate linear.” The collection of semilinear maps from X to Y may be denoted by Sem(X, Y ) . It forms a complex linear space, with the linear operations in it being defined just as for Hom(X, Y ) . The principal example is given by hermitian conjugacy operations in which one proceeds from a matrix of complex entries, of any dimensions, and proceeds to the transposed matrix of complex conjugate entries. If, in particular, Xis the space of complex column-matrices, with entries a, , ... , a, , the map which takes this into the space of complex row-matrices, taking the may be viewed as a semilinear map from X into its above into (6, , ... ,in), dual X * , if row matrices act on column matrices by matrix multiplication. Notes for Chapter 1 As a general reference we cite the text of Greub (1967); the third edition uses the term “vector space” rather than “linear space.” The terminology concerning external and internal direct sums follows the usage of Greub (1967), who, however, defines the internal direct sum for families of subspaces that need not be finite.

CHAPTER 2

BILINEAR AND MULTILINEAR FUNCTIONS

2.1 Multilinear Functions In the case of homomorphisms of linear spaces we were dealing with functions of a single variable. In many areas of mathematics, it is important to extend the consideration to similar functions of several variables. We shall begin with the general definition for any number of variables and then specialize to the case of two variables. In Chapter 4, we return to the general case in connection with tensor products. Let G1 , ... , Ck,Hall be linear spaces, over the same field K. We consider functions gk),

f ( g l , . - *9

grEGr,

= l,...

9

k,

(2.1.1)

which take their values in H, so that f acts from the Cartesian product G, x ... x Gk to H. We say that f is “multilinear,” or “k-linear,” if it is a linear function of each variable gr individually, when the other variables g l , ... ,g r - l , ... ,gk are fixed. For each r, we must therefore have f k l

9

***

9

+ gr’,

gr

=f(gl

3

Y

**.

gr,

gk)

... gk) + f ( g l 9

20

7 * * *

>

gr’

9

*** 9

gk),

(2.1.2)

2.1 for all pairs g , , g,' f(g1 7

E

21

MULTILINEAR FUNCTIONS

G,, and

' ' -3

cgr, * * '

9

gk) = d ( g 1

9

*"

7

gr,

*'*

3

gk).

(2.1.3)

In an obvious notation, we denote the set of all such functions by Mult(G, , ... , G,; H).

(2.1.4)

In the case k = 1, Mult reduces to Hom. As for Hom, we have 2.1.1. The set of functions (2.1.4) is a Zinear space. THEOREM The proof is similar to that of Theorem 1.2.2. We add functions in (2.1.4) by adding the values which they assign to the same set of arguments; multiplication by scalars is defined similarly. The zero element of (2.1.4) assigns the value 0 E H to all sets of arguments in G, , ... , Gk. As in the case of Hom, our definitions here are purely algebraic; no restrictions of the nature of continuity are imposed. These would not constitute restrictions in the finite-dimensional cases of prime interest here. In extension of Theorem 1.7.2 we calculate the dimension of the space (2.1.4). THEOREM 2.1.2. If G , , ... , ck , H are finite dimensional, then dim Mult(G, , ... , G,; H ) =

(ndim G,)dim H. k

1

(2.1.5)

To avoid triviality, we assume that all dimensions are positive, and write dim G, = m , , dim H = n. We claim that a basis of (2.1.4) is given by certain linearly independent elements, which we denote by

L,...uku, 1I

ur

s

m, , 1 I u I n .

(2.1.6)

The number of these elements is, of course, given by the right-hand side of (2.1.5). The definition of the functions (2.1.6) will be in terms of arbitrary bases and

grsrEGr,

h,EH,

Sr

=1,***,mr,

t = 1 ,... , n .

(2.1.7) (2.1.8)

The function (2.1.6) assigns the value h, to the set of arguments g , , ,...,gk,, . To sets of arguments g , , , ,...,gk,,, in which s, # ur for one or more r, the function assigns the value 0. This defines the value of (2.1.6) for sets of

22

2

BILINEAR AND MULTILINEAR FUNCTIONS

arguments chosen from the basis sets (2.1.7). By multilinearity, we extend its domain to the whole of G1 x x Gk. Explicitly, to a general set of arguments

the function (2.1.6) assigns the value (2.1.10) The proof that the elements (2.1.6) are linearly independent, and that they generate (2. I .4), is similar to that given in Section 1.7 for Hom. Suppose first that (2.1.11) vanishes. It must therefore assign the value 0 to every set of arguments

g l s l,...,gksk. This shows that

n

Since the hu must be linearly independent, we have that the coefficients in (2.1.11) must all be zero. It remains to check that functions (2.1.6) generate (2.1.4). Let f be any element of the latter. If the G, are given in the form (2.1.9), by the multilinear property, we have that

(17 c r s , ) f ( g l s l ~ k

f(g1,

9

gk) = S I , . . .Sk

1

*** 9

gksk);

(2.1.12)

this follows by repeated application of (2.1.2) and (2.1.3). We now write (2.1.13) Substituting (2.1.13) in (2.1.12) we obtain

In view of (2.1.9) and (2.1.10), this constitutes a representation o f f in the form (2; 1.1 1). This completes the proof of Theorem 2.1.2.

2.2

23

BILINEAR FUNCTIONS

2.2 Bilinear Functions

k

We now specialize to the case of two argument-spaces, given by taking

= 2 in the last section; discussion of the general case will be resumed in

Chapter 4. Suppose that we have three linear spaces G, , G 2 , and H ; the space Mult(G, , G,; H ) consists of functions f(g, ,g,), g, E G,, r = 1,2, with values in H, such that if either argument is fixed, f depends linearly on the other. Such functions are termed “bilinear.” Bilinear functions are often considered as “products”; the bilinear property requires that the “product”,f(g, ,g,) have the distributive property expected of a product. Thus, the scalar product of two vectors in real euclidean n-space is a bilinear function. The product of a linear operator, acting from a space X to a space Y, and a vector in X , constitutes a bilinear function from Hom(X, Y ) and X , to Y. The defining property of a bilinear function is slightly amplified in THEOREM 2.2.1. Let G1, G 2, H be linear spaces. Then there are isomorpliisms between Mult(G1, G2 ; H ) and the spaces Hom(G1, Hom(G,, HI),

(2.2.1)

Hom(G, , Hom(G, , H ) ) .

(2.2.2)

It will be sufficient to deal with the case of (2,2.1), and so to construct a n isomorphism Mult(G, , G , ; H ) -+ Hom(G, , Hom(G,, H ) ) .

(2.2.3)

There are three steps. First, one must construct the map (2.2.3). Second, it should be shown that the map is linear; this will be omitted. Finally, we show that the map is onto, and has zero kernel. We define (2.2.3) by saying that it takes a bilinear function f(gl, g,), with values in H , into the map described diagrammatically by g1

-+

(8 ,

-+fkl?g2));

(2.2.4)

in other words, with each g, , we associate the map taking g, intof(g,, g,). We show next that the map is onto. Let q be an element of the right of (2.2.2), so that with every g , E G I we have a linear map qg,from G, to H ; let the image of g, E G, under this map be qg,(g2).We then claim that this is a bilinear function from G1, G, to H ; it is certainly linear in g,, and is seen to be linear in g,, since the right of (2.2.3) consists of linear maps on

24

2

BILINEAR AND MULTILINEAR FUNCTIONS

G , . One checks further that if we apply to this bilinear function cp,,(g,) the map (2.2.3) we recover cp. Thus (2.2.3) is onto. Suppose finally that f(g, ,g,) is mapped into zero by (2.2.3). This means that for every g, , the map g, +f(gl, g,) is zero, so that f must be the zero bilinear function. This completes the proof. Next consider the nature of the image (2.2.4) of an individual bilinear functionf(g, ,g,). If this image is monomorphic, that is, iff(g, ,g2) = 0 for all g, implies that g, = 0, we say thatfis “nondegenerate” in its first argument. If, to impose a more drastic requirement, the image off under (2.2.3) is an isomorphism, so that every linear map from G, to H is obtainable by precisely one value of g, in g, +f(gl, g,), we say that f is “nonsingular” in its first argument. Similar definitions apply with respect to the second argument. We term f nondegenerate, or nonsingular, without reference to either argument, if the property holds with respect to both arguments. In a similar fashion to Section 1.3, we can form composite or induced maps involving bilinear functions. Thus if f E Mult(C, , G2 ; H ) , and A E Hom(H, H’), the function 4f(gl, g,) is clearly an element of Mult(G, , G2 ; H ’ ) . One has, that is, a map

Hom(H, H ’ ) + Hom(Mult(G, , G, ; H ) , Mult(G, , G2 ; H’)). (2.2.5) Consider next transformations of the argument spaces. Let A E Hom(Gl , G1’). If f E Mult(G,’, G2 ; H ) , the expression f(Ag,, g,), where g, E G , , g, E G 2 , is an element of Mult(G, , G, ; H ) . We thus obtain a map Hom(G1, C , ’ ) + Hom(Mult(G,’, G2 ; H ) , Mult(G,, G, ; H)).

(2.2.6)

As in the case of (1.3.8), this induced map is “contravariant,” in that the order of G1, G , ’ differs on the two sides. Naturally, there will be specializations of (2.2.5) and (2.2.6) in which H’ = H , GI’ = G I , with Hom being replaced by End, and a variant of (2.2.6) in which G2 undergoes transformation instead of G I .

2.3 Bilinear Functions on a Single Space In this, the first of two specializations of the last section, we take it that the argument-spaces G I , G, are the same; we denote them by G . Instead of Mult(G, G ; H ) we shall write Bil(G, H ) ; its elements are functions f ( g , g’), with g, g’ E G , with values in H , and linear in each argument. In accordance with Theorem 2.3.1 there are two isomorphisms Bil(G, H ) 3 Hom(G, Hom(G, H ) ) .

(2.3.1)

2.3

BILINEAR FUNCTIONS ON A SINGLE SPACE

25

Any element of the left will thus have two images in the space on the right, and it is appropriate to ask when these coincide. Explicitly, for a given functionf(g, g’), we associate with g” E G the map g -,f(g, g”) and the map g -,f(g”, g). These coincide if and only if f(g, g”), f(g”, g) coincide. We call a bilinear function “symmetric” if f(g, g’) = f k ’ , g),

g 9

g’ E G.

(2.3.2)

The set of such functions forms the subspace Sym(G; H ) of Bil(G, H). It is obvious that for a symmetric function f(g, g‘) = 0 for all g and fixed g’ implies the same forf(g‘, g). Thus the notions of nondegeneracy with respect to the first and second arguments coincide in this case. For any bilinear function f(g, g’), g, g’ E G, we can define an associated “quadratic” function

F O

= fkY g),

g E G.

(2.3.3)

One verifies immediately the homogeneous property that F(cg) = c2F(g),

c E K,

(2.3.4)

and the parallelogram^' property F(g

+ g’) + F(g - g’) = 2F(g) + 2F(g’).

(2.3.5)

The question arises of whether the bilinear function f can be recovered from the associated quadratic function F. The answer is affirmative iff is symmetric.

THEOREM 2.3.1. I f f E Sym(G; H), and F is given by (2.3.3), then f(g, g‘) =

+ g’) - m - g‘>>.

(2.3.6)

In view of the bilinearity, we have f(g

+ g‘, g + g’> = f(g, g) + f(& g’) + fk’Y g) + f k ’ , s’),

whence, using the symmetry; F(g

+ g’) = f(g, g) + 2fkY g’) + f k ‘ , g’),

We then obtain (2.3.6) from this and the corresponding result with g‘ replaced by -g‘. We pass to the notion of induced maps. In accordance with (2.2.6), there are two maps (2.3.7) End G 4 End(Bil(G; H)).

26

2

BILINEAR AND MULTILINEAR FUNCTIONS

Explicitly, an endomorphism A of G associates with a bilinear function f ( g , g ’ ) the bilinear functions f ( A g , g’) and f ( g , Ag’). It may happen that these coincide, or that

f(&,

g’) = fk, Ad),

g, g‘ E G.

(2.3.8)

In this case, A is said to be symmetric, with respect to the bilinear function f. The set of such symmetric endomorphisms forms a subspace of End G which includes, for example, the identity map and its scalar multiples. The notions of a symmetric bilinear function, and of an endomorphism symmetric with respect to a bilinear function, should be supplemented by those of skew symmetry. The bilinear function f ( g , g’) is said to be “skew symmetric,” or “alternating,” if

fk,g’) = -fk’,d,

s, g’ E G*

(2.3.9)

These also form a subspace of Bil(C, H ) . An endomorphism A of C is said to be skew symmetric, with respect tof, if m g , g‘) = 2.4

-fk, A&?’),

g, g’

E

G.

(2.3.10)

Bilinear Forms

We make now the additional specialization that the linear space in which the bilinear functions take their values is the field K itself. In this case, the term “form” is used, in place of “function.” Instead of Mult(G, G; K ) or Bil(G; K ) we shall simply write Bil G. Thus Theorem 2.2.1 now asserts that there are two isomorphisms Bil G

+

Hom(G, G*),

(2.4.1)

where G* is the dual space to G, while in place of (2.2.6) or (2.3.7) we have two maps (2.4.2) End G + End Bil G. It is evident from dimensional considerations that the last two maps are not, in general, isomorphisms. However, a similar question does arise. For any bilinear form f ( g , g’), an endomorphism A of G yields the induced bilinear functions f ( A g , g’), f(g, Ag’). Thus for fixed f, varying A , we have two maps (2.4.3) End G + Bil G. If C has the finite dimension n, both these spaces have dimension n2, by Theorems 1.7.2 and 2.1.2. It is reasonable to ask whether the maps (2.4.3) are isomorphisms.

2.5

SESQUILINEAR FUNCTIONS

27

THEOREM 2.4.1. If G is finite dimensional, and f(g, g‘) is nondegenerate in itsfirst argument, the map taking f(g, g’) into f(Ag, g’), A E End G, defines an isomorphism (2.4.3). Since the spaces in (2.4.3) have the same dimension, it is sufficient to prove the map to be monomorphic; we pass over the proof that it is linear. The kernel of the map is given by the set of A E End G such thatf(Ag, g’) = 0 for all g,g’. By the nondegeneracy assumption, this is possible only if Ag = 0 for all g, and so if A = 0. Thus, if, in a finite-dimensional linear space, we select a basic nondegenerate bilinear form, or “inner product,” we can set up a one-to-one correspondence between bilinear forms and endomorphisms. The following notions are relevant to the case that the field is ordered, and in particular that it is the real field. A bilinear formf(g, g’) on the space G is “positive definite” if f k , g ) > 0,

g E G7

g

z

0,

(2.4.4)

and if the associated quadratic form is positive for nonzero arguments. It is immediate that such a form is nondegenerate in both arguments. It is “positive semidefinite” if f(g,g) 2 0, g E G, (2.4.5) The terms “negative definite,” and “negative semidefinite” have the obvious meanings. The form is “indefinite” if the associated quadratic form can take both positive and negative values. 2.5

Sesquilinear Functions

We recall from Section 1.10 that for the complex field, and indeed for any field with a nontrivial involution, one can introduce semilinear or conjugate-linear maps, which differ from linear maps with respect to the homogeneity property. This raises the possibility of considering functions of several variables, which are linear in some of the variables and semilinear in the remainder. A function of two variables, linear in its dependence on one and semilinear in its dependence on the other, is termed “sesquilinear,” or If-linear. Supposing that both arguments lie in the same complex linear space G, and the values in a second such space H , a sesquilinear functionf(g,g‘), g , g’ E C, will be additive in both arguments, so that

28

2

BILINEAR AND MULTILINEAR FUNCTIONS

while for any complex c we have (2.5.3) (2.5.4)

Under the rules specified for Theorem 1.1.1, the set of such functions forms a linear space. It may be denoted by Ses(G; H ) . If the functions are scalar valued, so that H i s the complex field, we term them “forms.” The set of sesquilinear forms in G may be denoted by Ses G. As for bilinear forms, a sesquilinear form f ( g , g’) may be termed “nondegenerate,” in its first argument, if f ( g , g ’ ) = 0 for fixed g and all g‘ implies that g = 0, with a similar definition for the second argument, and simply nondegenerate if this is the case for both arguments. We associate a quadratic form f ( g , g) with every sesquilinear form f ( g , g’), g , g’ E G. As for bilinear forms, there is a class of sesquilinear forms for which this form can be recovered from the quadratic form. In analogy to the notion of a symmetric bilinear form, we term a sesquilinear form f ( g , g‘) “hermitian” if f ( g , g’) = f k ’ g, ) ,

g , g’ E G.

(2.5.5)

The set of such f is closed under addition and under multiplication by real scalars. It thus forms a real linear space, which may be denoted by Herm G. We have 2.5.1. In order that the sesquilinear form f ( g , g’) be hermitian, THEOREM it is necessary and suficient that the associated quadratic form f ( g , g ) be real valued.

The necessity is evident upon taking g‘ = g in (2.5.5). Suppose then that f is sesquilinear, and that f ( g , g) is real valued. Thus, f ( g + cg’, g cg’) is real for every complex c, and so

+

(2.5.6)

and cf(g’, g)

+m

’ , g)*

must also be real. Subtracting, we have that Z{f(g,g’) - f ( g ’ , g ) } is real, and since this is true for all c, it must be zero. Thus we derive (2.5.5), which completes the proof.

2.5

SESQUILINEAR FUNCTIONS

29

A formula permitting the recovery of the values of a hermitian sesquilinear form from the associated quadratic form is

f k ,g’) = S { f k + g ’ , g + g ’ ) - f ( g - g’, g - g ’ ) + if(g + ig‘, g + ig‘) - if(g - ig’, g - i g ‘ ) } ;

(2.5.7)

the proof is similar to that of (2.3.6). We use the term “hermitian quadratic form” for one that is associated with a hermitian sesquilinear form. A hermitian quadratic form f ( g , g ) is termed “positive-definite” if one has f ( s , g ) > 0, g z 0. (2.5.8) The set of such positive-definite forms is closed under addition, and also under multiplication by positive scalars; it forms what we shall refer to as a convex cone. Similarly, a hermitian quadratic form is termed “negative definite” if f(g, g ) < 0 when g # 0, and “indefinite” f ( g , g ) can take both signs. It may be termed “positive semidefinite” if f ( g , g ) is nonnegative in any case, can be positive, and can vanish for nonzero g . A positive-definite hermitian quadratic form has the property that the associated sesquilinear form is nondegenerate. The following important inequality is associated with the names of Cauchy and Schwarz. 2.5.2. Let the hermitian sesquilinear form f be positive dejinite, THEOREM or semideJnite. Then (2.5.9) Ifk,g’)I2 5 f k ,g l f k ’ , g’). We use the fact that, for any complex c, or that

f(g

+ cg’, g + cg’) 2 0,

f ( & g ) + 2 ke{?ff(g,g ’ ) } + IcIzf.(g’,g ’ ) 2 0.

(2.5.10)

Here we put c = te“, where t is real, and a is chosen so that e - ’ ” f ( g , g ’ ) is real and positive. Then (2.5.10) gives

and since this holds for all real t, we obtain (2.5.9).

30

2

BILINEAR AND MULTILINEAR FUNCTIONS

2.6 Sesquilinear Forms and Endomorphisms

Let f ( g , g’) be a sesquilinear form on G, and A an endomorphism of G . In this section, we are concerned with the maps

-+f(& g’), -f(s, A d ) ,

(2.6.1) (2.6.2) which associate with A two “induced” sesquilinear forms on G. For fixed J; these define two maps of the form A

A

(2.6.3) End G -+ Ses G, which may be used either to identify A , or to classify it in respect of some property of the induced forms. On the first point we have THEOREM 2.6.1. Let the complex linear space G be finite dimensional, and let f E Ses G be fixed, and nondegenerate in its first argument. Then (2.6.1) defines an isomorphism (2.6.3). The map (2.6.1) is a linear space map (2.6.3), which is a monomorphism between spaces of the same finite dimension, as in the proof of Theorem 2.4.1. This means that if we adopt as basic some fixed nondegenerate sesquilinear form on a finite-dimensional space, we can identify endomorphisms with associated induced sesquilinear forms. We pass now to the classification of endomorphisms, with respect to a form. The endomorphism A is said to be “hermitian” with respect to the sesquilinear form f if (2.6.4) f(& g’) = fk,A d ) . For fixedf, the set of A that are hermitian with respect to it forms a real linear space. We have THEOREM 2.6.2. Let f be hermitian. Then in order that A be hermitian with respect to it, it is necessary and sufficient that the forms (2.6.4) be hermitian, or that the quadratic form f ( A g , g) be real valued. Suppose first that A is hermitian, so that (2.6.4) holds. We then have that f ( A g , g) = f(g, Ag). Since these numbers are complex conjugates, by the hermitian assumption forf, they must both be real; the forms (2.6.4) are then hermitian, by Theorem 2.5.1. Suppose conversely that f ( A g , g) is real valued, so that the formf(Ag, g’) is hermitian. We thus have

f(4 g‘) ,= f(At?’,g), and then (2.6.4) follows, sincefis hermitian. We now combine Theorems 2.6.1 and 2.6.2. Suppose that G is finite

2.7

THE ZEROS OF HERMITIAN FORMS

31

dimensional, and that f is a nondegenerate hermitian form; it may, for example, be positive definite. The maps (2.6.1) and (2.6.2) coincide on the set of A hermitian with respect tof, and define an isomorphism of this set of A onto the set of hermitian forms on G, considered as a real linear space. Thus, taking as basic a fixed suchf, we can identify hermitian forms and endomorphisms that are hermitian with respect to f. We note some further classifications. An A E End G is positive definite, with respect to f, if the induced form f(Ag, g) is hermitian and positive definite; other definiteness notions may be extended similarly to End G. As an easy consequence of (2.6.4), or Theorem 2.6.2, we have A

THEOREM2.6.3. I f A, B are hermitian with respect to f, then so are real, and AB, ifAB = BA.

+ B, cA i f c is

Only the last assertion calls for proof. One hasf(ABg, g’) = f(BAg, g‘) = f(Ag, Bg’) = f(g, ABg‘), so that the requirement (2.6.4) holds for AB. 2.7 The Zeros of Hermitian Forms

In the remainder of this chapter we treat some less formal aspects of hermitian forms, or endomorphisms, of a complex linear space. Let G be such a space, and f(g, g) a hermitian quadratic form on it. By a “zero” of this form we shall mean an element g E G such that f(g, g) = 0; the term “nontrivial” will exclude the case g = 0. We note first THEOREM 2.7.1. The set of nontrivial zeros of a hermitian quadratic form is polygonally connected. By this we mean, as in Section 1.9, that two distinct nontrivial zeros of f(g, g), say g, , g,, can be joined by a finite number of straight segments, every point of which is a nontrivial zero off. If the space G is finite dimensional, this will imply that the set of such zeros is arcwise connected, in the usual topology. Suppose first that g, , g, are linearly dependent, so that g, = cg,, for some nonzero complex c. Then, as in the proof of Theorem 1.9.1, we can join them by at most two segments, consisting of nonzero multiples of g, , which will necessarily be nontrivial zeros off. Suppose next that g,, g2 are linearly independent zeros off. For some nonzero complex c, to be chosen later, we can join g , to cg, by a segment (1

- t)g,

+ tcg,,

0 I t I 1.

(2.7.1)

If g has the form (2.7.1), then f(g, g) is equal to (1

- t)2f(g,, 81) + r(l - 0 2 Re{cf(g2, g d ) +

t21CIZf(gz,g2).

(2.7.2)

32

2

BILINEAR AND MULTILINEAR FUNCTIONS

Here the first and last terms vanish. We now choose c, not zero, so that cf(g,, 8 , ) has zero real part, and then all of (2.7.2) will vanish. We have then connected g , to cg, by a straight segment, within the set of nontrivial zeros off. By at most two further such segments, we can then connect cg, to g , , and the proof is complete. In a special case we can say more.

THEOREM 2.7.2. The set of zeros of a positive semidefinite hermitian form constitute a subspace. The set in question is obviously closed under multiplication by scalars, and so we have only to show that it is closed under addition. Suppose that g , , g, are two zeros of f ( g , , g ) . By the semidefiniteness, we have f ( g , cg,, g, cg,) 2 0, for any complex c. Expanding, and dropping zero terms, we have 2 Re{cf(g,, g d > 2 0,

+

+

and since c is arbitrary, we must have f ( g , , g l ) = 0, and so f ( g , + g,, g , g,) = 0. In fact, the semidefinite form vanishes identically, as a sesquilinear form, on the subspace of zeros of the quadratic form. In a related topic, we consider the range of values of a hermitian quadratic form f ( g , g ) , when g varies through the set of nonzero elements of the linear space G . We note the following possibilities.

+

2.7.3. The range of values of the hermitian quadratic form THEOREM f ( g , g ) has one of the following forms: (i) (ii) (iii) (iv)

the open positive half-axis, or the open negative half-axis, the closed positive half-axis, or the closed negative haljlaxis, the entire real axis, the point 0 only.

The first possibility describes the case that f is positive definite, or negative definite. If f ( g , g ) takes some positive value, then by replacing g by cg we can make it take any positive value, and so the range will include the open positive half-axis. Case (ii) describes the case that f is positive, or negative, semidefinite. Case (iii) arises when, and only when, f is indefinite. It is immediate that i f f ranges over the whole axis, then it must be indefinite. Conversely, suppose that .f is indefinite, and so f ( g , g ) takes both positive and negative values. Then, since the set of nonzero elements of G is polygonally connected, and sinceJrg, g ) is a continuous function of t along any segment of

2.8 PAIRS OF HERMITIAN FORMS

33

the form (2.6.1), we have that f ( g , g) must have a nontrivial zero, so that zero is included in the range. Furthermore, if it takes any positive and negative values, it must take all such values, and so its range is the real axis. 2.8 Pairs of Hermitian Forms

Later on, it will be important to obtain results similar to those of the last theorem for collections of hermitian forms defined on several complex linear spaces. At this point we take up the next simplest case, that of two hermitian forms f , ( g , g), s = 1,2, on a single space G; one may consider this pair as a single hermitian vector-valued form, taking its values in R2. We need to discuss its range, for nonzero arguments, that is to say the set described by (f&, g>,fz(g, &?I), g E G, g z 0. (2.8.1) We denote this set by U,and have the following general property. 2.8.1. The set U is convex, and closed under multiplication by THEOREM positive scalars. The last remark follows, as previously, on replacing g by cg, when (2.8.1) is multiplied by lcI2. It remains to prove the convexity. We have to show that for any nonzero pair g, ,g, , the 2-vectors (2.8.2) (2.8.3) can be joined by a segment lying in U. The case that g , , g , are linearly dependent is covered by the last remark of the theorem, and so we suppose them linearly independent. Letfbe a nontrival linear combination of f,,f, such that f(g,, 8,) = f(g,, g,). We obtain a segment satisfying the requirements if we make g in (2.8.1) describe the curve (1 - t)"2g1

+ t"%g,,

0 It I1,

(2.8.4)

where c is chosen subject to IcI = 1, so that Ref(cg,, 8,) is zero. The calculations are similar to those of (2.7.1) and (2.7.2); we find that

f(& g) = (1 - t l f k l , 81) + tf(g, gz), Y

(2.8.5)

which is independent of t, and so (2.8.1) describes a segment joining (2.8.2) and (2.8.3).

34

2

BILINEAR AND MULTILINEAR FUNCTIONS

We are concerned with the situation that U does not contain the origin. Thus, if G is not zero, U will consist of a collection of rays from the origin, but not including it, each ray being determined by its intersection with the unit circle. Let C be the intersection of U with this circle. Then C cannot contain two diametrically opposite points, for then by the convexity property U would contain the origin. Also, for any two points of C, U will contain the chord joining them, and so C will contain the shorter of the two arcs joining these points. Thus C is a single arc on the unit circle, possibly a single point, but of length not exceeding n ; furthermore, if its length is n, this arc can contain at most one of its endpoints. From this geometric argument we can draw certain conclusions about pairs of hermitian forms; we take first the simpler and more relevant case that dim G < 00. THEOREM 2.8.2. Let f,(g, g), s = 1, 2, be a pair of hermitian quadratic forms on thejinite-dimensional complex linear space G. Then in order that these forms should not vanish together, except when g = 0, it is necessary and su8cient that some linear combinations of the forms be positive definite. If the forms have a common nontrivial zero, then so will every linear combination of them, and no linear combination can be positive definite. Thus the condition is clearly sufficient. We assume therefore that the forms do not vanish together, when g # 0, or that the set U does not contain the origin. It must therefore consist of a sector, whose intersection with the unit circle is an arc C, either of length less than n, or of length .n and not closed. In the finite-dimensional case, the last eventuality can be ruled out. Introducing a norm in the space G, we consider the range of (2.8.1) as g describes the compact set llgll = 1. The image of this set will then be a closed and bounded set in R2, which does not contain the origin. If this set is projected from the origin into the unit circle, we shall therefore obtain a closed set, which will coincide with C. It then follows from the fact that C has length less than n that there exists a line through the origin such that C , and U, lies strictly to one side of this line. This is equivalent to the assertion of the theorem. In view of the remarks of Section 2.6, this result may equivalently be expressed in terms of hermitian endomorphisms, with respect to some fixed hermitian sesquilinear form. The statement will be that if A , B are two hermitian endomorphisms, on a finite-dimensional space, then some linear combination of A , B is positive definite if and only if the hermitian quadratic forms associated with A , B do not vanish together, other than trivially. For completeness, we quote the result for the general case, when G need not be finite-dimensional.

2.9

35

THREE HERMITIANFORMS

THEOREM 2.8.3. In order that the two tiermitian jormsf,(g, g), not have a common nontrivial zero, it is necessary and suficient that there be a linear combination such that Ckflk,g)

+

@2f&

and a second linear combination such that tken Plfl(g, g)

g ) 2 0,

(2.8.6)

if equality holds here, with g

+ B2f2(g, g) > 0.

# 0,

(2.8.7)

In other words, one linear combination must be at least positive semidefinite, with the second positive definite on the subspace of zeros of the first. The conditions are clearly sufficient; they are violated iff,(g, g), s = I , 2, vanish together with g # 0. For the necessity, we go back to the geometric argument. If the arc C has length less than n, the proof of the last theorem indicates that there is a linear combination of the f, which is positivedefinite; the specification of the second linear combination is then redundant. This is also true if C has length n, but does not contain either of its endpoints. The full form of the conditions is appropriate if C has length n, and contains one of its endpoints; it cannot contain both, in view of the convexity of U and the hypothesis that U does not contain the origin. In this case, there is a line through the origin which has C to one side of it, but not strictly, intersecting it at one point. Any other second line through the origin will then have this intersection strictly to one side of it and this is equivalent to the result.

2.9 Three Hermitian Forms This case also has sufficient special features and relevance to warrant a separate treatment. It is a question of three forms f,k,

$;I,

s = 1, 2, 3,

g E G,

(2.9.1)

where the formsf, are hermitian, and G is a complex linear space; in a special form, this topic arises in the literature under the heading “numerical range” of a linear operator on a hilbert space. Effectively, it also includes the topic of two forms, of which only one is assumed to be hermitian, and the other is resolved into hermitian and “skew-hermitian” parts. We consider (2.9.1) here in a symmetrical manner, without allotting a special status to any one of the forms.

36

2

BILINEAR AND MULTILINEAR FUNCTIONS

It is natural to ask whether the hypothesis that the three forms do not vanish together, except when g = 0, implies that some linear combination of the forms is positive definite. This is trivially the case when dim G = 1, and less trivially when 3 I dim G < co, but can fail when dim G = 2. We shall not prove this now, but will establish a result of a similar character. 2.9.1. Let the range U of the triple of forms (2.9.1) in R3, as g THEOREM ranges through nonzero elements of the finite-dimensional complex linear space G, not include the origin. Let there be at least one nonzero vector p E R3 such that neither of p, - p is in U . Then the forms have a positive-dejinite linear combination, and U is convex. As before, it is evident that if U contains any nonzero vector, it also contains all multiples of that vector, and so consists of a collection of rays. We write p = ( p l , p 2 , p3), and choose two linearly independent sets of real scalars or19 or29 or39 r = 1 , 2, (2.9.2) such that 3

C arsp, = 0,

Then the forms

s= 1

r = 1,2.

(2.9.3) (2.9.4)

cannot vanish together, when g = 0; for if they did, the set (2.9.1), which cannot be zero, would be a nonzero multiple of ( p l , p2 , p3), and this is excluded. By Theorem 2.8.2, there must be a linear combination of the forms (2.9.4) which is positive definite, and so we have a result of the form g # 0.

(2.9.5)

This proves the first statement of the Theorem. At this stage, we comment that the argument has shown that for any nonzero vector p’ E R3 such that neither of p ‘ , - p ‘ is in U,there i s a plane containing p’ such that U lies strictly on one side of this plane, or in one of the two open half-spaces separated by the plane. We now propose to prove that U is the intersection of a family of open half-spaces. Since half-spaces are convex, and the intersection of convex sets is convex, this will show that U is convex, and complete the proof of the theorem. More precisely, we show that if a vector p ‘ is not in U, then there is a closed half-space, containing p’, that does not intersect U. This will show

2.10

GENERAL REMARKS ON THE RANGE OF FORMS

37

that the complement of U is a union of closed half-spaces, so U is an intersection of open half-spaces. Let (xl, x2, x 3 ) denote a typical element of R3. We have shown that the half-space described by

c 3

1

?,X,

(2.9.6)

I0

does not intersect U . Thus, the preceding statement is certainly true if p’ is the zero vector. For other cases, we write p’ = (a,, a 2 , a3).If ~ : t s a5, 0, then p’ lies in the half-space (2.9.6), and the statement is proved in this case. Suppose finally that c:?,a, > 0. Then x;rs(-as) < 0, and so - p ’ # U. Thus if p‘ 4 U, there is a closed half-space not intersecting U and containing p’. This completes the proof. 2.10 General Remarks on the Range of a Set of Forms Suppose now that we have a set of p hermitian forms f,(g,g),

t = 1,

... ,P, g E G ,

(2.10.1)

where G is a complex linear space; we postulate that the forms should not vanish simultaneously when g # 0, and denote by U the subset of RP formed by the range of thep-tuple (2.10.1), for varying nonzero g . As before, we can say that U consists of a collection of rays, being closed under multiplication by positive scalars; it may be described as a “cone,” without a vertex. If G is finite dimensional, we can say in addition that the intersection of U with the unit sphere is closed. The question of whether U need be convex was determined affirmatively in the case p = 2, and in the case p = 3 with a supplementary assumption: U should not contain, for some vector p, both p and - p . While this latter proviso can be removed if dim G > 2, we shall not do this here. In this section, we make two types of observations concerning the convexity of U . One of these is that on dimensional grounds, we cannot expect U to be necessarily convex if p is unrestricted. The second is that we can choose the forms (2.10.1) so that U can assume the form of certain types of prescribed convex sets. Let dim G = n < 00. We need to determine the dimension of the real linear space formed by the set of hermitian quadratic forms on G . If g, , ... ,g , is a basis of G , a hermitian form f ( g , g ) may be specified in terms of the n real numbers f k r , gr),

r = 1,

9

n,

(2.10.2)

38

2

BILINEAR AND MULTILINEAR FUNCTIONS

and the -)n(n - 1) complex numbers f ( g r , g,),

15 r <sI

11.

(2.10.3)

By the hermitian property, the values (2.10.3) determine the values when r > s, and so the sesquilinear form f is determined for all pairs of basis elements, and thus for all elements of G. Since (2.10.3) involve the specification of n(n - 1) real numbers, the total number of real scalars to be specified is nz. These can be specified at will. Thus the set of hermitian forms has dimension nz as a real linear space. It follows that, ifp 5 n2, we can choose the forms (2.10.1) to be linearly independent, and U will not then be contained in any proper subspace of RP.If, in addition, U were to be convex, it would have to contain interior points, as a subset of RP,and so, at some points this range would have local dimension p . On the other hand, consider the dimension of the domain of the map G + U given by (2.10.1). Each element of G, a complex linear space of dimension n, can be specified by means of 2n real parameters. However, (2.10.1) is unaffected if g is replaced by eidgfor any real a, and so the map proceeds in effect from a space with local dimension 2n - 1. These rough arguments suggest that if 2n-1 q ; the rejection of this eventuality will complete the proof. Let APg = 0. By the descent property, we can write A P - ’ g = A’s’, and so have A p + + ’ g= ’ 0. From the ascent property we then deduce that APg’ = 0, so that A p - ’ g = 0. This shows that Ker AP c Ker A p - ’ , so that these two subspaces are equal. This contradicts the definition of p as the ascent. T h u s p = q. The situation that the ascent and descent are both finite, and so equal, certainly obtains if G is finite dimensional. In this case, the subspaces Ker A“, Im A” are likewise of finite dimension, and from (3.1.1) and (3.1.4) we have 0 Idim Ker A Idim Ker A 2 I

dim G 2 dim Im A 2 dim Im A 2 2

Idim G,

2 0.

(3.2.3) (3.2.4)

44

3

DECOMPOSITION OF FINITE-DIMENSIONAL ENDOMORPHISMS

Since these sequences are bounded, they are ultimately constant, and when the dimensions become equal, the corresponding subspaces will coincide. Of particular value here is the following consequence. 3.2.2. Let the endornorpliism A of the linear space G have THEOREM finite, and so equal, ascent and descent p . Then there is a direct sum decomposition (3.2.5) G = Ker A P 4 Im AP ; in particular, this is true if G is finite dimensional. Also, the indices p in (3.2.5) can be replaced by larger numbers.

We must show that for any g E G, a representation g = g,

+ g,,

g , E Ker A P , g , E Im A P ,

(3.2.6)

is both possible and unique. We prove first the uniqueness, or that Ker AP n Im A P = 0.

(3.2.7)

Suppose that g is in both these subspaces, so that APg = 0, g = APg’. It then follows that AZPg’= 0. Since 2p 2 p , we have by the ascent property that APg’ = 0, and so g = 0, as was to be proved. To achieve (3.2.6) for given g , we note that by the descent property we can write APg = AZPg’for some g ’ . We then write g , = APg‘, so that g , E Im A P , and g , = g - g , . We then have so that g 1 E Ker A P .This completes the proof. The decomposition is unaffected by replacing the indices p by larger numbers, since, in view of the definitions of ascent and descent, this does not affect the subspaces in question. The decomposition separates off, so to speak, the singularity of A . We have 3.2.3. Under the assunzptions of Theorem 3.2.2, A induces an THEOREM automorphism of Im A P .

We note first that A maps Im A P onto itself. For any element APg in Im A P can be put in the form A P + ’ g ’ ,by the descent property, and so in the form A(APg‘),and thus lies in the image of A acting on Im A P .We must also show that A acts monomorphically on Im A P . For if A(APg)= 0, we have APg = 0, by the ascent property. This completes the proof.

3.3

EIGENSUBSPACES A N D ROOT SUBSPACES

45

3.3 Eigensubspaces and Root Subspaces We now take a fixed endomorphism A of the linear space G, and apply the previous considerations to the endomorphism A - AI, where I is the identity map of G and 1 E K is at our disposal. We are interested in the A values for which A - AI fails, in some way, to be an isomorphism. It will be sufficient for our purposes to emphasize the case that A - AI fails to be rnonomorphic. We call A an “eigenvalue” if Ker(A - AZ) # 0. The set of g E G such that ( A - AI)g = 0 will be called the “associated eigensubspace.” The ascent of A - AI, which will be at least 1 for an eigenvalue, will be called the “rank” of the eigenvalue 1;it is thus the smallest integer p such that Ker(A

- AI)P = Ker(A -

(3.3.1)

or else is considered as infinite, if this equality does not hold for any p . The set of g E G such that ( A - AZ)llg = 0, for some integer q > 0, which may depend on g, is called the “root subspace” associated with an eigenvalue A. It clearly includes the eigensubspace, this consisting of those g for which we can take q = 1. It is easily seen that it is a subspace. For if ( A - 1Z)llg = 0, then clearly ( A - AZ)4(cg) = 0 for all c E K, while if ( A - AZ)qg = 0, ( A - 11)4’g’ = 0, then ( A - AI)4”(g g’) = 0 if q” = max(q, 4’). If the eigenvalue A has the rank p c co, the root subspace is, of course, Ker(A - AI)p. If we restrict A to a root subspace, we find that it has only the one eigenvalue.

+

THEOREM 3.3.1. Let A be an eigenvalue of A , and H the associated root subspace. If / i # I., the endomorphism A - p I induces an isomorphism of H . We must show that A - p I is both monomorphic and epimorphic. On the first, suppose that g E H , so that ( A - 1.1)4g = 0 for some q, and suppose also that ( A - p1)g = 0. If we write the first of these in the form ((A

-p o f

(p

- 4Z}%

= 0,

(3.3.2)

expand, and use the fact that ( A - pZ)g = 0, we obtain that (p - 1Yg = 0. Thus, g = 0, as was to be proved. Next we show that ( A - pZ) maps H onto itself; we pass over the verification that A does map H at least into itself. We must show that if ( A - 11)4g = 0, then there exist g‘, q’ such that ( A - A1)4’g‘ = 0, g = ( A - PI)&?‘.

46

3

DECOMPOSITION OF FINITE-DIMENSIONAL ENDOMORPHISMS

We now write (3.3.2) in the form

here we have ( A - pZ)r-lg E H , r = 1, ... ,q, and so we have a representation of g in the form ( A - pZ)g', g' E H . This completes the proof. It follows at once that two root subspaces associated with distinct eigenvalues can have only zero intersection. We pass to a more general property for several root subspaces, confining our attention to the case of finite ranks.

3.4 The Splitting Off of Root Subspaces The notion of the rank of an eigenvalue 1 extends to general scalar A, as the ascent of A - 1.Z; this quantity will be zero if A does not happen to be an eigenvalue, and will be positive or infinite otherwise. This definition has a counterpart in the definition of co-rank, the descent of A - AZ; as was shown in Section 3.2, if both rank and co-rank are finite for some 1, then they are equal. The most interesting case is that in which they are finite and positive, so that 1 is an eigenvalue with finite and equal rank and co-rank. In this case, we have by Theorem 3.2.2 a nontrivial direct sum decomposition of the space, in the form G = Ker(A - AZ)p $ Im(A - l.Z)P,

(3.4.1)

where the integer p is equal to or greater than the common value of the ascent and descent of A - 1Z. Both these spaces on the right of (3.4.1) are invariant under A , or, more precisely, are mapped into themselves by A. This result extends to the situation where we have several such eigenvalues. We have

THEOREM 3.4. Let A,, ... , Am be distinct eigencaliies of the endomorphism A of the linear space G , having $finite ranks and co-ranks q1, ... ,qm, respectively, Write

B,

=

( A - A.rZ)q',

r

=

1 , ... , m .

(3.4.2)

Then we have a direct sum decomposition G

m

= 1

Ker B,

4 Im(B, ... Bm).

(3.4.3)

We prove this by induction over m. If m = 1, the result is simply (3.4.1).

3.5

THE FINITE-DIMENSIONAL CASE

41

We suppose therefore that for some t, 1 I t < m, G =

t

1

Ker B,

4

Im(B,

(3.4.4)

B,),

+

and aim to deduce the corresponding result with t 1 for t . For this we consider the endomorphism of the last summand in (3.4.4), which is induced by A - A,+,Z. This operator induces an automorphism of the subspaces Ker B,, ... , Ker B,, by Theorem 3.3.1, and so its ascent and descent on the remaining subspace in (3.4.4) will be the same as on the whole space G. Thus on applying (3.4.1) to the subspace Im(B, ... B,), with A = A,+, and p = q t + l ,we obtain Im(Bl ... B,) = Kert B,+l4 Imt B,+, ;

(3.4.5)

here the dagger t signifies that Ker and Im are formed with B,+, having its B,). domain restricted to Im(B, From Theorem 1.4.1 it now follows that G = Z K e r B, 1

4

KertB,+,

Thus in order to realize (3.4.4) with t and that

p ImtB,+, .

(3.4.6)

+ 1 in place of t we must show that

Kert B,,, = Ker B r + , , Imt B,+, = Im(B,

... B,+,).

(3.4.7) (3.4.8)

For the first of these we must show that if B,+,g = 0, g E G , then B,). This will follow from Theorem 3.3.1, which shows that the g E Jm(B, operators A-].,I, r = 1,..., t (3.4.9) all induce automorphisms of the root subspace associated with A,+, ; the same is therefore true of any product of powers of (3.4.9). Thus B1 B, has is, it induces an automorphism of Ker B , , , . Thus the property-that Ker B , + , c Im(B, B,), which proves (3.4.7). Finally, we note that (3.4.8) is trivial. The left is the range of B,+, as Bg,g E G ; since the applied to the collection of elements of the form B, B, commute, this is the same as the right of (3.4.8). This completes the proof of Theorem 3.4.1. 3.5 The Finite-Dimensional Case

If the space G on which the endomorphism A acts is finite dimensional we have automatically that the eigenvalues, if any, are of finite ranks, with

48

3

DECOMPOSITION OF FINITE-DIMENSIONAL ENDOMORPHISMS

equal finite co-ranks, so that the decomposition theorem of the last section is certainly available. To round out this statement, we must discuss the existence of eigenvalues. In general, for an endomorphism of an infinitedimensional space, we cannot assert the existence of an eigenvalue. With a supplementary assumption for the finite-dimensional case we have THEOREM 3.5.1. Let the linear space G ocer the$eld Kbe finite dimensional, and let K be algebraically closed. Then an endomorphism A of G has at least one eigenvalue. We exclude here the trivial case that G is zero. We shall follow the dimensional proof of this result. Another proof would use determinants, identifying the eigenvalues with the roots of an algebraic equation det(A - AZ) = 0, the “characteristic equation.” The proof considers the map that associates with formal polynomials n

f(t>= C Crt‘, 0 the endomorphism-valued functions f(A) =

cr E K,

f crAr 0

(3.5.1)

(3.5.2)

We ask whether we can find n 2 1 and coy... , cn E K, not all zero, such that

f ( A ) = 0.

(3.5.3)

In the first stage of the proof, we argue that this is possible if n 2 (dim G)2. To see this, we consider the transition from (3.5.1) to (3.5.2) as a homomorphism of linear spaces. The first of these, involving n + 1 coefficients c, has dimension n + 1; here the linear operations on polynomials are the obvious, while a polynomial is considered as zero if and only if all its coefficients are zero. In this map from (3.5.1) to (3.5.2), the range space is End G, which by Theorem 1.7.2 has dimension dim G * dim G. Thus if n 2 (dim G)’, the map from (3.5.1) to (3.5.2) goes from a space to one of lower dimension and so, by Theorem 1.6.2, has nonzero kernel. This means that there exist co , ... , c, , not all zero, such that (3.5.3) holds. We may thus suppose that (3.5.2) and (3.5.3) hold nontrivially, and without loss of generality may take it that c, # 0. We now appeal to the algebraic closure of the field K, to the effect that (3.5.1) can be factorized in the form n

j(t) = cn fl (t - as), 1

(3.5.4)

3.5 THE FINITE-DIMENSIONAL CASE

49

for various a, E K, not necessarily all distinct. It then follows that

n ( A - uJ), n

( A ) = cn

(3 5 5 )

1

and since f(A) = 0, the factors A - a,l, s = 1, ... ,n, cannot all be automorphisms. Thus at least one of them must have nonzero kernel, and so at least one of the as must be an eigenvalue. We use this result to complete the discussion of the decomposition (3.4.3) in the finite-dimensional case.

THEOREM 3.5.2. Let the distinct eigenvalues of A , an endomorphism of a finite-dimensional space G over an algebraically closed jield K, be A, ,...,A,, and let their ranks be q l , . .. ,4,. Then there holds the direct sum decomposition G =

m 1

Ker(A - Arf)qr.

(3.5.6)

The proof is a matter of showing that the last subspace in (3.4.3) is zero. If not, A as restricted to this subspace would have an eigenvalue, say A,,, and the subspace would contain some nonzero element of Ker(A - A&. However, A,, would have to be one of the set A,, ... ,A,,,,say A,, and so Ker(A - 1,Z) would be contained in Ker B,. This conflicts with the directness of (3.4.3), and completes the proof. We now clarify the question of what polynomials (3.5.1) have the property that the corresponding operator-valued polynomials (3.5.2) are the zero endomorphism.

THEOREM 3.5.3. With the assumptions and notation of the last theorem, in order that (3.5.3) should hold, it is necessary and suficient that f ( 5 ) should be divisible by

fi (5 -

2r)q-e

1

(3.5.7)

Suppose that f(5) is divisible by (3.5.7). Passing over the case in which f ( 5 ) = 0, we note that its expression in the form (3.5.4) will contain the factors (3.5.7) to the appropriate multiplicities or higher, so that f ( A ) can be expressed as a product of the form (3.5.9, containing the factors (A

- L rZ)qr,

r = 1, ... ,m.

(3.5.8)

Since all the factors commute, and since each of (3.5.8) annihilates one of the summands in (3.5.6), we have that f ( A ) annihilates G, and is zero.

50

3

DECOMPOSITION OF FINITE-DIMENSIONAL ENDOMORPHISMS

Conversely, suppose that f ( A ) = 0. Once again passing over the case thatf(() = 0 (when the result is trivial), we supposef(A) factorized in the form (3.5.5), and consider its action on the summand in (3.5.6) corresponding to A,, which must, of course, be zero. By Theorem 3.3.1, all factors A - a,Z will act as automorphisms of Ker(A - A,Z)+, except those for which a, = A r . Thus this subspace must be annihilated by the factors in (3.5.5) for which a, = A r . By the definition of rank, there must be at least qr of these. This completes the proof. The polynomial (3.5.7) is the “minimal polynomial” of the endomorphism A . It generates the (principal) ideal in the algebra of polynomials in 5 that have the property that the corresponding polynomialsf(A) vanish.

3.6 Several Commuting Operators Let A , , ... , ‘4, be endomorphisms of a linear space G , such that A,A,

=

A,Ar,

r , s = 1,

... ,k.

(3.6. I )

For such a set we.have the notion of a simultaneous eigenvalue, a set of scalars a , , ... , ak such that the endomorphisms A, - @,I, r = 1, ... ,k , are not only all singular, but annihilate some common nonzero element; that is, we require that

n Ker(A, - a,Z) # 0. k

r= 1

(3.6.2)

If 0 < dim G < 00, and the field K is algebraically closed, simultaneous eigenvalues certainly exist, and all of them can be found in the following manner. We choose any eigenvalue a , of A , . We then note that A2 induces an endomorphism of Ker(A, - ~ ~ For 1 ) if. A , g = x,g, then A 2 A , g = a1A2g, and so, by (3.6.1), A I A L g= a , A 2 g , so that A2g E Ker(A, - %,I). We then choose a2 to be an eigenvalue of A2 acting as an endomorphism of Ker(A, - %,I), and then have the situation (3.6.2) with k = 2. If k > 2, we continue the process. We have that A3 induces an endomorphism of Ker(A, - % , I ) , Ker(A2 - cxzZ), and so, therefore, of their intersection. We then take a j to be an eigenvalue of A 3 , acting as an endomorphism of this intersection, and so on. We go on to extend to this situation such ideas as “eigensubspace,” “root subspace,” and “rank.” The eigensubspace can be defined to be that appearing on the left of (3.6.2)-the set of elements annihilated by all the A , - zrI. The possibly larger root subspace can be defined to be the set of

3.7

51

THE HERMITIAN CASE

elements annihilated by some power of each of A , - CCJ;in other words, it will be the union of the intersections

n Ker(Ar - IX,~)", k

r=l

n

=

1, 2,

... .

(3.6.3)

If this nondecreasing sequence of subspaces is constant from some I I onward, this value may be taken to be the rank of the simultaneous eigenvalue a ] , ... aL. The rank will certainly be defined if the space G is finite dimensional. It is possible to introduce a more elaborate notion of rank, in which the powers of the operators in (3.6.3) vary with r, but we are mainly concerned with hermitian situations, in which the rank is 1, so this would have little relevance for us. By a simple extension of Theorem 3.5.2, we have a decomposition theorem for the finite dimensional case. THEOREM 3.6.1. Let 0 < dim G < co,and let thejield K be algebraically closed. Let the endotnorphisins A l , ,.. , A , of G commute. Then tliere holds a direct sim decomposition of G itito the root siibspaces. Here we are to take all the simultaneous eigenvalues, and for each to form the associated root subspace; two simultaneous eigenvalues, k-tuples of scalars, are considered distinct if they differ in at least one pair of corresponding entries. It will perhaps suffice to describe the proof. Using the endomorphism A ] , we take the direct sum decomposition of G whose existence is assured by Theorem 3.5.2, which expresses G as a direct sum of root subspaces of A . For each of these root subspaces of Al, we take the endomorphism thereof induced by A , , and decompose the root subspace of A l into root subspaces of A , . The resulting subspaces at this stage will have the form Ker(A, - z1l)"'n Ker(A2 - a21)nz ; here we are at liberty to take i i , , i i 2 equal if we make them sufficiently large, for example, as large as dim G. The process is continued by decomposing these subspaces into root subspaces with respect to A , , and so on.

3.7 The Hermitian Case Let now A be an endomorphism of a complex linear space H. We will take the case that .4 is "hermitian," by which we shall mean that there is defined on H a positive-definite sesquilinear form or inner product ( , ),

52

3

DECOMPOSITION OF FINITE-DIMENSIONAL ENDOMORPHISMS

such that the induced quadratic form ( A x , x ) , x E H , is real-valued. This specialization results in considerable simplification. We have

THEOREM 3.7.1. Oor 1.

With the preceding assumptions, the ascent of A is either

If Ker A = 0 we have the first case. To complete the proof we show that Ker A = Ker A’ in any case. Thus we must show that A Z h = 0, h E H implies that Alt = 0. Since A2/t = 0, we have (A2/?,It) = 0, and so by the hermitian property (Theorem 2.8.1), (Ah, Ah) = 0. Thus Ah = 0, since the product ( , ) is positive definite. Next we have

THEOREM 3.7.2.

If A is hermitian, its eigenvalues are realand have rank 1 .

Let I be an eigenvalue, and suppose that Ah = Ah, h # 0. Then (Ah, h) = I(h, h). Here (Ah, / I ) is real, while (It, / I ) is real and positive, and so I must be real, as was to be proved. The rank of R will be the ascent of A - 21. The latter is hermitian, since I is real, and so has ascent that is 0 or 1, by the last result. Since A - 21 has nonzero kernel, R being an eigenvalue, the ascent must be 1. This means that in the hermitian case, the notion of a root subspace coincides with that of an eigensubspace. We can take advantage of this to simplify statements such as those of Theorem 3.5.2 regarding direct sum decompositions.

THEOREM 3.7.3. Let A be a hermitian endomorphism of a finite-dimensional complex linear space H witlt positive-definite inner product ( , ). Let the distinct eigenvalues of A be I , , ... , A,,,. Then there holds the direct sum decomposition in

H

=

1Ker(A - A,[) .

r=1

(3.7.1)

This results directly from (3.5.6), on noting that all the ranks qr are unity. A corresponding simplification applies of course to the case of several commuting endomorphisms A , , ... , A , of a complex linear space H, which are hermitian with respect to a positive-definite inner product ( , ) in H . Defining the notion of a simultaneous eigenvalue as in Section 3.6, we can conclude at once that these are real, in the sense that all k entries of each such eigenvalue are real, that the rank is 1 in each case, and that the root

3.8

53

ORTHOGONALITY

subspaces coincide with the eigensubspaces. Thus from Theorem 3.6.1 we have THEOREM 3.7.4. Let A l , ... ,A, be commuting hermitian endomorphisms of the jinite-dimensional nonzero complex linear space H. Then there holds a direct sum decomposition of H into the eigensubspaces.

In view of its importance we formulate this result more specifically. Let the distinct simultaneous eigenvalues be A(P)

, ... ,A?),

p = 1,

... , m ,

(3.7.2)

where two such eigenvalues are considered distinct if they differ in at least one entry. The eigensubspaces are then H

n Ker(A, k

=

I=

(3.7.3)

1

the simultaneous eigenvalues (3.7.2) being characterized by the nonvanishing of the right of (3.7.3). The result then asserts that we have the direct sum decomposition H =

m

p=

1

(3.7.4)

H(p).

3.8 Orthogonality A further special feature of the hermitian case is that the direct sum decompositions of the last section can be expressed more explicitly in terms of orthogonal or orthonormal bases. This is based on the orthogonality properties of eigenvectors, which we prove in THEOREM 3.8.1. Let A be a hermitian endomorphism of a complex linear space H with positive-dejinite inner product ( , ). Then elements h, h‘ of the eigensubspaces associated with distinct eigencalues R, A’ are orthogonal, in the sense that (h, h’) = 0.

The assumption is that Ah = Ah, Ah’ = A’h’, 3. # A’. Upon taking inner products with A’, h we obtain (Ah, 12‘) = A(h, h’),

(Ah’, 12)

=

I’(h‘, 12).

(3.8.1)

Upon taking complex conjugates in the second of these we obtain (h, Ah’) = A’(h, h’), where we have used the fact that I‘ must be real, by Theorem 3.7.2. Using the hermitian property of A, we can replace this by (Ah, 12’) =

54

3

DECOMPOSITION OF FINITE-DIMENSIONAL ENDOMORPHISMS

i’(h, h‘). We compare this with the first of (3.8.1), and deduce that (h, h ’ ) = 0, since i# %’. This automatic orthogonality of eigenvectors associated with distinct eigenvalues may be supplemented by specially constructed orthogonalities for eigenvectors associated with the same eigenvalue so as to produce a convenient explicit version of Theorem 3.7.3. With each eigenvalue Ar of A , we associate a “multiplicity” p, , the dimension of the eigensubspace Ker(A - p J ) ; if p , = 1, the eigenvalue 1, may be said to be “simple.” For each r , 1 5 r 5 m, we choose for Ker(A - i r Z ) a basis of pr vectors

(3.8.2) which are mutually orthogonal. Then the entire set h11 ... hlpl, .“ , 1zm1 9

9

9

3

hmp,,, 9

(3.8.3)

will be mutually orthogonal-by construction if a pair of vectors is associated with the same eigenvalue, and by Theorem 3.8.1 otherwise. Under the conditions of Theorem 3.7.3, an arbitrary element I7 E H can be expressed as a sum of elements taken one from each of the spaces Ker(A - AJ), and these in turn can be expressed in terms of the basis elements (3.8.2). Thus for any h E H we have an expression r = l s=l

(3.8.4)

where, in virtue of the orthogonality, the coefficients c,, are given by

On combining these we obtain (3.8.6) The result can be simplified in various ways. We can normalize, the It,,, applying to them scalar factors so as to arrange (hrs,hrS) = 1. Another procedure is to number the eigenvalues according to their multiplicity, without requiring them to be distinct; in the set of eigenvalues, each eigenvalue can appear repeatedly, according to its multiplicity. In this way, (3.8.4) and (3.8.6) can be written as simple summations. If we do this we obtain

THEOREM 3.8.2. Let A be an endomorphism of a complex linear space H , of finite positive dimension n, with positive-definite hermitian inner product

3.9

55

SOME MODIFICATIONS

( , ), with respect to which A is herinitiun. Tlien tliere is a set of eigenvalues

i., and eigenvectors

ti,,

r

=

1,

( A - ArI)lir (11, , I i J Every element

ti E

... , i t , such that = 0, r = 1, ... ,n, =

(r # s),

0

>O,

(3.8.7)

(r

=

s) .

(3.8.8)

H adinits a unique expression (3.8.9)

Here, of course, we have departed from the previous notation, in that the

I., do not denote the distinct eigenvalues, unless all eigenvalues are simple; each eigenvalue appears in the set I,, , ... , A,, a number of times corre-

sponding to its multiplicity. Similar remarks apply to the case of several commuting hermitian endoniorphisnis. We obtain

THEOREM 3.8.3. Let H he as in Tlteorcm 3.8.2 and let A l , ... , ‘4, be conitnwtiiig hesniitiurt endoniorpliisnu of H . Tlieii tliese is a set of siniiiltaneous eigencaIiies I.?), ... , A f ’ an(/ eigcncec‘toss / I , , r = I , . .. ,11, such that ( A , - l.F)l)hr= 0,

s = 1,

... , k ,

r

= 1,

... , 11,

(3.8.10)

aid siich that (3.5.5) and (3.8.9) tiold. Here likewise we have departed from the notation of (3.7.2), in which (3.7.2) denoted the distinct eigenvalues. In the result just formulated, eigenvalues are repeated according to multiplicity, this multiplicity being the dimension of the eigensubspace (3.7.3).

3.9 Some Modifications Sometimes an eigenvalue problem arises not in the form that A - I I is to be singular, but rather in the form that A - 3,B is to be singular, where A, B are fixed linear operators from a space G to a space H . Then, much as before, an eigenvalue can be defined as a scalar I for which Ker(A - 1.B) # 0. One sees at once that if B is an isomorphism, this is the same as requiring that Ker(B-’A - I I ) # 0. Thus A values for which A - %B has nonzero kernel are simply eigenvalues of B-’A in the sense introduced previously: here B”A is an endomorphism of G. The same 1. values will also be the eigenvalues of AB-’, an endomorphism of H .

56

3

DECOMPOSITION OF FINITE-DIMENSIONAL ENDOMORPHISMS

In particular, suppose that H is a finite-dimensional complex linear space with hermitian inner product ( , ), and that A , B are hermitian endomorphisms of H , and that furthermore, B is positive-definite. We are concerned with eigenvalues, in the sense of scalars 1 such that A - 1 B is singular, and with eigensubspaces, in the sense of the kernels of the A - 1B. We can first note that the eigenvalues are real. This can be seen either by repeating the argument of Theorem 3.7.2, with minor adaptations, or by transforming the situation to the standard hermitian one. In one procedure, we note as before that we are concerned with eigenvalues of B - 'A , where B-' certainly exists, since we are asking that B > 0. We then note that B - ' A is hermitian in a suitable sense. We introduce in the space H the new inner product, associating with each pair x, y E H the scalar product (3.9.1)

( ( x , Y ) ) = ( B x ,Y ) = ( x , BY).

To see that B - ' A is hermitian with respect to this product we observe that ( ( B - 'A X , x)) = (B(B-'AX), X ) = (AX,

X)

(3.9.2)

is real-valued. By our assumption concerning B, we also note that the product (3.9.1) is positive definite, so it can serve in place of the original product ( , ) for the purposes of Theorems 3.7.1, 3.7.2, and 3.8.1. The eigenvectors of B - l A , that is, elements h E H such that ( B - ' A - 1Z)h = 0 for some 1,will also be characterized by ( A - 1B)h = 0. When associated with different eigenvalues, they will be mutually orthogonal not with respect to the original product ( , ), but with respect to the product (3.9.1), since it is with respect to this product that B - 'A is hermitian. For ease of reference, we formulate the resulting modifications explicitly, as they affect the completeness of eigenvectors. 3.9.1. Let H be a finite-dimensional complex linear space with THEOREM sesquilinear form ( , ), and A , B two endomorphisms of H of which both are hermitian with respect to the product ( , ), and B is positive definite. Then there is a set of eigenvalues A, and eigenvectors h, , r = 1 , ... , n, where n = dim H, such that r = 1 , ... , n , (3.9.3) ( A - 1,B)k, = 0, (Bh,, h,) = 0 ,

( r # s), > O

(r = s).

(3.9.4)

Every h E H admits the representation (3.9.5)

This comes immediately from Theorem 3.8.2.

3.10

REDUCTION OF PAIRS OF HERMITIAN FORMS

57

The corresponding result for the case of several commuting endomorphisms becomes

THEOREM 3.9.2. Let the space Hand the endomorphism B be as in Theorem 3.9.1, and let A , , ... , Ak be a set of endomorphisms of H, all hermitian with respect to ( , ), such that A,B-’A, = A,B-lA,,

1 5 r, s 5 k .

Then there is a set of simultaneous eigenvalues h,, r = 1, ... ,n, such that (A,

- Ag“ B)h, = 0,

(3.9.6)

nf), ... , ,If) and eigenvectors

s = 1, ... , k , r = 1 ,

... ,n,

(3.9.7)

and such that (3.9.4) and (3.9.5) hold.

We need only remark that the condition (3.9.6) is equivalent to the condition that the endomorphisms B-’A,,

s = 1, ... , k,

(3.9.8)

all commute. We can then apply Theorem 3.8.3, using, of course, the inner product (3.9.1). 3.10 Reduction of Pairs of Hermitian Forms

In the last three sections we operated with a complex linear space that was endowed with a basic sesquilinear form or product ( , ), and we considered the reduction of endomorphisms that were hermitian with respect to this product or perhaps positive definite; here the terms “hermitian” and “positive definite” refer to the behavior of the sesquilinear forms that these endomorphisms induce by means of the given inner product ( , ). There is an alternative formulation in which we work in terms of pairs of forms, and only secondarily in terms of endomorphisms. Let cp, # be a pair of hermitian forms on the complex linear space H. For simplicity, we take the, case in which one of them, I) say, is positive definite. We can then define an eigenvalue as a scalar I such that for some nonzero h E H we have d h ,g) -

4w,g) = 0,

(3.10.1)

for all g E H. Here h is naturally called an “eigenvector.” To put the matter another way, an eigenvector h has the property that the range of the pair (cp(h,g), $(h, g)), as g varies in H, is one dimensional. On putting g = h in (3.10.1) one has immediately that the eigenvalues are real. It is easily seen

58

3

DECOMPOSITION OF FINITE-DIMENSIONAL ENDOMORPHISMS

that eigenvectors h, h' associated with distinct eigenvalues 1,1'are orthogonal in the sense that cp(h, h') = 0, I)(h, h') = 0. One can develop the theory of pairs of hermitian forms either independently of that of hermitian endomorphisms or by reduction to it. To achieve the latter, in the finite-dimensional case, we note there is an endomorphism A of H such that

in view of our assumption that I) is positive definite. Since we assume cp to be hermitian, so that cp(g,g) is real valued, it follows that I)(Ag,g) is real valued, so that A is hermitian with respect to the inner product I). Furthermore, eigenvalues and eigenvectors in the sense (3.10.1) are precisely eigenvalues and eigenvectors of A in the usual sense. In this way we obtain, from Theorem 3.8.2,

THEOREM 3.10.1. Let H be a complex linear space of Jinite positive dimension n, and cp, I// a pair of hermitian forms on H , of which $ is positive deJinite. Then there is a set of n real eigenvalues Al , ... , A, and nonzero eigenvectors h, , .,. , h, sucli that cp(hr,g ) - Ar$(hr, g )

=

0,

g E H , r = 1,

... , n,

(3.10.3)

which are orthogonal in the sense cp(h,, 11,) = $(hr, 11,) = 0,

r # s.

(3.10.4)

For arbitrary g , ft E H we have

(3.10.5)

Here the orthogonality cp(hr, 11,) = 0 is the same as (3.8.8), with t,h playing the role of ( , ); the remaining case cp(h,, h,) = 0 of (3.10.4) follows from (3.10.3). In (3.10.5) we have taken the opportunity to formulate the Parseval equality. To obtain it, we use the expansions

of the arbitrary vectors g , h, substitute in I)(g, / I ) , and use the orthogonality (3.10.4).

NOTES FOR CHAPTER

3

59

Notes for Chapter 3 For a more general treatment of the topic of ascent and descent see Taylor (1966). The decomposition of a finite-dimensional space into the rootsubspace of an endomorphism, or “generalized eigensubspaces,” is treated in almost every text on linear algebra; see, for example, Greub (1967, Chapter 13).

CHAPTER 4

THE TENSOR PRODUCT OF LINEAR SPACES

4.1 Introduction

It will be an essential part of our treatment of simultaneous eigenvalue problems posed in separate spaces, that we can transpose the problems so as to act in the same space, namely their tensor product. Apart from this application which is our special concern here, the tensor product operation, which we denote by 0, forms along with 0 and Hom one of the more basic operations on linear spaces. It is useful to compare it with these two operations. The operations 0 and 0 have certain affinities with those of addition and multiplication, as the notation, of course, suggests; for example, in their effect on dimension, these operations do indeed behave in this way (Theorems 1.7.1 and 4.3.2). However, 0 is more recondite than 0 . For example, in the case of the direct sum of two spaces, we could form a linear space out of formal sums, or really juxtapositions, of elements taken one from each space; in the case of the tensor product, on the other hand, the formal products of vectors, taken one from each space, do not generally form a linear space, but rather generate a linear space. Another aspect of the comparison between 0 and 0 is that the topology of the spaces may enter into the composition of the tensor product space. 60

4.2

THE DEFINITION BY MEANS OF FUNCTIONALS

61

In the case of the direct sum, the topologies of the summand spaces may be used to define a topology on the direct sum space, but do not affect the stock of elements in the space. In the general theory of the tensor product, it can become necessary to enlarge the space generated by products, and to complete it with respect to some topology. Here there is a contrast with Hom, for which one often restricts the crude set of all linear maps between two given spaces to the subset satisfying some topological restriction, such as continuity. However, this aspect will not in fact concern us, either for 0 or for Hom, since we are mainly concerned with finite-dimensional spaces. A further feature of the tensor product is that there are several ways of defining it; in the present finite-dimensional cases, these definitions all lead to essentially the same result. Each definition represents some important property of the tensor product and so, whichever definition we adopt as primary, we must work around to the others. As definitions we have first the algebraic definition, in which we form the linear space generated by formal products of vectors from the spaces in question, subject to equivalence relations, which amount to imposing distributivity on the product. A second definition proceeds in terms of bases ; here it is necessary to indicate how the expressions for elements of the space transform under charge of basis, as in the classical theory of tensors. A third definition, which we shall use here, proceeds in terms of multilinear functions; this, while perhaps open to the criticism of being rather indirect, is nevertheless both concise and intrinsic. 4.2 The Definition by Means of Functionals

We give a direct definition of the tensor product of several linear spaces; we will assume the spaces finite dimensional, though much that we say will retain sense more generally. Let G I , ... , Gk be linear spaces, over the same field K, such that Of is a functional of arguments

h, E G,*,

given by (A:f)(hl

9

... , j r , ...

s

3

hk)

# r, j , E H,*, =,f(hl

* * *

,jrAr,

(4.6.5)

...

9

hk);

(4.6.6)

here j,A, denotes the image of j , under (4.6.3). It is easily checked that this defines a multilinear function, and that the map (4.6.2) so defined is linear. In the second method of defining (4.6.2) we prescribe that A:(g, 8

'**

8 gk)

= gl

8

'*'

@ A,gr 8

"'

8 gk-

(4.6.7)

This map of decomposable tensors is then to be extended to the whole of G1 8 8 Gk by linearity. One can check that the two methods agree. By means of either definition, we have an important result which, somewhat inexactly phased, is THEOREM 4.6.1.

The induced homomorphisms (4.6.2) commute.

For example, if r # s, the effect of ArtA: on a decomposable tensor g , 8 ... 8 gk is to apply A , to the factor g , , and A, to the factor g, ; it is clearly immaterial in which order we perform these operations. Likewise, if we work in terms of (4.6.4) and (4.6.6), it is a question of transforming the rth and the sth arguments off in (4.6.4) by means of A, and A,, acting dually, and again this may be done in either order to give the same result. The above result, that A:A> = AStA:, r # s, needs the supplementary comment that the actions of A,, A , on the two sides of this equation are different; A , , for example, acts in either case by way of A , acting on the rth factor in the tensor product, but the remaining factors in the tensor product may not coincide. To clarify the matter we exhibit the domains and ranges explicitly in the case k = 2. The statement is that there is a commutative diagram

71

4.7 EXACTNESS PROPERTIES

In the above, iterates of Art are not defined, since this is the case also for A , . However if all the A , are endomorphisms, the spaces H, being identified with the G,, respectively, then the A: all become endomorphisms of the tensor product G, and form a set of commuting endomorphisms. 4.7 Exactness Properties

We must now ask how properties of operators between linear spaces are reflected in the properties of induced operators in tensor products. At the moment is simply a matter of relating kernels and ranges in the two cases. We continue to assume that we are dealing with finite-dimensional spaces, and have first

THEOREM 4.7.1. In order that the induced operator A: in (4.6.2) should be a monomorphism, an epimorphism, or an isomorphism, it is necessary and suficient that this be the case for A , . To simplify matters, we present the proof in the case that r = 1 . If A l annihilates some nonzero g,, E G1,then, by (4.6.7), A l t annihilates any product g,, 0 g, 0 ... 0 gk. Assuming that none of the spaces is zero, we can choose nonzero elements g 2 , ... ,g k , and then g,, 0 g , 0 ... 0 gk will not be zero, by Theorem 4.3.3, and will be annihilated by Alt. Thus if A , has nonzero kernel, so does.Alt. In other words, for A l t to be monomorphic it is necessary that A , be monomorphic. For sufficiency, so far as monomorphisms are concerned, we suppose that Al is monomorphic, and use the notation (4.2.8) for bases in the G,. Then the elements Alglt,,

tl

=

‘5

”’

Y

(4.7.1)

n17

are linearly independent and, by Theorem 4.3.3, so are Algl,,

0 g212@

0 gk,,,

f, = 1,

... , n,, r

=

1,

... , k.

(4.7.2)

72

4

THE TENSOR PRODUCT OF LINEAR SPACES

Thus the image of Alt acting on G , 0 0 Gk has the same dimension as the latter space and so, by Theorem 1.6.1, Al has zero kernel. This completes our discussion of whether A , , A,+ are monomorphic together. Next suppose that A , is epimorphic. Then the elements (4.7.1) generate H I , and either they, or some subset of them, will form a linearly independent basis of Hl. Thus either (4.7.2), or a subset thereof, will form a basis of Hl 0 G2 0 0 G,, and so A l t will be onto. If again A , is not onto, there will be some nonzero j , E HI* such that j l A l = 0. Then the multilinear functionals, which are in the image of AIt, in the sense of (4.6.6), will assign the value zero to every set of arguments j l ,h2, ... ,h k , where h, E Gr*, r = 2, ... , k . However, the space Hl 0 G2 0 0 Gk contains multilinear functionals that do not have this 0 gk, special property; for example, the decomposable element f,0 g , 0 where f l E H I , g2 E G, , ... ,g, E Gk, will not have this property, and so will not be in the range of A,+, if j , fl # 0, and g, # 0, r = 2, ... ,k . Thus, for Alt to be a monomorphism or an epimorphism, it is necessary and sufficient that this be the case for A , . The case of an isomorphism follows on combination of these two cases. Concerning kernels we need a more precise result. . - a

4.7.2. The set of elements annihilated by all the induced THEOREM operators (4.6.2) is the tensor product of the kernels of the operators (4.6.1). The set in question is described by It is clear that this will be so if g = g, 0

**.

0 g,,

(4.7.4)

where the g, E G, are such that A,g, = 0,

r

= 1,

... ,k .

(4.7.5)

Furthermore, (4.7.3) will hold if g is any linear combination of decomposable elements of this form. This shows that k

(IKer A:

r= 1

3

Ker A , 0

0 Ker A , .

(4.7.6)

We must show that equality holds here, which may be done by showing that the reverse inclusion also holds. To see this, we suppose bases (4.2.8) of the G, chosen so that the first m, elements constitute a basis of Ker A , . We take it that m, 2 1, 1 5 r 5 k, since

4.8 UNIVERSAL PROPERTY

73

otherwise both sides of (4.7.6) are zero, and the result is trivial. If n, > m,, the elements Argrt,, mr tr Inr, (4.7.7) will be linearly independent. Suppose now that g is represented in the form (4.3.9) as a sum of tensor products of basis elements, and that it is annihilated by all the A:. If we apply A: to g , and so apply A , to the factor appearing in the rth place in the products of basis elements, we obtain a linear combination of tensor products of elements taken one from each of the sets gsls,

t,=1,

..., n,, s = l , ..., k, s # r ,

(4.7.8)

along with one of (4.7.7); here we have taken account of the fact that A,g,,, = 0, t, = 1, ... , m,. Since the sets (4.4.7) and (4.4.8) are linearly independent, the same is true of their tensor products, by Theorem 4.3.1, and so a vanishing linear combination of them must have zero coefficients. This implies that in the expression (4.3.9) for g , terms for which m, < t, In, must have zero coefficient. Hence g must be a linear combination of tensor products formed from the sets (4.2.8), restricted to g,,,, 1 It, Im,, r = 1, ... , k, and so to Ker A,. This completes the proof. 4.8

Universal Property

We have chosen here to define the tensor product of linear spaces in terms of multilinear functionals on the duals of these space. The tensor product is however also connected with the topic of multilinear functions on the spaces themselves, rather than on their duals. For finite-dimensional linear spaces G1, ... , Gk, we consider an element F = F ( g l , ... ,gk) of Mult(G,, ... , Gk ; H ) , where H is a further linear space. We claim that we can use this function to construct a linear map Ft, say, from the tensor product G, @ 0 Gk = G to H, and in fact that we have a linear map

...

Mult(G1, ... , Gi ; H ) + Hom(G, @ -.* @ Gk, H ) .

(4.8.1)

We first define the value of Ft on decomposable elements of G, setting Ft(gl @

@ gk) = & l ,

"' 9

gk).

(4.8.2)

We note that the function is so far well defined; if we replace g,, g , , r # s, on the left by cg,, c - l g S , for some nonzero scalar c, the argument of Ft is unaffected, as noted in Section 4.2, while the value of the right of (4.8.2) is likewise unaffected, by the multilinearity of F. We assert now

74

4

THE TENSOR PRODUCT OF LINEAR SPACES

THEOREM 4.8.1. The function (4.8.2) extends to a linear map from G to H. We choose bases (4.2.8) of the Gr and then, with g in the form (4.3.9), define

This certainly defines an element of Hom(G, H ) . Let us show that it agrees with (4.8.2) for decomposable arguments. Such an argument can be taken in the form (4.3.12), which is equivalent to (4.3.9) subject to (4.3.11). Thus the prescription (4.8.3) gives (4.8.4) and this, by the multilinearity, is the same as (4.8.5) thus we obtain (4.8.2), with

This is what we had to prove. This shows that any multilinear map F from G, , ... , Gk to H can be @ g k , from considered to be a composite of a map ( g , , ... ,g k ) + g , @ the Cartesian product of the G, to the subset of decomposable elements of their tensor product, followed by a linear map Ft from this tensor product to H . Briefly, a multilinear map can be factored over the tensor product. In particular, with k = 2, and H = K , the field, we have the proposition that a bilinear functional on G,, G2 defines a linear functional on the tensor product GI @ G2 ; the converse is also true. 1..

4.9 Bilinear Forms and Tensor Products

In Section 4.6 we carried out an extension of homomorphisms in separate spaces to homomorphisms of tensor products. We now perform a similar operation in respect of bilinear forms. Suppose that we are given k pairs of

4.9

BILINEAR FORMS AND TENSOR PRODUCTS

... ,k,

and for each pair a

gr E Gr , hr E Hr *

(4.9.1)

finite-dimensional linear spaces Gr, Hr , r = 1, scalar-valued bilinear function, or functional, $rkr

9

hr),

75

Their product

(4.9.2) then defines a function of the 2k arguments, g , , ... ,gk, h , , ... ,hk, which is linear in each argument, and so is a multilinear function of 2k arguments. It follows from Theorem 4.8.1 that this function extends to a linear functional on GI 0 ... 0 Gk 0 H , 0 ... 0 H k . A more special statement of relevance to later work is H

THEOREM 4.9.1. There is a bilinear function $ on G = G1 0 H 1 0 ..- 0 Hk, such that f o r decomposable arguments

0 G,,

=

k

$ ( g l ~ . " ~ g k , h l O " ' O h k ) = ~ $ r ( g r , h r ) . (4.9.3) 1

This may be proved in the same manner as Theorem 4.8.1. We choose bases in the spaces G r , H r , use (4.9.3) as a definition of $ for tensor products of basis elements, extend $ to the whole of G x H by bilinearity, and then show that the result agrees with (4.9.3) for general decomposable arguments. The result extends to linear combinations of products of bilinear forms. Later, we shall be much concerned with determinants of arrays of operators and of forms. For the latter case we note

THEOREM 4.9.2. For each of the k pairs ofjinite-dimensional linear spaces G r , H r , r = 1, ... ,k, let there be k bilinear forms $rs(gr,hr)y

grEGr, hrEHr, r = l , . . . , k *

(4.9.4)

Then there is a bilinear f o r m on the tensor products G , H which f o r decomposable arguments as in (4.9.3) assumes the value

det

$rskr

3

hr).

(4.9.5)

Here the determinant is to be evaluated in the usual manner, as the sum of k ! terms of the form (4.9.2) with appropriate signs. The result then follows from the last theorem. Actually, the case of bilinear forms is less relevant for us than the slightly more involved case of sesquilinear forms, which gives rather similar results, and to which we now proceed.

76 4.10

4

THE TENSOR PRODUCT OF LINEAR SPACES

Products of Sesquilinear Forms

We modify the situation of Section 4.9 by assuming that the finitedimensional linear spaces G, , H,, r = 1, ... , k are now over the complex field, and that the forms (4.9.1) are now sesquilinear, in the sense given in Section 2.5, with scalar or complex values. It is immediate that the product (4.9.2) is linear in each of the g , , and conjugate-linear in each of the h,. We wish to deduce that it extends to a function on the tensor products G, H which are linear in the first argument, and conjugate linear in the second. THEOREM 4.10.1. There is a sesquilinear form with arguments in G , H that for decomposable arguments satisfies (4.9.3). The proof is similar to that of Theorems 4.8.1, and 4.9.1, but will be sketched. We choose bases in the G,, H,, say and

grl, *.. ,grmr,

hrl,

(4.10.1)

9

For general arguments in G , H , say g = C "' SI

C aSl...SkglSl @

"*

@ gkSkY

(4.10.2)

Sk

and (4.10.3) we define h

This defines a sesquilinear form on G , H. The verification that it agrees with (4.9.3) for decomposable arguments is similar to that given in the proof of Theorem 4.8.1 and will be omitted. We can apply this result to the more special situation that each space H , is the same as G,, r = 1, ... ,k, and, more specially still, to the case that the sesquilinear forms I//, are hermitian, in the sense that the quadratic forms $,(g,,g,) take only real values. For its relevance to later work we quote this as 4.10.2. Let G , , r = 1, ... ,k be finite-dimensional complex THEOREM linear spaces, and let $r(gr,gr'),

gr,gr'EGr, r = 1,

,k,

(4.10.5)

4.10

77

PRODUCTS OF SESQUILINEAR FORMS

be hermitian sesquilinearforms on the Gr . Then there is a hermitian form $ on the tensor product G such that, for decomposable arguments,

n k

$(gl @

’.*

@ gk gl I @

**‘

@ gk’) =

1

$rkr>

gr‘) *

(4.10.6)

The only additional information here, beyond that given in the last theorem, is that the form $ is hermitian. It is obvious that if the decomposable arguments on the left of (4.10.6) coincide, then the right is real. That $(g, g ) is real, whether g is decomposable or not, can be seen from (4.10.4), if we identify the two sets of bases in (4.10.1), the coefficients in (4.10.2) and (4.10.3), and use Theorem 2.3.1. The determinantal variant will be of considerable importance in later chapters. The result, a trivial consequence of the last one, is THEOREM 4.10.3. Let the spaces G, be as in Theorem 4.10.2 and on each Gr let there be defined k hermitian forms $rskr,

gr?,

gr, g,’ E Gr

3

s = 1,

*..

k*

(4.10.7)

Then there is a hermitian form on the tensor product G which,for decomposable @ &’ assumes the value arguments g , @ ..-@ g,, g, 6 I

det $r&r

gr’)*

(4.10.8)

For the proof we need only remark that the determinant, when expanded, is the sum of signed products which have the form of the right of (4.10.6). The last two theorems pave the way for further questions relating to positive definiteness. In the case of determinants (4.10.8) these questions turn out to be nontrivial, and will occupy us later. At this point, we take up the case of a product of forms; this can be viewed as a special case of (4.10.8), by taking the array of forms to be diagonal. The result is important in the topic of tensor products of hilbert spaces. We have

THEOREM 4.10.4. Let the hermitian forms (4.10.5) all be positive dejinite. Then the inducedproduct forb (4.10.6) is also positive definite.

=-

We must show that $(gyg) 0 if g E G, g # 0. This is immediate from (4.10.6) if g is also decomposable. It is not apparent from this that the result must therefore hold also for nonzero indecomposable g, though in fact such a line of reasoning will be developed later in the determinantal case. To obtain the conclusion we choose bases in the G,, grl,***,grmr, r = l Y . . . , k ,

(4.10.9)

78

4

THE TENSOR PRODUCT OF LINEAR SPACES

which are orthonormal with respect to $, ; this can be done, since the $, are positive definite. Thus

It then follows that the tensor products g,,,

0 ... @ g k t k ,

1 5 t, 5 m,, r

=

1, ... ,k,

(4.10.11)

are orthonormal with respect to $, by (4.10.6). Thus if g has the form (4.10.2), we have 4%

g) =

c .*.c l%l. .srl SI

which is, of course, positive unless g

=

z9

(4.10.12)

Sk

0.

Notes for Chapter 4

The tensor product is sometimes termed the “direct product,” or the “Kronecker product.” As references dealing with the tensor product in varying situations and from various viewpoints, one may mention Bellman (1960), Berezanskii (1969, Bourbaki (1948), Chambadal and Ovaert (1968), Gel’fand (1948), Godemont (1968), Greub (1967), Grothendieck (1955), Halmos (1958), Jacobson (1953), Lang (1965), Maclane (1963), Schatten (1950, 1960), van der Waerden (1966). See also Gelbaum (1962). The basic property of operators acting on different floors of the tensor product, that they commute, according to Theorem 4.6.1, has prompted the question of what cases of commuting linear maps arise in this way. On this see de Boor and Rice (1964), and Davis (1970). It is evident that the induced maps (4.6.7) preserve decomposability; on this aspect see Westwick (1967). Concerning applications of multilinear algebra to matrix theory see Marcus (1960, 1964) and Marcus and Minc (1964). Theorem 4.7.2 constitutes a special case of the “Kiinneth tensor formula”; see Maclane (1963), Greub (1967).

CHAPTER 5

TENSOR PRODUCTS AND ENDOMORPHISMS

5.1

Introduction

The basic theory of the tensor product is concerned with the extension to the product space of phenomena occurring in the factor spaces of which the product is composed. In this chapter, we discuss the relation between endomorphisms of these factor spaces and the endomorphisms that they induce in the tensor product space with special reference to eigenvalue decompositions. Let G I , ... , Gkbe finite-dimensional linear spaces over a field K, and let G = GI 0 0 Gk be their tensor product. We suppose that for each G, we have an endomorphism A, of G,. According to Section 4.6, with each A , , we can associate an induced map A:, acting on the rth factor in tensor products of vectors taken one from each of G I , ... , Gk; this induced map will be an endomorphism of G. A particularly important property of these induced maps is that they commute; we have A , + A , ~= A,~A:, r z s. (5.1.1) This was noted in Theorem 4.6.1. In (5.1.1) all operators are endomorphisms of the same linear space G, and the distinctions between various domains 79

80

5

TENSOR PRODUCTS AND ENDOMORPHISMS

and ranges drawn in (4.6.8) in the case k = 2 become unnecessary. Of course, (5.1.1) is true also if r = s; generally however, if we have two endomorphisms of G,, the endomorphisms of G that they induce will not commute. In particular, the identity endomorphism of G,, I, say, will induce an endomorphism of G that is easily seen to be the identity endomorphism Z of G . Similarly, the zero endomorphism of G, will induce that of G . We take a brief look at the matrix versions of these constructions. Let n, = dim G,, r = 1, ... , k , and let n = n f n , denote the dimension of G. If bases are introduced, an endomorphism A, of Gr will be representable by a square matrix of order n,, while the induced endomorphism A,+ will be represented by a square matrix of order n. Specifically,let bases of the G, be given by (4.2.8), with the tensor products (4.3.1) being taken as a basis of G . Then elements of G, are given by sets of scalars c , , ~, ... , c,,,,,, and A, will be represented by an array of scalars 1 Iu, v I n,; the action of A, will be to take the above set of scalars into the set n,

C ar,uuCr,u, u=l

u = 1, ... nr. 9

(5.1.2)

With respect to the basis (4.3.1), elements of G will be represented by arrays of scalars c,, ... u k , 1 I V , I n,, s = 1, ... ,k, (5.1.3) in the manner (4.3.9). A general endomorphism of G will be represented by an array bu,. . . u k u l ... v k y 1 Iu,, U, I n,, s = 1, ... , k, (5.1.4) with action that takes the array (5.1.3) into

In terms of (5.1.4) and (5.1.5) the endomorphism A; induced by A,, acting as in (5.1.2), is given by taking bUl

if

U, = v , ,

"'

UkVl

'.. U k - 'r,U,"r

s =

Y

1 , ... , k, s # r ;

(5.1.6) (5.1.7)

otherwise the left of (5.1.6) is zero. 5.2

The Hermitian Case

Since in this case the eigenvalues and associated decompositions have special properties, it is important to be able to relate the hermitian character

5.2

81

THE HERMITIAN CASE

of induced endomorphisms to that of the original operators. The term “hermitian” for finite-dimensional endomorphisms will be interpreted here as requiring a special relationship with regard to a hermitian sesquilinear form, the field being complex. Therefore, we must start by supposing there exist such forms in the spaces G,, and introducing a corresponding form in the tensor product G. It would be possible to introduce bases in the spaces, and to proceed in terms of hermitian matrices; however, an intrinsic formulation will be much more concise. We have, supposing all spaces finite-dimensional,

THEOREM 5.2.1. Let the endomorphisms A , of G,, r = 1, ... , k, be hermitian with respect to hermitian forms $, on G,, r = 1, ... , k , respectively. Then the induced endomorphisms A: are hermitian with respect to the product form $ defined by (4.10.6). We use criterion (2.6.4) for an endomorphism to be hermitian; in the present case, we write what we must prove as * ( A h , g’> = %4

g , g‘ E G.

A:&?’),

(5.2.1)

It will be sufficient to prove this for decomposable g , g ’ ; the general case will then follow by linearity. Writing then g = gl @

“’

@ gk,

g’ = gl’ @

*’*

@ gk’,

(5.2.2)

the two sides of (5.2.1) become, in view of (4.10.6) and (4.6.7), $lkl,

g l ’ ) * ” $r(Argr, gr’) ’ * *

$ k k k ~gk’)

$lkl

g l ’ ) * ’ * $r(gr~

$k(gk, gk‘),

and Y

’**

and these are equal in view of the hermitian character of A,. It follows from Theorem 2.6.2 that any linear combination of the A: with real coefficients will likewise be hermitian. Since the A: commute, it follows from Theorem 2.6.3 that the products of the A: will also be hermitian. Generally we have THEOREM5.2.2. In the circumstances of Theorem 5.2.1, any polynomial in the A,+, ... ,A:, with real coeficients, is hermitian. We shall be concerned later with the case that for each Gr,r = 1, ... ,k, we have k hermitian endomorphisms A r l , ... , A r k . These induce an array

82

5

TENSOR PRODUCTS AND ENDOMORPHISMS

(5.2.3) of G, which will also be hermitian. From this array we can form a “determinant” A, say, in the same way as a scalar-valued determinant is formed from a square array of scalars; such an expression is written explicitly in (6.2.2). We note here that A will be hermitian; it is a sum of products of hermitian endomorphisms, those in any one product commuting, with real coefficients, namely 1 or - 1.

5.3 Eigenvalues and Ranks Going back from the special case of hermitian endomorphisms and the complex field, we note here the very straightforward relation between the eigenvalues of an endomorphism and those of an induced endomorphism on a tensor product.

THEOREM 5.3,l. Let A, be an endomorphism of the jinite-dimensional linear space G, , r = 1, . . . , k . Then the eigenvalues of the A , and their ranks 0 C,. coincide with those of the induced endomorphisms Art of G = GI @ This statement comes immediately from Theorems 4.7.1 and 4.7.2. For A: - AI to have nonzero kernel as an element of End C it is necessary and sufficient that A, - AIr have nonzero kernel; here I, denotes the identity map of C,. The kernel of Art - AZ is, in fact,

G, 0 ... 0 Ker(A, - l l r )0

... 0 G,,

(5.3.1)

and this is nonzero if and only if Ker(A, - ,?Ir)# 0; here we exclude the case that one or more of the spaces Cr is zero. Thus the eigenvalues of the A , and the Art coincide. When talking of “rank” and “root subspaces” we are concerned with the least integer q such that ’

Ker(A; - 11)“ = Ker(A? - I J ) “ + ’ ;

(5.3.2)

this least integer q, which certainly exists when the spaces G, are finitedimensional, will be the rank of 1 as an eigenvalue, whereas the spaces in (5.3.2) will be the root subspace associated with 1. By Theorem 4.7.2, the kernel of (A: - JLZ)” will be

G, 0 ... 0 Ker(A, - Al,.)” 0 ... 0 Gk.

(5.3.3)

5.4

83

DECOMPOSITION

As 11 increases, this gives a nondecreasing sequence of subspaces of G, which

attains its maximum first when n = q, the rank of A;: this is clearly the same as the least n for which the factor Ker(A, - AI,)” attains its maximum, and this is the rank of A as an eigenvalue of A,. The root subspace (5.3.2) of I as an eigenvalue of Art is the tensor product of the root subspace of A as an eigenvalue of A, with the remaining spaces G, , ... , Gr.-l , G,, , ... , G,. As a corollary we have, for the hermitian case, 5.3.2. In addition to the assumptions of Theorem 5.3.1, let the THEOREM A, be hermitian with respect to positive-dejk’te hermitian forms $, defined on the G,. Then the eigenvalues of the induced endomorphisms A: are real and of rank 1.

This follows from Theorem 3.7.2, which shows that the eigenvalues of A , are real and of rank 1. It also follows from the same theorem, as applied to Art directly; we know by Theorem 5.2.1 that A,’ is hermitian with respect to a product form which, by Theorem 4.10.4, is positive-definite. 5.4 Decomposition

If the field K is algebraically closed, we have for each A, E End G,, r = 1, ... ,k , a direct sum decomposition of G, into root subspaces; these can be combined to give a decomposition of the tensor product space. We thus have 5.4.1. THEOREM

Let K be algebraically closed, and let

dim G,

=

n,,

0 < n,
-

~

~

n

... ~

n Ker(Ak+~ ~ &,kl)qskk. f

~

(5.6.7) ~

In order to show that this is annihilated by F", it will be sufficient to show that it is annihilated by some factor of F", namely, (B -

PSI..

.,,om.

(5.6.8)

On reference to (5.6.5) and the definition of B as (5.5.1), we see that this can be written as

(5.6.9) Expanding this by the multinomial theorem, we obtain a linear combination of terms of the form

(5.6.10) If m is suitably large, the last condition implies that pr 2 q,,, for at least k, and then the product (5.6.10) will annihilate the subspace one r , 1 5 r I (5.6.7). This completes the proof that F" = 0 for large m. We now appeal to Theorem 3.5.3. This shows that the polynomial

1

88

5

TENSOR PRODUCTS AND ENDOMORPHISMS

is divisible by the minimal polynomial of B ; in view of the form of this polynomial, given by (3.5.7) in the appropriate case for that section, we see that the eigenvalues of B must be among the quantities ps,...Sir. This completes the proof of the theorem. 5.7 Separation of Variables

This notion is found principally in connection with partial differential equations. One requires a function u(x, y) of say two independent variables x, y, which should satisfy a linear partial differential equation and supplementary conditions. In certain special, but highly important cases, it is useful to seek solutions of the form u(x,y) = u(x)w(y), since one is thereby led to the theoretically simpler case of ordinary differential equations in x and y separately. Here we are close to tensor product notions; the product of functions of x and y forms a special type of function of x and y, in much the same way as a decomposable tensor forms a special type of element of a tensor product. The relation between the partial differential equation, in separable cases, and the ordinary differential equations which arise from it, is similar to that between a Kronecker sum and the endomorphisms of separate spaces which give rise to it. Here our discussion of tensor products will be confined to finite-dimensional cases, and so cannot treat the case of differential operators. There is, however, an analogous theory of the separation of variables in partial difference equations ; here the variable will have as its range some discrete set. The theory is applicable to cases in which the partial difference operator is representable as a Kronecker sum.

A linear equation

BU = g,

(5.7.1)

where B E End G, G being a finite-dimensional linear space, with g given and u to be found, will be amenable to treatment by the method of the separation of variables if, first, G can be represented as a tensor product, in a nontrivial manner, and second, B can be represented as a Kronecker sum, with respect to this same tensor-product representation of G. For simplicity, suppose that G is represented as a product of two spaces, say (5.7.2) G = Gi 0 Gz, and that (5.7.3) B = A I t AZt,

+

where A l , A , are endomorphisms of G1, G,; in the notation (5.5.6), we

5.8

THE TENSOR PRODUCT OF IDENTICAL FACTORS

could write equally

B = A1 6 1,

+

11

6A z .

89

(5.7.4)

As a preliminary observation, we remark that, given g, (5.7.1) will be uniquely soluble for u if B does not have zero as an eigenvalue; by Theorem 5.6.1 this is equivalent to requiring that no sum of an eigenvalue of Al and an eigenvalue of A , vanishes. If Al , A , are hermitian, with respect to positive-definite hermitian forms $ 1 , t,hz, so that B is hermitian with respect to the positive-definite hermitian product form $ on G , we can solve (5.7.1) in terms of the eigenvectors and eigenvalues of B. By recourse to the “separation of variables,” it is enough to have the same entities for the separate endomorphisms A , , Az or the lower-dimensional spaces G I , G, . The formalities are straightforward. Let dim G, = n,, r = 1, 2, and let A, have the eigenvalues A r l , ... ,A,,,*, and eigenvectors g,, , ... ,g,,, ; here we assume that the eigenvalues are repeated according to multiplicity and that the eigenvectors are orthonormal, with respect to the product $r in the space G,. It then follows that the tensor products g1s1

6g z s 2 ,

1

sr 5

nr,

r = 1,Z

(5.7.5)

are orthonormal with respect to the product $, and are the eigenvectors of B, with respective eigenvalues A,, + I,, . The solution of (5.7.1) can then be written immediately as

5.8 The Tensor Product of Identical Factors In the foregoing, we considered the tensor product of k unrelated linear spaces, for each of which we had an endomorphism. The same constructions can be applied in the much more special case that we have a single space X , say, and an endomorphism A of it; the greater specialization makes possible a much more detailed structure. An extensive treatment of this topic would constitute a digression, so our remarks will be brief. For a positive integer k , and finite-dimensional linear space X , we can form the tensor product

G

=X

6

6X ,

( k factors)

(5.8.1)

which we term the “tensor product of k copies of X ” ; a corresponding notion was introduced in Section 1.4 for the direct sum. If now A E End X ,

5

90

TENSOR PRODUCTS AND ENDOMORPHISMS

we can form k induced endomorphisms of G by applying A to the first, .: If second, ... , kth factors in (5.8.1); let us denote these by A,+, ... , A we denote the identity map of X by Zx, and use the notation (5.5.6) for ( 5 . 5 3 , these induced maps can be written A,' = A Q Zx Q =

Akt =

'**

0 Ix,

Zx Q A Q Zx Q Zx Q

***

* * a

(5.8.2)

(5.8.3)

Q I,,

Q Zx Q A.

(5.8.4)

It follows from Theorem 5.3.1 that the A: have the same eigenvalues as A , and that the ranks of the eigenvalues are the same in the two cases. Thus if the distinct eigenvalues of A are (5.8.5)

with ranks (5.8.6)

we have the direct sum decomposition

G

=

P Sl=l

P

1 Ker(A - 1,1Zx)4s1 Q ... Q Ker(A -

1,1x)4sk,

(5.8.7)

Sk=l

as a specialization of (5.4.4). We obtain a hermitian structure if we suppose that in X there is defined a positive-definite hermitian form 50, with respect to which A is hermitian. As in Section 4.10, this defines a product form, $ say, on the tensor product (5.8.1) that is hermitian and positive definite, and with respect to which the A: are hermitian; the latter property also obtains for polynomials in the A:, with real coefficients, notably the Kronecker sum and product. In this hermitian case, the eigenvalues are, of course, all real, with rank 1, so that the indices in (5.8.7) can all be replaced by unity. In the hermitian case, we can express (5.8.7) more explicitly, in terms of orthonormal sets of vectors. Let dim X = I, and let a complete set of eigenvectors of A be gl 7

*** 7

gl

(5.8.8)

7

where these are to be chosen to be orthonormal with respect to the product cp. Then a complete set of eigenvectors of the A,, orthonormal with respect to is given by $y

gr, 0 *.. Q gt,,

1 It, I

I, r

= 1,

... ,k.

(5.8.9)

5.9

INDUCED MAPS OF SYMMETRY SUBSPACES

91

For any g E G we have an expansion

5.9 Induced Maps of Symmetry Subspaces

Our remarks concerning induced maps of the tensor product of k copies of a linear space have so far merely been specializations of general results for tensor products of unrelated spaces. The novelty that we obtain in return for this restriction to a special case lies in the properties of the Kronecker sum Alt

and product

+

Alt

9..

+ A:,

(5.9.1)

Akt,

(5.9.2)

and more generally of “symmetric functions” of the A:. As invariant subspaces of G, these operators admit the eigensubspaces and the root subspaces appearing in (5.8.7), as does any polynomial in the Art. In addition, in the case of the last section, (5.9.1) and (5.9.2) admit as invariant subspaces certain symmetry subspaces of (5.8. l), which are independent of the endomorphism A . The study of the action of (5.9.1), (5.9.2), and other symmetric functions on these subspaces leads to important quantities associated with an endomorphism, such as its trace and its determinant; whereas these are well known from more explicit definitions, the approach in question yields a valuable supplementary viewpoint. The simplest nontrivial subspace of G, invariant under maps of the form (5.9.1) and (5.9.2), for any A E End X , is perhaps the subspace of symmetric tensors; these are multilinear functionals f(yl9

... yk), 3

y, E X * ,

= 1, ... 9 k ,

(5.9.3)

where X * is the dual space to X , which are unaffected by interchange of any two arguments, and so by any permutation of the arguments. If we apply (5.9.2) to (5.9.3) in accordance with the prescription (4.6.6) for the action of induced operators, we obtain the multilinear functional f ( y l A , ... ,YkA),

y, E X * , r = 1, ... ,k.

(5.9.4)

It is immediately apparent that if (5.9.3) is invariant under a permutation of the arguments, then so is (5.9.4), and so the Kronecker product (5.9.2) maps the subspace of symmetric tensors into itself.

92

5

TENSOR PRODUCTS AND ENDOMORPHISMS

The same holds for the Kronecker sum. If we apply (5.9.1) to (5.9.3) we obtain f(ylA,

Y2 9

*** 3

yk)

+ f ( y 1 , yZA, Y3 ... ,yk) + + . f ( ( v l , *.. yk-1 ykA)* ***

3

(5.9.5)

Suppose, for example, that in (5.9.5) we interchange the arguments y l , y,. If we assume (5.9.3) symmetric, the last k - 2 summands are then unaffected in value, whereas if we interchange the first two arguments in the first two terms, and also interchange the latter, we obtain the original form of (5.9.5). Thus (5.9.1) also has the subspace of symmetric tensors as an invariant subspace. Next take the case of skew-symmetric tensors, given by multilinear functionals (5.9.3) with the property that if two arguments are interchanged, the effect on the functional is to apply a factor - 1. If this property holds for (5.9.3), it clearly also holds for (5.9.4), and so (5.9.2) has the subspace of skew-symmetric tensors as an invariant subspace. Likewise, for the Kronecker sum, it is easily checked that if (5.9.3) is skew-symmetric, then so also is (5.9.4). Perhaps the most interesting special case is that in which k = dim X , and in which we consider the subspace of skew-symmetric tensors, that is,

xA

A ( k factors)

x.

(5.9.6)

This space is noteworthy for being one dimensional, so that any endomorphism of it is given by multiplication by a scalar. In the case of (5.9.1) this scalar turns out to be the trace of A, and for (5.9.2), the determinant.

Notes for Chapter 5 For a treatment of the Kronecker sum and product, along with other Kronecker functions and the associated properties concerning eigenvalues, we refer to Bellman (1960), Friedman (1961), and Grobner (1966). Concerning spectral resolutions of Kronecker sums and products in a Hilbert space setting, see Berezanskii (1965), Cordes (1953, 1954-1955a, b), Carroll (1963, 1965), and Brown and Pearcy (1966). For applications of the procedure of “separation of variables,” in a discrete setting, with advantages from the viewpoint of numerical analysis, see Lynch et al. (1964a, b; 1965). See also Jaeger and Kuo (1967). For a discussion of symmetry classes of tensors we refer to Greub (1967, Chaps 5-8). The study of the endomorphism of the set (5.9.6) of skew-symmetric

NOTES FOR CHAPTER

5

93

tensors, which takes (5.9.3) into (5.9.4), for general k, constitutes an abstract version of the study of the “minors” of A, when expressed as a matrix, with respect to a basis in X; of course, only the determinant itself will be independent of the choice of the basis. This presents the possibility of a basis-free treatment of topics involving minors, such as determinantal identities or, in a different direction, total positivity [see, for example, Gantmaher and Krein (1960), Karlin (1968)], and Markham (1970). The spectral resolution of Kronecker sums is closely connected with the theory of the matrix equation AX + XB = Y, to be solved for X (see Bellman, 1960). For a recent complex variable treatment, covering more general equations in Banach spaces, see Daleckii and Krein (1970, Chapter I).

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PART

2

MULTIPARAMETER PROBLEMS FOR MATRICES

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CHAPTER 6

SIMULTANEOUS EQUATIONS IN LINEAR SPACES

6.1 Introduction We turn now from preliminaries to our main theme-the title of this chapter. While motivation for this study comes primarily from differential equations, there are at least three areas of linear algebra of suggestive value for us. The first is that of linear equations in scalar unknowns, with scalar coefficients. There are two useful cases for us. If the number of equations is the same as the number of unknowns, we obtain the situation of a square array of coefficients ars,

1 I r, s I k,

(6.1.1)

for which we consider, in the first place, the set of solutions xl, ... ,xk of the homogeneous equations k

z a r s x , = 0,

s= 1

91

r = 1, ... , k .

(6.1.2)

98

6

SIMULTANEOUS EQUATIONS IN LINEAR SPACES

We are interested in operator analogs of this situation, and in analogs of the proposition that (6.1.2) has nontrivial solutions if and only if the determinant (6.1.3)

det a,, = 0,

one knows also that if (6.1.3) fails, the corresponding inhomogeneous equations are always soluble, and it is desirable to extend this statement also. Alternatively, one can say, in a determinant-free formulation, that the inhomogeneous equations, corresponding to (6.1.2), are always soluble if the homogeneous equations have only the trivial solution, and the extension of this statement also deserves consideration. We are also concerned, indeed even more so, with the case of homogeneous equations in which the number of equations is one less than the number of unknowns. That is, we have a rectangular array of scalar coefficients

r = 1, ... , k, s = 0,... ,k ,

a,,,

(6.1.4)

and are concerned with the homogeneous equations k

arsx, = 0,

r = 1,

... , k .

(6.1.5)

S=O

Here again determinants can play a key role. We know that the set of solutions of (6.1.5) is one-dimensional if and only if the rank of the array (6.1.4) is k , that is, if at least one of the k by k matrices to be formed from (6.1.4) has nonzero determinant. Again, we are interested in operator analogs of this statement. The second relevant area of linear algebra is that of the eigenvalues of endomorphisms of finite-dimensional spaces. Here we are asking that the endomorphism

A

- II

(6.1.6)

be singular, where A E End G, I is the identity on a linear space G . This may be viewed as an analog of (6.1.5) in the case k = 1; a more general analog would be the topic of the singularity of AIR, + A,&, where Al ,A , are linear operators from one space to another, and A l , A, are scalar parameters. The third area we wish to cite here is a development of the second in the tensor product direction, which was taken up in Chapter 5. If we have endomorphisms A, of spaces G,, and denote the identity on G, by I,, we can

6.1

99

INTRODUCTION

arrange these operators in an array

3

(6.1.7)

in which the operators in any one row act on the same domain space, while those in any one column are to be associated with the same one of a set of scalar parameters. It is our task to extend results for (6.1.7) to more general arrays

[Aka :A10

All

”‘

Akl

Alk

(6.1.8)

A]’

in which again those in any one row act on the same space. There will be corresponding arrays of induced operators, in which all the operators act on the same tensor product space. We now develop operator analogs of the homogeneous equations (6.1.2); much that we say will apply with the obvious alterations to the case of (6.1.4) and (6.1.5). It turns out that if we replace the coefficients arsin (6.1.1) and (6.1.2) by linear operators, there are several analogs of the homogeneous equations, and it becomes necessary to consider the relation between them. We suppose given k pairs of linear spaces C,, H,, all over the same field K,nonzero, and of the same finite dimension for each pair, so that 0 < dim G, = dim H, < co;

(6.1.9)

dim G, = dim H, = v,,

(6.1.10)

we write allowing the dimension to vary with r. For each pair, we suppose given a set of k linear operators A r l , ... ,A,,

E Hom(G,,

H,),

r = 1, ... ,k.

(6.1.1 1)

We can then pose a number of questions concerning this square array of operators. The first of these, of eigenvalue type, does not involve the tensor product construction :

100

6

SIMULTANEOUSEQUATIONS IN LINEAR SPACES

QUESTION I. Do there exist scalars I , , ... ,1, , not all zero, such that all k operators

c A,,&, k

s= 1

r = 1, ... , k,

(6.1.12)

have nonzero kernel? It may be that the A,, are specified as square matrices, of order v,. In any case, we can introduce bases in C,, H , so that the A , should be so represented. Question I is then equivalent to 1'. Do there exist scalars 1,, ... ,A,, not all zero, such that QUESTION the k polynomials

det

c A,,&, k

= 1,

I'

s= 1

... , k,

all vanish ?

(6.1.13)

c:=l

In the determinants (6.1.13), we are to represent A J , as a square matrix by means of the matrix representation of the A,,; this matrix will have as its entries homogeneous linear forms in the A,. On expansion, the determinant then becomes a poJynomial in the A, of degree v,. We are asking whether these k polynomials have a nontrivial common zero. Further analogs of (6.1.2) come to light if introduce the tensor product

G

=

GI 8

a * *

@ Gk.

(6.1.14)

We can then pose two modifications of Question 1, the first being formally more demanding than the second. 11. Do there exist scalars QUESTION the operators

c A,:&, k

S=

1

A1, ... ,A, not all zero, such that

r = 1, ... , k,

(6.1.15)

have kernels with nonzero intersection, where A:s denotes the map of G induced by A,, [see (4.6.2)]? QUESTION 111. Do there exist scalars A,, ... , A,, not all zero, such that the operators (6.1.15) all have nonzero kernels? In the tensor product we can also pose a question in which the unknowns in (6.1.2) are replaced not by parameters, but by vectors; this is

6.2

DETERMINANTAL MAPS

101

QUESTION IV. Do there exist elements fl,

,heG,

(6.1.16)

not all zero, such that

2 A:f, k

s= 1

= 0,

r

= 1,

... , k?

(6.1.17)

Such a question cannot be posed in the original spaces G, , .. . , Gk, since the operators in different rows of the array (6.1.1 1) do not have comparable actions; all the questions become indistinguishable in the scalar case. We are not yet in a position to clarify fully the relation between these questions; an additional variant, an analog of (6.1.3), will be introduced in the next section. However, by means of results so far achieved, we can deal at least partly with the matter. THEOREM 6.1.1. Questions I , II, and III are mutually equivalent, and a positive answer to any of them implies a positive answer to Question IV. It follows from Theorem 4.7.1 that Questions I and 111 are equivalent. It is trivial that a positive answer to 11 implies one to 111. We complete the proof of the equivalence of I, 11, and I11 by showing that a positive answer to I implies a positive answer to 11; this follows from Theorem 4.7.2. Suppose now that I1 has a positive answer, and that f E G, f # 0, lies in the intersection of the kernels of (6.1.15). Since the 2, are not all zero, we obtain a nontrivial solution of (6.1.17) on taking

f, =

s = 1,

... , k.

(6.1.18)

This completes the proof of Theorem 6.1.1. The argument will be completed in Chapter 8, taking into account also a determinantal formulation.

6.2 Determinantal Maps We next enlarge the formal machinery in order to provide for an analog of condition (6. I .3) for the nontrivial solubility of (6.1.2). Equations (6.1.17), subject to (6.1.9), can, of course, be viewed as a set of scalar equations, equal in number to k(dim G), and so treated by determinantal methods. However, we need a different construction, involving certain operatorvalued determinants. For (6.1.1 I), with finite-dimensional spaces G,, H,,

102

6

SIMULTANEOUS EQUATIONS IN LINEAR SPACES

(6.2.1) and mean by this that the right-hand side is to be expanded in the usual manner for determinants, the products of the entries being interpreted as the composites of the operators concerned. Explicitly we have (6.2.2) where 0 runs through permutations of 1, ... ,k, and 8, is 1 or - 1 if Q is even or odd, respectively. Each induced operator A?,,,) takes a factor G, in a tensor product into H,, and so the combined action is A0 1

G = GI @

* - a

@ Gk + H = HI

@ Hk.

(6.2.3)

By Theorem 4.6.1, the entries in (6.2.1) commute when in different rows; however, those in the same row cannot be composed. The result is that some, but not all, of.the familiar properties of determinants with scalar entries carry over to this case. The following theorem can be proved as in the standard case. THEOREM 6.2.1. For a determinant (6.2. l), we have the following properties.

(i) If two columns are interchanged, the sign of the determinant is reversed. (ii)

If two columns are identical, the determinant vanishes.

(iii) The value of the determinant is unchanged i f a scalar multiple of one column is added to another column. It will be noted that at the moment we do not assert any such result concerning rows in place of columns. Further results, parallel with those of the standard case, concern the use of cofactors. We denote by A,,, 1 5 r, s 5 k, the expression derived from (6.2.2) by deleting the factor A:s from all terms in which it appears, and dropping those in which it does not appear. Much the same expression is given as an operator-valued determinant of the form (6.2.1), by deleting the row and column containing A?,, and attaching a sign-factor (- IT+,. We then have

6.3

SINGULAR DETERMINANTAL MAPS IN THE CASE

k =2

103

THEOREM 6.2.2. For 1 I s, t I k ,

For the proof, we note in the usual way that each term in (6.2.2) contains exactly one factor from the sth column so that, if t = s, (6.2.4) gives the reconstitution of A. according to factors from the sth column; if t # s, (6.2.4) gives the expansion, according to terms in the sth column, of a determinant in which the tth column has been replaced by a duplicate of the sth column, and so is zero. Strictly speaking, it should be mentioned that the first equation in (6.2.4) involves different actions of the operators concerned. Rather as in (4.6.8), these actions are as indicated in the commutative diagram

As an easy consequence of these identities we have THEOREM 6.2.3. A necessary condition f o r (6.1.17) to have a nontrivial solution is that A. be singular. On applying to (6.1.13) the operators Art, r = 1, ... ,k , respectively, and summing, we obtain in view of Theorem 6.2.3, that (6.2.6) Aofr=O, t = l , ..., k. This is one half of the analog of the standard proposition concerning (6.1.2) and (6.1.3). The other half, that if A. is singular, then (6.1.17) has a nontrivial solution. seems more difficult. In our case, we note as QUESTION V. Is the operator-valued determinant (6.2.1) singular? So far we have that Questions I, 11, and I11 are equivalent, that an affirmative answer to I11 implies an affirmative answer to IV, and an affirmative answer to IV implies an affirmative answer to V. In the next section, we complete the chain in a special case.

6.3 Singular Determinantal Maps in the Case k = 2 By way of illustration, and as partial progress, we shall elucidate the points just raised in the case k = 2.

104

6

SIMULTANEOUSEQUATIONS IN LINEAR SPACES

THEOREM 6.3.1. Let 0 < dim G,

=

dim H, < co, r

=

I , 2, and let

A,, , A,, be linear mapsfrom G, to H, , r = 1, 2. Let tliejield K be algebraically

closed. Write

(6.3.1)

A. = A f l A I 2 - A f 2 A i 1 ,

where the ATs are maps induced on tensor products by the A,,, and A. acts from G = GI 0 G2 to H = H , 0 H2. Then for A. to be singular, it is necessary and sufficient that for some scalars A1, A 2 , not both zero, the operators

(6.3.2) The sufficiency of the condition is covered by Theorems 6.1.1 and 6.2.3. We pass to the necessity, and assume that for no 2, , A 2 , not both zero, are both of (6.3.2) singular. Thus the polynomials det(ArlA,

+ Ar2A2),

r = 1, 2,

(6.3.3)

have no common nontrivial zero. Thus neither vanishes identically, and so each vanishes for only a finite number of values of the ratio A, : A 2 . We can therefore choose a scalar a such that Arl

+ aAr2,

r

=

1, 2,

(6.3.4)

are both nonsingular; we assume throughout that the field is of characteristic zero, and so infinite. We can then write (6.3.1) in the form

so that

A0 = (At1

(Ail

+ aAf,)At2

- At2(Ai1

+ MA;,),

(6.3.5)

+ aAf2)-'(Ail+ a A i 2 ) - ' A 0

= (At1

+ aAf2)-'Ai2

- (A?, + ~Af,)-'Af2.

(6.3.6)

Here the left is an endomorphism of G, which is singular or nonsingular along with A. . The right-hand side of (6.3.6) has the form of a Kronecker sum; the two terms on the right of (6.3.6) are endomorphisms of G I , G 2 , respectively. Theorem 5.6.1 now shows that the eigenvalues of (6.3.6) are given by an eigenvalue of the first term on the right, less an eigenvalue of the second term on the right. Thus A. will be singular if and only if these two terms have a common eigenvalue. Denoting such a common eigenvalue by p, we have that A. will be singular if and only if there is a p such that the operators 4 2

are both singular.

- P(Ar1 + ~ A r 2 ) ,

r = 1,2,

(6.3.7)

6.4

105

REACTANGULAR ARRAYS

We observe now that in (6.3.7) we have a pair of operators of the form (6.3.2), with A, = -p, A, = 1 -up, so that A1, A, are not both zero. We have thus shown that if A. is singular, then there is a nontrivial pair of the form (6.3.2) which are both singular. This completes the proof of the necessity. Thus, in the case k = 2, Questions I-V are equivalent, subject to K being algebraically closed and of characteristic zero. As an incidental consequence we have THEOREM 6.3.2. Let A. be as in Theorem 6.3.1. Then if the kernel of A. includes any nonzero element, it includes a nonzero decomposable element.

To obtain such an element we choose a pair A1, A, such that (6.3.2) are both singular, with A,, A, not both zero. A nonzero decomposable element in the kernel of A. is then given by forming the tensor product of two nonzero elements, chosen from the kernels of (6.3.2). We extend this result in Chapter 8 to general k. 6.4 Rectangular Arrays

We pass now to the subject of operator analogs of homogeneous equations (6.1.5) with one more unknown than equation. In this scalar case, if one of the k by k determinants that can be formed from the k by (k 1) array (6.1.4) is not zero, then the general solution of these equations can be expressed in terms of these k by k determinants. We wish to set up corresponding results involving operator-valued determinants of the type introduced in Section 6.2. For k pairs of spaces (6.1.9) over a field K of characteristic zero, we now suppose given k sets of k + 1 linear maps,

+

Are, ... , A r k :G,

r = 1,

+ H,,

... ,k.

(6.4.1)

We postpone discussion of the analog of (6.1.13) and go straight to that of (6.1.17). Supposing the spaces finite dimensional, and denoting by G the tensor product of the G,, and by A!s the operators induced by the A,,, we consider the solutions

of the simultaneous equations

2 AJsfs = 0, L

s=o

I’

= 1,

... ,k .

(6.4.3)

106

6

SIMULTANEOUS EQUATIONS IN LINEAR SPACES

We wish to extend to this situation the machinery of “Cramer’s rule,” which solves (6.1.5) in terms of determinants, except in degenerate cases. We begin by introducing notation for various determinants and cofactors, which will be operator-valued, as in Section 6.2. We write

As

=

(-1)”det

(6.4.4) Aio

AiS

e - 9

**’

Alk

where the caret ( A ) indicates omission, for the signed minor of the full array of the A?s obtained by deleting the (s 1)th column. As in Section 6.2, this will be a map from G to H . We also write

+

Asu,,

0

< s,

v 2 k , s # c,

1 5 u

< k,

(6.4.5)

for the cofactor of A:, in As. We then have the formal results given in

THEOREM 6.4.1.

We have the identities k

(6.4.6)

k

CAsu,AL

u= 1

=

-4,

(6.4.7)

s # v,

and (6.4.8) Here the order of the factors in the sums on the left can be reversed, subject to proper interpretations of domain and range, as in (6.2.5). We note that (6.4.6) is equivalent to the first case of (6.2.4), being the expansion of a determinant according to elements from a column. In (6.4.7) we are, in effect, deleting from As the column containing At,, and replacing it by that containing A/s; this gives us A,, subject to an interchange of columns. In (6.4.8), the left is the expansion of a determinant with two identical columns; we have deleted from As the column containing A!, and replaced it by that containing AL . From these identities we deduce

THEOREM 6.4.2. A solution of (6.4.3) satisfies Apfq - Aqfp = 0,

p,q

=

0,

... ,k.

(6.4.9)

6.5

DEFINITENESS REQUIREMENTS

107

For the proof we apply the operator AP,,, to (6.4.3) and sum over 1, ... ,k. This reduces to (6.4.9)in virtue of (6.4.6)-(6.4.8). We pass now to the nonformal matter of whether solutions of (6.4.9) necessarily satisfy (6.4.3).

r

=

6.5 Definiteness Requirements

As already mentioned, we shall consider the operator analog of the principal case of (6.1.4)and (6.1.5),which we take to be that in which the matrix (6.1.4)has rank k. This means that at least one of the k by k determinants to be formed from (6.1.4)is not zero. In the operator analog, this assumption can be paralleled in various ways; we note the following, in order of increasing generality : (i) One of the determinants As in (6.4.4)is nonsingular. (ii) Some fixed linear combination of the As is nonsingular. (iii) For every nonzero f E G , there is a linear combination of the A, that does not annihilate it. In addition, there is the question of whether we require these operators to be nonsingular in their application to G generally, or only as applied to decomposable tensors, or indeed whether this makes any difference. We shall emphasize here Case (ii). We shall assume that for some set ,uo, ... ,Pk the expression k

(6.5.1)

defines an operator that is an isomorphism. We can write this as a determinant, namely as

here the interpretation is similar to that of (6.2.1).We expand (6.5.2)in the usual way, and take products of operators to mean composites of operators. This comes to the same thing as interpreting (6.5.2)as acting on K @ GI @

a * *

@ Gk.

Since we have not yet dealt in general with the singularity of determinantal maps in relation to decomposable and general tensors, we shall be assuming

108

6

SIMULTANEOUS EQUATIONS IN LINEAR SPACES

that (6.5.1) i.e. (6.5.2) is nonsingular in its application to G generally. However, we can dispose of this point in the case k = 2, in virtue of Section 6.3. THEOREM^.^.^. L e t k = 2,andletO < dim G, = dim H, < w , r = 1,2. Then in order that there exist scalars p o , p,, p z such that

(6.5.3) 4

0

At,

At2

does not annihilate nonzero elements of G, it is necessary and sufficient that such an expression exist that does not annihilate nonzero decomposable tensors. The necessity being trivial, we assume that we can choose the ps so that (6.5.3) does not annihilate any decomposable element of G other than zero. The pcscannot all be zero; it will be sufficient to treat the case that p o # 0 and indeed that in which p o = 1. We can then write (6.5.3) in the form (6.5.4) This does not annihilate a nonzero decomposable tensor, and so is nonsingular, by Theorem 6.3.2, which proves the result. 6.6 Solutions for Rectangular Arrays We now go back to the solution of (6.4.3) by the method of Cramer's rule.

THEOREM 6.6.1. Let 0 < dim G, = dim H, < coy r = 1, ... , k, and let there exist scalars p o , ... , pk such that (6.5.1) and (6.5.2) are nonsingular. Then the solutions of (6.4.3) and (6.4.9) coincide. The general solution of (6.4.3) is given by

f, = A-'ASf,

s = 0 , ... ,k ,

(6.6.1)

for arbitrary f E G. The argument is dimensional. We have shown that the set of solutions of (6.4.3) is included in that of (6.4.9). Furthermore, we have, from (6.4.9) on

6.6

109

SOLUTION FOR RECTANGULAR ARRAYS

multiplication by pp and summation over p , that

c k

A& = Aq

p=o

(6.6.2)

PPfP'

It thus follows from (6.4.3) that

f = pc= o P p f p . k

fs =

A - 1 Asf9

(6.6.3)

Thus the set of solutions of (6.4.3) is included in the set described by (6.6.1), for varying f E G . Now let v denote the dimension of G , so that in fact k

v =ndimG,.

(6.6.4)

1

We consider the dimensionality of the set of solutions fo, ... ,hof (6.4.3); this set can be considered as a subspace of the direct sum of k 1 copies of G. The subspace described by (6.6.1) has dimension v, and so we have that the set of solutions of (6.4.3) has dimension at most v. On the other hand, consider the map

+

(6.6.5) which gives a linear map from the direct sum of k + 1 copies of G to the direct sum of k copies of G.In (6.4.3) we are asking for the kernel of this map. However, in (6.6.5) we are going from a space of dimension (k l)v to one of dimension kv. The kernel of this map must have dimension at least v , being the decrease in dimension as between domain and range, by (1.6.2.) It thus follows that the solution-space of (6.4.3) has dimension precisely v. Since this solution-space is included in the set described by (6.6.1), and since this set also has dimension v, we have that the solution-space of (6.4.3) coincides with the set given by (6.6.1). This proves the last statement of the theorem. We have to show also that the solution-space of (6.4.3), and the set given by (6.6.1), both coincide with the set described by (6.4.9). In fact (6.4.9) follows from (6.4.3)) according to Theorem 6.4.2, while (6.6.1) follows from (6.4.9), as shown in (6.6.2), (6.6.3). Thus the set of solutions of (6.4.9) includes the set of solutions of (6.4.3), but is included in the set described by (6.6.1). Since the last two sets coincide, the first must coincide with them.

+

110

6

SIMULTANEOUS EQUATIONS IN LINEAR SPACES

6.7 Nonforrnal Determinantal Properties The result of the last section, that the homogeneous equations (6.4.3) have the general solution (6.6.1), provided that some expression (6.5.1) is nonsingular, is trivial in the scalar case (6.1.5); the solution can be verified by direct substitution, based effectively on the expansion of a determinant according to the entries in a row. In the general case, we argue in the opposite sense. Having established by other arguments that (6.6.1) solves (6.4.3), we use the substituti n procedure to derive determinantal identities. 6.7. THEOREM

.

Let the conditions of Theorem 6.6.1 hold. Then k

A!sA-l As = 0,

r

=

1,

S=O

... , k.

(6.7.1)

This follows from Theorem 6.6.1 on substituting (6.6.1) in (6.4.3), and noting that the result is true for all f E G. The action of the two sides of (6.7.1) is as in (4.6.2). Formally, the result can be written as

where the factor A - l is to be interposed in the place indicated when multiplying out the determinant. The statement is trivial in the scalar case, since then the factor A - l can be taken outside, leaving a determinant with two identical rows. By very similar arguments we obtain the important

THEOREM 6.7.2. Under the assumptions of Theorem 6.6.1, the operators, endomorphisms of G, A-'AS,

s = 0,

... ,k ,

(6.7.3)

all commute. We have that (6.6.1) provides a solution of (6.4.3) and so also of (6.4.9). This shows that A,A-'A,f = AqA-lAPf,

6.8

111

EIGENVALUES FOR A RECTANGULAR ARRAY

and since this is true for all f E G we have APA-‘Aq = AqA-’Ap,

p, q

=

0, ... ,k ;

(6.7.4)

we obtain the assertion of the theorem on applying A - l to both sides. 6.8 Eigenvalues for a Rectangular Array

For the situation (6.4.1) we shall use the term “eigenvalue” to denote a (k 1)-tuple of scalars l o ,... ,lk,not all zero, such that the k operators

+

k

(6.8.1)

all have nonzero kernel. Assuming that (6.1.9) holds, we may, as explained in Section 6.1 in a similar case, require equivalently that det or again that

c Ar,ls k

s=o

Ker

= 0,

c A:&, # 0, k

s=o

r = 1,

... , k,

(6.8.2)

r = 1,

... , k,

(6.8.3)

or that these kernels should have nonzero intersection. We do not distinguish between a pair of eigenvalues of the form l o ,... ,l k and d o ,... ,d k . In this section we relate these eigenvalues to those of the commuting operators (6.7.3). THEOREM 6.8.1. eigenvalue we have

Let the assumptions of Theorem 6.6.1 hold. Then for an

c k

s=o

p,& # 0.

(6.8.4)

The eigenvalues are, except for a proportionality fuctor, the simultaneous eigenvalues of (6.7.3), that is to say the sets l o ,... ,1, such that

n Ker(A-’ A, - lsZ)# 0. k

s=o

(6.8.5)

Suppose that Ao, ... , is an eigenvalue, and let f # 0 be in the intersection of kernels (6.8.3), or that k

cA:sAsf = 0 ,

s=o

r = 1,..., k.

(6.8.6)

112

6

SIMULTANEOUSEQUATIONS I N LINEAR SPACES

It then follows from Theorem 6.4.2 that

(ApAq - AqAp)f = 0, p, q = 0, ... , k, and so, multiplying by pp and summing, (AAq - Aq

k

1Appp)f = 0,

q

p=o

=

0,... , k.

(6.8.7)

(6.8.8)

Let us suppose if possible that (6.8.4) is false. It then follows that AqAf = 0, q = 0,... ,k, and since the Aq are not all zero, Af = 0. This is impossible since A is nonsingular, and f # 0. It now follows from (6.8.8) that the set k

Aq’

=

q = 0,... , k ,

A q { x AppP}-’, 0

(6.8.9)

is a simultaneous eigenvalue of (6.7.3), since f lies in the kernel of &‘I - A-‘Aq, q = 0, ... ,k. In the reverse direction, suppose that l o ,... , Ak is a simultaneous eigenvalue of (6.7.3), and thatf # 0 lies in all the kernels in (6.8.5). Thus

&f

= A-’ASf,

s

= 0,

... ,k.

(6.8.10)

It then follows from Theorem 6.6.1 that (6.8.6) holds, and so Ao, ... , j.k is an eigenvalue of the given array. The requirement for an eigenvalue that the As should not be all zero is here fulfilled. On multiplying (6.8.10) by ,us, summing over s and using (6.5.1), we have in fact that psAs = 1.

cko

6.9 Decomposition

Having related the eigenvalues of the array (6.1.8) to those of the commuting operators (6.7.3), we now consider the relation between the corresponding eigenvectors and associated notions. For an eigenvalue A,, , ... ,Ak of (6.1.8), we will understand by the eigensubspace the set of solutions f E G of (6.8.6); any such f will be termed an “eigenvector.” In particular, if

2 A,&,

= 0,

g, E G , ,

s=o

then

f = g, 0 * - .

0 g,

r = 1, ... , k ,

(6.9.1) (6.9.2)

will satisfy (6.8.6), and will be termed a decomposable eigenvector. By Theorem 4.7.2, the eigensubspace is generated by decomposable eigenvectors.

NOTES FOR CHAPTER

6

113

According to arguments of the last section, we have, from Theorems 6.4.9 and 6.6.1, THEOREM6.9.1. With the assumptions of Theorem 6.6.1, the eigensubspace associated with an eigenvahe 10, ... , & is the same as

n Ker(A-' A, - 1,Z). k

s=o

(6.9.3)

It is a particularly important question whether the eigensubspaces for various eigenvalues generate the whole of G , so that the eigenvectors are complete. The answer to this depends on whether the ascent of the operators A-lA, - 1,I is always 1 whenever 1, is an eigenvalue of A-lA,. In general, of course, this need not be so, and we must extend our consideration to root subspaces. By the root subspace associated with an eigenvalue &, .. . , & we mean the subspace

n Ker(A-' k

s=o

A, - A,Z)'",

(6.9.4)

where m is so large that (6.9.4) is not increased when m is replaced by m We then have, by Theorem 3.6.1,

+ 1.

THEOREM 6.9.2. Under the assumptions of Theorem 6.6.1, the space G admits a direct sum decomposition into the root subspaces. If the ranks of eigenvalues are 1 in all cases, we have the simpler situation that the eigensubspaces, the solutions of (6.8.6), provide a direct sum decomposition of G. In Chapters 7 and 10 we investigate certain hermitian situations in which this is the case.

Notes for Chapter 6 The eigenvalue problem of Section 6.8 has been discussed by Carmichael (1921). Since an endomorphism of a linear space does not induce an endomorphism of a direct sum of which it is a summand, we cannot transpose the operators (6.8.1) so as to act on the direct sum of the spaces concerned. What we can do is transpose them so as to act on the tensor product. It is then possible to regard (6.4.3) as a matrix equation, in which a "matrix of operators" A?, acts on a direct sum of k + 1 copies of this tensor product. There is slight contact here with the topic of matrices of operators, though

114

6

SIMULTANEOUS EQUATIONS IN LINEAR SPACES

in a special manner, in that the operators in different rows of the matrix are to commute. See Anselone (1963), Dunford (1965-1966). Some of the results of this chapter will be found in Atkinson (1964); however certain aspects taken up in the next chapter, concerning positive definiteness on decomposable tensors and orthogonalization, were not dealt with there.

CHAPTER 7

SIMULTANEOUS EIGENVALUE PROBLEMS FOR HERMITIAN MATRICES

7.1 Introduction

In this chapter we specialize the field K to the complex field, and will impose conditions of, roughly speaking, hermitian symmetry. The actual formulation can be put in terms of matrices, operators, or hermitian quadratic or sesquilinear forms. In the first of these versions, we suppose given an array of k rows of k + 1 square matrices (7.1.1)

In each row, all matrices are to be of the same size; the size may, however, vary from row to row. The entries in these matrices are to be real or complex, the matrices being hermitian. The matrices in the rth row can be considered as endomorphisms of a complex linear space G, whose elements are column matrices of the corresponding size. 115

116

7

SIMULTANEOUS EIGENVALUE PROBLEMS

We write, as before,

G = GI @

**-

@ Gk,

(7.1.2)

and denote by A!, the endomorphism of G induced by A,, acting on G,. In matrix terms, G is a linear space of multidimensional rather than column matrices, while A?, is a Kronecker product of A,, with k - 1 unit matrices of various sizes. However, we make only minimal use of bases here. As compared with the formalism of the last chapter, we have identified the spaces G,, H, of (6.1.6) and (6.1.7), and so built in the requirement that 0 < dim G, = dim H, < co. Thus all our operators, including the induced operators and their determinantal combinations, will be endomorphisms, and the spaces in commutative diagrams such as (6.2.5) will all be the same. The formulation in terms of matrices implies not only a notion of hermitian symmetry, but also a hermitian conjugacy operation, which we indicate by (*), which takes column matrices into row matrices and taking their entries into their complex conjugates. Thus, for any pair g,, g,’ E G,, we can write g,*A,,g,’ for the product of the row matrix-the complex conjugate transpose of g,, the square matrix A,,, and the column matrix g,’. This defines a sesquilinear form on G, which is in fact hermitian, in the sense that g,*A,g, is real valued. It is possible to adapt the formulation of the last chapter to this situation. One postulates that (6.1.6) and (6.1.7) are supplemented by a set of semilinear maps (*) from C, to the dual H,* of H,, r = 1, ... ,k , and that A,, is hermitian in the sense that g,*A,,g, is real valued, where g,* E H,* is the image under (*) of g, E C,. However, we shall not use this formulation. One can avoid the conjugacy operator (*) by using a scalar product notion ( , ), according to which for g , , g,’ E G,, (g,, g,’) would have the same meaning as gi*g,, and g,*A,,g,’ could be written (A,&’, g,). We shall use such a notion in Chapter 11, but here will keep to the matrix notation given earlier. We have mentioned that the operators or matrices (7.1.1) give rise to a corresponding set of k(k + 1) hermitian forms. This suggests a useful alternative formulation, in which we suppose given, as before, k complex linear spaces G I , ... , G k , and on each a set of k + 1 hermitian sesquilinear forms

This version is particularly suitable for procedures in which we restrict G, to subspaces. From this version we can if we wish go over to a matrix version, provided of course that the G, are finite dimensional. We choose a basis of G,, and

7.2

117

THE FIRST DEFINITENESS CONDITION

introduce a scalar product ( , ) by demanding that the chosen basis elements be orthonormal. We then introduce operators or matrices A,, by requiring that Yrskr’, Sr) = (Arsgr’, gr). 7.2 The First Definiteness Condition and Its Consequences

The situation of an array (7.1.1) of hermitian matrices is, of course, a specialization of that considered in the last chapter from Section 6.4 onward, and carries with it a specialization to the complex field. In return for this we shall seek on the one hand more precise results, and on the other, will investigate different hypotheses concerning the array of maps. In the last chapter, we relied on a hypothesis that some linear combination of operatorvalued minors of the array, of the form (6.5.1) and (6.5.2), was nonsingular. I n the present case, an alternative type of restriction relates to the behavior of the associated quadratic forms. A number of hypotheses are of use, and these are formally distinct, though not necessarily so in fact. Formally, the weakest of these restrictions is what we shall term 1. For all DEFINITENESS CONDITION

s, E Gr,

Rr

sets

# 0, r = 1,

... k , 9

(7.2.1)

1I’C IlU”

(7.2.2) Thus for all sets (7.2.1) at least one, not necessarily the same one in each case, of the k by k minors to be formed from the k by ( k +1) matrix in (7.2.2) is not zero. We now take up consequences for eigenvalues and eigenvectors. The definition of an eigenvalue is, as in Section 6.8, that of a nonzero ( k 1)tuple of complex numbers L o , ... , 2, such that the matrices (6.8.1) are all singular. We say that an eigenvalue is “real” if for some x # 0, the equivalent eigenvalue a&, ... , xL, consists only of real numbers. We have then

+

THEOREM 7.2. I . real.

/f Definiteness Condition I holds, then all eigenvalues are

If i.o,.. . , A, is an eigenvalue, then for some set (7.2.1) we have (6.9.1 ),

7

118

and so

SIMULTANEOUS EIGENVALUE PROBLEMS

k

z(g:A,,g,)&

=

s=o

0,

r = 1,

... ,k.

(7.2.3)

Here the matrix of coefficients has rank k and is real, since the A,, are hermitian. Hence A. , ,1, must be a multiple of some real (k l)-tuple, as was to be proved. A more precise result is

+

...

THEOREM 7.2.2. Under the same condition, $ A o , this ( k 1)-tuple is proportional to

+

g*AOg,

where g

= g, @

”. > g*Akg,

(7.2.4)

... @ gk, the g , are nonzero solutions of (6.9.1), and gl*AIOgl

g*A,g

... ,1, is an eigenvalue,

=

det [gk*AkOgl

*.*

’

***

gl*dlsgl

”’

gl*Alkgl

I

,

gk*dksgk

“*

gk*Akkgk

(7.2.5)

Here the caret ( A ) indicates omission of the column in question, as in (6.4.4), and the determinant is an ordinary scalar-valued one. The assertion, with the interpretation (7.2.5), follows immediately from (7.2.3) by standard determinantal manipulations, in similar fashion to the operator analog (6.4.9). It remains to discuss (7.2.5) itself. We note first that the *-operation on each space G, serves to define a positive-definite hermitian sesquilinear function gr*gr’ [or (gr’, g,)] on G,. By Theorem 4.10.2, this leads to a similar form (g’, g ) on G , which we now write as g*g’, and which, for decomposable arguments, is given by the product of the separate forms, as in (4.10.6). On this basis one has, taking for simplicity the case s = 0, g*AOg = g* =

z U

EuAlu(l)gl @

1Eu(g?Alu(l)gl)

”*

***

@ Aku(k)gk

(gk*Aku(k)gk)?

where E and 0 are as in (6.2.2), and this gives (7.2.5). This gives rise to the notion of a “signed eigenvalue.” We can, by Theorem 7.2.1, require eigenvalues to be real (k + 1)-tuples; by our homogeneity assumption, they are then determinate except for a nonzero real factor. We can then go further and demand that the A, have the same signs as the g*Asg, s = 0, . . . , k , where g has the decomposable form required in Theorem 7.2.2. Here two further comments are needed. One is that the signs are not dependent on the choice of such g , since the sets of g , , ... ,gk in question are connected. Another is that the signs remain the same even if

7.3

ORTHOGONALITY OF EIGENVECTORS

119

g is not decomposable, but is a general nonzero eigenvector in the sense of Section 6.8; however this need not be proved now. In particular, if A. is positive definite, at least when applied to decomposable arguments, we can require that A. = 1, and so go over to an inhomogeneous formulation of the notion of an eigenvalue, as is usual in standard cases.

7.3 Orthogonality of Eigenvectors We continue to investigate consequences of Definiteness Condition I. We use the definition of an eigenvector, decomposable or otherwise (given in Section 6.9), and have THEOREM 7.3.1. Let Definiteness Condition I hold, and let lo,... ,& and lot, ... ,1,' be distinct eigenvalues. Let g, g' E G be associated eigenvectors. Then g*A,g' = 0, s = 0, ... ,k. (7.3.1) We prove this on the supposition that g, g' are decomposable; the general case will then follow by linearity. Along with (6.9.1), we suppose that

c Arsls'gr' k

s=o

=

r

0,

=

1, ... , k .

(7.3.2)

On applying g,* we obtain k

1 g,*A,,g,'l,'

s=o

=

r

0,

=

1, ... , k.

(7.3.3)

... , k.

(7.3.4)

In a similar way, we have from (6.9.1) that

c g;*A,,g,il, k

s=o

=

0,

r

=

1,

We suppose, as we may, that all the A,, As' are real. Then on taking complex conjugates in (7.3.4) we have k

1 g,*A,,g,'l,

s=o

= 0,

r

=

1,

... , k.

(7.3.5)

On comparing (7.3.3) and (7.3.5) we see that the same system of equations has solutions A s , As' which are not proportional, this being our interpretation of the distinctness of eigenvalues. Hence the matrix of coefficients must have rank less than k , and all k by k minors vanish. This gives (7.3.1) in the decomposable case, and so establishes the result generally.

120

7

SIMULTANEOUS EIGENVALUE PROBLEMS

We derive some rather partial results concerning the eigenvectors and eigenvalues; in both of the next two theorems we assume that Definiteness Condition I holds. THEOREM 7.3.2. A set of nonzero decomposable eigenvectors associated with distinct eigenvalues is linearly independent. Let the eigenvectors in question be t = 1,

h ( ') ,

... ,m,

(7.3.6)

and suppose that for scme scalars c, we have rn

C c,h(') = 0.

1=1

(7.3.7)

Then for any u, 1 I u Im, we have

C cIh(u)*Ash(l) = 0, rn

s = 0, ... ,k.

t=O

(7.3.8)

Since the A(') are associated with distinct eigenvalues, the orthogonality yields c,h(")*A&(U)= 0, s = 0, ... , k . (7.3.9) Since h'") # 0, Definiteness Condition I implies that the coefficients of c, in (7.3.9) are not all zero, and so we have c, = 0, which proves the result. The second is a bound on the number of distinct eigenvalues. We write k

v = dim G = n d i m G,, r=l

(7.3.10)

and have THEOREM 7.3.3. The number of distinct eigenualues does not exceed v. This follows from the last result, together with the observation that to each eigenvalue there corresponds at least one decomposable eigenvector, of the form (6.9.1) and (6.9.2). Theorems 7.3.1-7.3.3 take us some way toward the proposition that there exists a set of decomposable eigenvectors, orthogonal in the sense (7.3.1), which are complete in G. We shall complete the proof of this result, on the basis of Definiteness Condition I, in Chapter 10. For the present, we shall continue using more restrictive hypotheses.

7.4

STRONGER DEFINITENESS CONDITIONS

12 1

7.4 Stronger Definiteness Conditions

We now retreat to less general assumptions, with a view to being better able to advance later on. We introduce two progressively stronger conditions; the strongest of the three conditions thus produced will enable us to complete the proof of the completeness of the eigenvectors. It should be mentioned that this condition covers at least one important case, that of the discrete analog of multiparameter Sturm-Liouville theory, which can serve as a foundation for the continuous case. In Definiteness Condition I, we postulated that at least one k by k minor of the matrix (7.2.2) was not zero, the choice of the minor not being specified, and possibly varying with the choice of (7.2.1). It will be a more restrictive 1 minors is not condition if we specify that some particular one of these k to vanish or, more symmetrically, that some fixed linear combination of these minors should not vanish. This is the content of

+

DEFINITENESS CONDITION 11. For somefixed set ofreal scalars p o , .. . , P k , and aI1 sets (7.2.1.), we have

(7.4.1)

We remark that if we assume this determinant not to vanish then, by the connectivity of the nonzero part of a complex linear space, we have that this determinant has fixed sign. This sign can be made to be positive by appropriate choice of the sign of the ps. With the notion (6.5.1), we can write (7.5.1) in the more succint form

to hold for all nonzero decomposable tensors g. It is formally a stronger requirement to ask that this hold for all nonzero g, decomposable or not. This we term DEFINITENESS CONDITION 111. There exist real scalars p o , ... , P k such that (7.4.2.) holds, with A given by (6.5. l), for all nonzero g E G . Definiteness Conditions I, 11, and I11 are formally in order of increasing strength. We discuss later whether they are really so. It turns out that

122

7

SIMULTANEOUS EIGENVALUE PROBLEMS

Condition I1 is more restrictive than Condition I if k 2 3 only, whereas Condition I11 is really equivalent to Condition 11. We shall proceed on the basis of Condition I11 and note that since A is here to be positive definite, it is nonsingular, so that the results of the last chapter, from Section 6.5 onward, are available. In particular, Section 6.9 provides a basis for asserting the completeness of the set of eigenvectors. At this time, we formulate the result in terms of the distinct eigenvalues Aot,

...

t = 1,

Akt,

.1.

y

m,

(7.4.3)

and the associated eigensubspaces, which we shall denote by G@),

t

1,

=

... ,m.

(7.4.4)

The number of (7.4.3) was shown to be finite in Theorem 7.3.3. In accordance with Section 6.9, C(‘) is the set off E C satisfying

c A:J,,f k

r = 1, ... ,k .

= 0,

s=o

(7.4.5)

THEOREM 7.4.1. Let Definiteness Condition 111 hold. Then there is a direct sum decomposition

G

c G(‘). m

=

i= 1

(7.4.6)

This follows from Theorems 6.9.1 and 6.9.2, subject to it being proved that the eigenvalues of A - l A , , s = 0, ... , k, all have rank 1. This follows from the fact that the inner product g*Ag is positive definite on G , together with the fact that the endomorphism A - l A , is hermitian with respect to this product, since g*A(A-’A,)g = g*Asg is real valued. We define the “multiplicity” of an eigenvalue I,,, , ... , & as the dimension of the corresponding eigensubspace, and so as

ndim Ker c A,,&. k

k

r=l

s=o

(7.4.7)

An eigenvalue is “simple” if its multiplicity is 1, so that all the kernels in (7.4.7) are one dimensional. We have THEOREM 7.4.2. If Definiteness Condition 111 holds, and f a l l eigenvalues are simple, then there exists a complete set of decomposable eigenvectors, orthogonal in the sense (7.3.1). Without assuming the simplicity of the eigenvalues, we have at present a weaker result.

7.5

SPLITTING OF MULTIPLE EIGENVALUES

123

THEOREM 7.4.3. Let Definiteness Condition III hold. Then there is a complete set of eigenvectors, orthogonal in the sense (7.3.1), where to each eigenvalue is associated a number of eigenvectors equal to its multiplicity. At this stage, we cannot assert that in the case of a multiple eigenvalue, there exists a mutually orthogonal set of decomposable eigenvectors, equal in number to the multiplicity. This will be taken up later. If an eigenvalue is simple, the eigensubspace is generated by a single decomposable eigenvector, and so Theorem 7.4.2 follows from Theorems 7.4.1 and 7.3.1. Concerning Theorem 7.4.3, we note that, by Theorem 7.4.1, if for every eigenvalue we select a linearly independent basis of the corresponding eigensubspace, we shall obtain a complete set of eigenvectors, in which those associated with distinct eigenvalues are orthogonal, by Theorem 7.3.1. Thus all we have to do is to arrange that for each eigenvalue, the associated basis elements of the eigensubspace are mutually orthogonal. We can arrange this by orthogonalization, with respect to the positive definite inner product A; however, there is no reason to suppose that the standard orthogonalization process will, in general, yield decomposable tensors. It still remains to be proved that if we orthogonalize an eigensubspace with respect to A, the resulting set will be orthogonal in the sense (7.3.1). We omit the details of this since we shall prove stronger results later on.

7.5 Splitting of Multiple Eigenvalues In the last section we noted, subject to Definiteness Condition 111, that if all eigenvalues were distinct, then there existed a complete set of orthogonal decomposable eigenvectors. We use this fact to establish a similar proposition in the case of multiple eigenvalues ; here a straightforward orthogonalization procedure need not yield decomposable eigenvectors, except in the case that not more than one of the kernels in (7.4.7) has dimension greater than 1. We discuss the problem by means of a perturbation method. We show that by arbitrarily small perturbations of (7.1.1) one can achieve that all eigenvalues are simple, for which case there will be a complete set of mutually orthogonal decomposable eigenvectors. A limiting process then yields this situation for the original array (7.1.1). In order to discuss continuous dependence on matrices, we use as a norm, in the space of matrices of some given size, the sum of the absolute values of the entries. We are concerned with Definiteness Condition 111.To minimize notational

124

7

SIMULTANEOUS EIGENVALUE PROBLEMS

requirements, we shall carry out the argument for the case that A, > 0 on G, or that p, = 1, pl = = ,uk = 0. We then show that the general case can be reduced to this by a transformation. As our first preliminary result we need

LEMMA7.5.1. Let A, > 0. Then the eigenvalues depend continuously on the matrices in the first column of (7.1.1), in a sense to be specijied.

To accomplish the latter, we require first that eigenvalues (A,, .. . , 2,) be normalized in some way. We require that all the As be real, as we can by Theorem 7.2.1, and that A, > 0, as we can, in view of Theorem 7.2.2, with our current assumption that A, > 0. We can then determine the A, precisely A: = 1. An eigenvalue then becomes a point on one half by asking that of the unit sphere in Rk+l,on which we use the topology induced by the usual topology on R k + ' . We depart now from the formulation (7.4.3) in terms of the distinct eigenvalues of an array, and suppose that each multiple eigenvalue is repeated in the set of eigenvalues according to its multiplicity. By Theorem 7.4.1, the total number of eigenvalues will be precisely v, the dimension of G, as in (7.3.10). Our assertion is that for any set of neighborhoods U1, ... , U, of the respective eigenvalues of (7.1.1), we can find neighborhoods V , , ... , Vk of the matrices A',, ... , A k O such that for hermitian matrices

1;

A : , € V,,

r

=

1, ... , k ,

(7.5.1)

the perturbed array

has eigenvalues that can be numbered, according to multiplicity, so that the mth eigenvalue occurs in Urn,m = 1, ... . It will be sufficient to prove this on the assumption that the Urncoincide in the case of coincident eigenvalues, and are disjoint otherwise. The proof goes by contradiction. We suppose that for a certain choice of neighborhoods Urnthere exist sequences of hermitian matrices

ArOn-tAro,

n = 1 , 2,..., r = 1,..., k,

(7.5.3)

7.5

SPLITTING OF MULTIPLE EIGENVALUES

125

such that the eigenvalues of the arrays

cannot be numbered in this way. Our first remark is that a convergent sequence of eigenvalues of (7.5.4) must have as its limit an eigenvalue of (7.1.1). For if we have

where the eigenvalue (pan, .. . , P k n ) is normalized as above and converges to (Ao, ... ,A k ) , we may choose a subsquence such that the gr, tend to nonzero limitsg,, r = 1 , ... , k , and then in the limit we have (6.9.1) so that (Ao, .. . ,2,) is an eigenvalue. It follows from this that, for large n, all eigenvalues of (7.5.4) lie in one or another of the Urn.What we must show is that they lie there in the correct numbers, that is to say in numbers equal to the respective multiplicities of the eigenvalues of (7.1.1) associated with these Urn. The proof will be by contradiction. We note that the total numbers of eigenvalues of (7.1.1) and (7.5.4) is the same, with regard to multiplicity. Thus if the assertion is not correct, there will be a deficiency of eigenvalues of (7.5.4) in some of the Urnand an excess in others. It will therefore be sufficient to dispose of the possibility that there is one Urn,and an infinite n-sequence, such that the number of eigenvalues of (7.5.4) in Urnexceeds the multiplicity of the eigenvalue of (7.1.1) which lies in Urn. Let CT be the multiplicity of the eigenvalue of (7.1.1) lying in Urn.We suppose that for an infinity of n, Urncontains at least CT + 1 eigenvalues of (7.5.4), these being counted according to multiplicity. For such n, we form an orthonormal set of eigenvectors hr,€G,

t=l,

in the sense

1

..., a + l ,

1

h;",A0lt,.,

=

(t

=

(7.5.6)

t'),

(7.5.7)

0

(t

# t'),

associated with eigenvalues which we may conveniently number Pot,,

...

9

Pkm,

=

**.

Y

CT

+

(7.5.8)

126

7

SIMULTANEOUS EIGENVALUE PROBLEMS

so that r=l,

... , k ,

t=l,

..., o + l .

(7.5.9)

We now make n tend to infinity through the sequence in question. Since eigenvalues can only tend to eigenvalues, the eigenvalues (7.5.8) must all tend to the eigenvalue of (7.1.1) associated with Urn.The elements of the orthonormal set (7.5.6) form a bounded set, from which we can extract a convergent subsequence. Then on proceeding to the limit in (7.5.7) and (7.5.9), we obtain an orthonormal set of o 1 eigenvectors, associated with the eigenvalue in question of (7.1.1). This contradicts the assumption that this eigenvalue has multiplicity CT,and completes the proof of the lemma. As a special case we have, heuristically expressed,

+

LEMMA7.5.2. If A. > 0, then a simple eigenvalue of (7.1.1) remains simple under suitably sniall perturbations of the Jirst column of (7.1.1). More precisely, a neighborhood U of a simple eigenvalue of (7.1.1), not containing any other eigenvalue of (7.1. l), has the property that there exist neighborhoods V, of the ArO such that if (7.5.1) holds, then U contains precisely one eigenvalue of (7.5.2)-this eigenvalue being simple. We now begin a procedure for “simplifying” multiple eigenvalues of (7.1.1) by perturbing the matrices in the first column. Suppose that A. , ... , A, is a multiple eigenvalue of (7.1. l), so that for some r, 1 r -< k , the kernel of A,,& has dimension greater than 1. We consider a perturbation of the form (7.5.10) ArO(E) = 4 0 + E C r , where E is real, and C, hermitian. We choose the C,, r for small real nonzero E, the operators k

=

1, ... ,k, such that (7.5.11)

all have one-dimensional kernel. Passing over those r such that this is so with C, = 0, we achieve this by requiring that C, be positive semidefinite, &As. The effect of this with one-dimensional kernel included in Ker ct=o is that for small E # 0, Ao, ... ,1, will be a simple eigenvalue of the array

[

AIO(E)

A k o (E)

All

‘**

Alk

!--

A]’

(7.5.12)

7.6

DECOMPOSABLE ORTHOGONAL EIGENVECTORS

127

Also, for

E in some suitably small neighborhood of the origin, any other simple eigenvalues of (7.1.1) will be perturbed into simple eigenvalues. Thus, for all suitably small real E # 0, (7.5.12) will have more simple eigenvalues than (7.1.1).

This procedure can be repeated until only simple eigenvalues remain, the number of repetitions being clearly less than v. We sum up the result as LEMMA 7.5.3. If A. > 0, then for any neighborhoods V, of Are, r = 1, ... , k , there exist (7.5.1) such that the eigenvalues of (7.5.2) are all simple. 7.6 Decomposable Orthogonal Eigenvectors

We can now take a further step in improving the result concerning the completeness of the eigenvectors of (7.1.1). This consists in dealing with the aspect mentioned at the end of Section 7.4 as to whether the eigenvectors could be assumed decomposable. Another improvement, concerning the replacement of Definiteness Condition I11 by Condition 11, will be undertaken in Section 7.9. We prove now

THEOREM 7.6.1. Let De$niteness Condition III hold. Then the eigenvalues can be numbered, according to multiplicity, as

A$), ... ,/If)

t =

1, ... , v,

(7.6.1)

with associated decomposable eigenvectors, none zero, h")

=

hy) 8

-.. 8 hf),

t = 1, ... , V,

(7.6.2)

... , k.

(7.6.3)

such thut

h(')*A&('') = 0,

t#

t',

s = 0,

The argument of Theorem 7.3.2 shows that the h(') are linearly independent, and so form a basis of G . We suppose first that Definiteness Condition I11 is specialized to do > 0, to hold on the whole of G. By Lemma 7.5.3, there exist sequences (7.5.3) such that the arrays (7.5.4) have only simple eigenvalues. Then, by Theorem 7.4.2, for these arrays the assertion of Theorem 7.6.1 is true. The result then follows for the array (7.1.1) on making n --f oc) through a suitable subsequence. To sketch the details, we write the eigenvalue equations for (7.5.4) in the form {A,o,/I$")

+ s = lA,,,/ISf.")}h!t,")= 0, k

r = 1, ... k,

(7.6.4)

128

7

SIMULTANEOUS EIGENVALUE PROBLEMS

with the orthogonality relations h("")*A 0k t'," = 0,

t # t',

(7.6.5)

where /I('>") is defined similarly to (7.6.2). We can normalize the eigenvalues, for example, as in the last section, and the eigenvectors, for example, by h$")*h!fs") = 1. We can then choose a subsequence such that all these entities converge. In the limit, we obtain the theorem, with s = 0 in (7.6.3); the remaining cases of (7.6.3) can be proved by a similar limiting process, or by Theorem 7.2.2. It remains to remove the specialization of A = psAs to Ao. Supposing A > 0, we make a linear transformation of the A,, setting

1;

2 mstxt, k

As

=

s

t=o

= 0,

... , k,

(7.6.6)

where the mSt have positive determinants, and will be chosen later. We introduce new elements of End(G,), r = 1, ... ,k, by k

Brt=CArs~st, t = O s=o

,..., k , r = l , ... k ,

(7.6.7)

and then have k s=o

k

A,,& =

1B , , X ~ ,

r

1,

=

t=O

... , k.

(7.6.8)

One checks easily that we may apply the usual rules for the multiplication of determinants, to the effect that

k

k

(7.6.9)

where the B,!s are the induced operators on G. In justifying this, we rely on the property of operator-valued determinants that one with two identical columns vanishes, together with the linear dependence on columns.

7.7

We now suppose the k

1 0

PS%O

=

129

A CONNECTEDNESS PROPERTY

clst

1,

chosen so that

c k

p s g , = 0,

s=o

t = 1,

... , k,

(7.6.10)

and so that they have positive determinant, which can certainly be done for k 2 1. We then reach the situation that the operator-valued determinant

is positive definite. We can now appeal to the special case already treated. The eigenvalues of the array (7.6.7) are related to those of (7.1.1) by the nonsingular linear transformation (7.6.6), while the eigensubspaces will be the same. We thus have that there exists a complete set of mutually orthogonal decomposable eigenvectors, the orthogonality being with respect to (7.6.1 l), and so with respect to A. It then follows from Theorem 7.2.2 and the fact that A > 0, that we also have orthogonality with respect to the As. This completes the proof of Theorem 7.6.1. As an incidental consequence we have a result for square arrays A,,, 1 5 r, s 5 k , of hermitian matrices, those in any one row being of the same size. With the previous notation we have

THEOREM 7.6.2. Let A. > 0. Then there is a complete set of nonzero decomposable tensors that are orthogonal with respect to Ao. This follows from the last theorem if we augment the given square array to a rectangular array (7.1.1) by the addition of an arbitrary first column of hermitian matrices of the appropriate sizes. 7.7 A Connectedness Property

We now move on toward the elucidation of the distinction we drew between Definiteness Conditions I1 and 111. It is a question of the positive definiteness of an operator-valued determinant on all tensors, or on decomposable tensors only. In this section, we prove a preliminary result that will be needed in one treatment of this matter.

130

7

SIMULTANEOUS EIGENVALUE PROBLEMS

We are now concerned with a square array All

*’*

(7.7.1) Ak] of hermitian matrices, those in any one row being of the same size, the entries being real or complex. As before, we may consider the A,, as endomorphisms of spaces G, of column matrices, and will denote by G the tensor product of the G,, and by ATs the induced endomorphisms of G. We depart from previous notations in writing Af1 A

=

-.*

det(~il:..

Ark

(7.7.2) Al)

for the associated operator-valued determinant. We shall consider A as a function of varying hermitian matrices A,. By saying that A is positive definite on decomposable tensors we shall mean that the kth-order determinant detk,*Arsgr)ls r , s _ c k

’OY

(7.7.3)

for all nonzero sets g,, ... ,gk of column matrices of the appropriate lengths. We have then

LEMMA 7.7.1. Tlie set of A’s that are positive definite on decomposable tensors is connected. We show that an arbitrary A with this property can be continuously varied, retaining the property into a special case in which the array (7.7.1) has diagonal form-the diagonal entries being unit matrices. In this latter case, A reduces to the identity map of G, and is positive definite on the whole of G, by Theorem 4.10.4. We choose some nonzero hl E G , , and write a,, = hl*Alsh1, It then follows that a,,

s = 1,

... , k.

(7.7.4)

1 (7.7.5)

7.7

for all nonzero sets g,, we have

A CONNECTEDNESS PROPERTY

... ,gk.

131

Thus if Il denotes the identity map of G1,

(7.7.6)

for all nonzero sets g, , ... , gk, since the determinant in (7.7.6) is the same as that in ( 7 . 7 3 , except for the top row, which is a nonzero multiple of that in (7.7.5). Thus if we write

(7.7.7)

we shall have that A' has the same form of a determinant of hermitian matrices, which is positive definite on decomposable tensors. We then note that A, A' can be connected within the set of such determinants. The endomorphism

(1 - z)A

+ TA',

0 I z I 1,

(7.7.8)

clearly forms a path connecting A, A', is positive definite on decomposable tensors, and can be written in the form of a determinant of hermitian matrices, in which the top row is a linear combination of the top rows of A,A', and which coincides with them in the other rows. We repeat the process, applying it next to the second row of A', so that A' can be connected to A", an endomorphism-valued determinant that arises from an array with multiples of 11,I2 in the top two rows, the remaining rows being the same as those of A, A'. In this way, we ultimately connect A to an endomorphism arising from an array of the form ursZ,,where I, is the identity on G,. As an endomorphism, this has the value Idet(u,,), where I is the identity on G. Here det(u,,) must be positive, since we remain throughout in the set of endomorphisms which are positive definite on decomposable tensors. We may therefore connect det(u,,) to 1 by a path through real nonzero scalars, and so connect Idet(u,,) to I. This completes the proof.

132

7

SIMULTANEOUS EIGENVALUE PROBLEMS

7.8 The Main Result on Positive Definiteness It will be convenient to prove first LEMMA 7.8.1. Let A have the form (7.7.2) arisingfrom the array (7.7.1)

of hermitian matrices, and let A be positive definite on decomposable tensors. Also let there be a set of v = dim G nonzero decomposable tensors

t

h(t),

=

1,

... ,v,

(7.8.1)

which are orthogonal with respect to A, so thut h(t)*Ah(t')= 0,

t # t'

(7.8.2)

Then A is positive definite on G.

We first note that the elements (7.8.1) are linearly independent, and so form a basis of G. For suppose that

h

c cth(') V

=

=

t= 1

0,

(7.8.3)

for some scalars c t . It then follows that

h* Ah =

5 l ~ , l ~ h Ah'" ( ~ ) * = 0. 1

(7.8.4)

By the positive definiteness assumption for decomposable arguments, we have h(')*Ah(')> 0, t = 1, ... , V , (7.8.5)

and so all the ct are zero, as was to be proved. It follows that a general element g E G can be written in the form

c dth"'. V

g =

1

We then have g*Ag

C Idr12h(')*Ah('), I

=

1

(7.8.6)

(7.8.7)

and by (7.8.5) this is positive unless all the d, are zero, and so unless g = 0. This proves the lemma. We can now prove the main result given by THEOREM 7.8.2. Let A be the endomorphism (7.7.2) arising from the array (7.7.1) of hermitian matrices. Then in order that A be positive definite on G it is suficient thut it be positive definite on decomposable tensors.

7.9

THE EIGENVECTOR EXPANSION

133

By Lemma 7.7.1, there exists a continuous path A(7), 0 I7 I 1, in End(G) formed by endomorphism-valued determinants that are positive definite on decomposable tensors, with A(0) = I, A(l) = A, each A(7) arising from an array of hermitian matrices. It is clear that A(0) is positive definite on G as a whole. Since the set of positive definite endomorphisms is open, A(7) will be positive definite for 7 in some right-neighborhood of 0. If it is not true that A(1) = A is positive definite, we define 7' as the largest number in (0, 13 such that A(7) > 0 for 0 I7 < 7 ' . We have 0 < 7 ' 5 1, and note that A(t') will not be positive definite; this is so if 7 ' = 1 since otherwise our contradiction hypothesis is violated. It is also so if 7 ' < 1, since if A(7') > 0, we should have A(7) > 0 for 7 in some right-neighborhood of 7 ' , violating the definition of 7 ' . We shall obtain a contradiction if we prove that the set of 7 such that A(T) > 0 is closed. By Theorem 7.6.2, for such 7 there will be a set of nonzero decomposable elements of G, say h(")(7),

u = 1,

... , v ,

(7.8.8)

which are orthogonal in the sense h'"'")*A(t)h'""(~) = 0,

u # u'.

(7.8.9)

We normalize them in some way, for example, by h'"'(r)*h'"'(r) = 1,

u =

1, ... , v.

(7.8.10)

Let t 1 , 7 2 , ... be a sequence of points such that A(7,,) > 0, n = 1,2, ... , with T,, + T,, as ii -+ OD. We put t = T ~ z 2, , ... in (7.8.8)-(7.8.10) select a subsequence such that we have convergence in the sequences h(")(T,,),

U =

1,

... , v,

(7.8.1 1)

and make ii --f OD. We thus find that A(T~)admits an orthogonal set of v nonzero decomposable elements of G. From Lemma 7.8.1 it then follows that A(70) > 0 on G, so that the set of 7 with A(7) > 0 is closed. This completes the proof. The special case of an array (7.7.1) of diagonal form, so that A,, = 0 if r # s, is essentially what was treated in Theorem 4.10.4. 7.9 The Eigenvector Expansion We can use the results of the last four sections to make certain improvements in Theorem 7.4.1, which asserted the completeness of the eigenvectors of (7.1.1). These improvements consist of a formal relaxation of the definiteness condition, and an assertion that the eigenvectors can be chosen both

134

7

SIMULTANEOUS EIGENVALUE PROBLEMS

mutually orthogonal and decomposable ; the latter improvement is material in the case of multiple eigenvalues, and has already been noted in Theorem 7.6.1. Summing up, we have 7.9.1. Let Dejiniteness Condition II (7.4.1) hold. Then there is THEOREM a set of v real eigenvalues (7.6.1), with eigenvalues repeated according to multiplicity, and a corresponding set of v decomposable eigenvectors, which form a basis of G, and which are orthogonal in the sense (7.6.3). As compared with Theorem 7.6.1, we have replaced Definiteness Condition 111 by Condition 11; that this could be done was the content of Theorem 7.8.2. Returning to our previous notation, we use A in the sense (6.5.1), which, according to Definiteness Condition 11, is positive definite on G. Thus, with the notation (7.6.2) for the eigenvectors, we shall have

h(I)*Ah(I')= 0

( t # t')

h(')*Ah('')> 0

( t = t').

(7.9.1)

We can use these to express the eigenvector expansion more explicitly. For an arbitrary g E G we have a unique expression (7.9.2) It then follows from (7.9.1) that h("*Ag = c,h(')*Ah('),

t =

1, ... ,

(7.9.3)

and so we have (7.9.4)

A useful version of this is the Parseval equality, which can be used for a limiting treatment of infinite dimensional cases. This is 7.9.2. For any g THEOREM

E

G, we have

g*Ag = C (g*Ah'")(h'"*Ag)/(h(~)*A~~(l)). V

1

(7.9.5)

This results on inserting the expansion (7.9.4) for g on the left, and using the orthogonality (7.9.1).

NOTES FOR CHAPTER

7

135

Notes for Chapter 7 The case of the finite-difference analog of multiparameter SturmLiouville theory was treated in Atkinson (1963, 1964); it can serve, by means of a limiting process, to establish the completeness of the eigenfunctions in the differential equation case [see Faierman (1969)l. This case has some simplifying features. First, the eigenvalues are necessarily simple, in the second-order case with one-point boundary conditions; thus appeal to Sections 7.5 and 7.6 of this chapter becomes unnecessary. Second, the operator-valued determinant A. has in this case entries which are diagonal matrices ; the positive-definiteness property of Theorem 7.8.2 can be established more simply in this case. Third, the completeness of the eigenfunctions, in the discrete case, can be established by an independent method. By oscillatory arguments, one can show that the number of eigenvalues is equal to the dimension of the tensor product space [Atkinson (1964), Sect. 6.10)]; since the eigenvectors are linearly independent, they must span this space. The discussion of the eigenvalue problem of this chapter becomes difficult if we drop all assumptions of a hermitian character concerning the matrices A,, . The topic seems to call for more elaborate algebraic machinery, namely that of graded modules over polynomial rings; a treatment was sketched in Atkinson (1968).

CHAPTER 8

THE SINGULARITY OF SQUARE ARRAYS

8.1 Introduction

We now return from the complex field to a general algebraically closed field K of characteristic zero, and deal with a point raised in Section 6.2, and dealt with in a special case in Section 6.3. For the set (6.1.2) of k homogeneous equations in k unknowns it is trivially necessary and sufficient for the existence of a nontrivial solution that the determinant of coefficients vanish. In the operator analog we have a variety of analogs of the homogeneous equations that lead trivially to the singularity of a certain determinant whose value is an operator; what is less trivial is that the singularity of this determinant ensures the nontrivial solubility of these analogs of the homogeneous equations. The resolution of this question will provide an alternate treatment of the question of the positive definiteness of a determinantal endomorphism, in the hermitian case, on decomposable and general tensors; this was dealt with in the last chapter-the concluding argument in Section 7.8. We consider an array A,,,

r , s = 1,...,

136

k,

(8.1.1)

8.2

EQUIVALENT SINGULARITY CONDITIONS

137

where A r l , ... ,Ark may be thought of as square matrices of the same size, possibly varying with r, or as linear maps from Gr to H,, where G,, H, are linear spaces of the same, positive, finite dimension; with the latter formulation, we write as usual G = G1 0 ... 0 G,, and indicate induced operators by the symbol t. 8.2 Equivalent Singularity Conditions

As already mentioned, the scalar homogeneous equations (6.1.2) have several analogs in the operator case, the relations among which have been partly settled in Section 6.1. Here, for completeness, we list them along with the determinantal condition whose relation to them is our main subject in this chapter. As our principal result on this matter we give

THEOREM 8.2.1. The following conditions on the array (8.1.1) are equivalent: (i) There exist A , ,

... ,Ak E K,not all zero, such that

Ker

c A,,& # 0, k

s= 1

(ii) The polynomials

det

c Ar,As, k

s= 1

r = 1, ... ,k.

r = 1,

... ,k,

(8.2.1)

(8.2.2)

have a common nontrivial zero. (iii) There exist A 1 , ... , A k , not all zero, such that

Ker

k

# 0,

s= 1

r = 1,

... , k .

(8.2.3)

(iv) There exist A,, ,.. , A,, not all zero, such that

n Ker c A!sAs # 0.

(0)

k

k

r=l

s=l

(8.2.4)

There exist f l , ... ,fk E G, not all zero, such that (8.2.5)

(vi) The map (6.2.1), which we denote now by A, is singular.

From Section 6.1 we know that (i)-(iv) are equivalent and imply (u),

138

8

THE SINGULARITY OF SQUARE ARRAYS

whereas in Theorem 6.2.3 we noted that ( v ) implies (vi). To prove the theorem we must show that (vi) implies (i). One of the possible proofs of the theorem is by induction, and it is this one that we shall follow. The theorem in the case k = 1 is trivial, and was proved in Section 6.3 for k = 2. We shall proceed by induction over k . 8.3 An Algebraic Lemma

We need the following result from polynomial algebra, and for completeness will include a proof. It concerns polynomials with scalar coefficients in A1, ... ,Ak which are homogeneous, in the sense that each term in the polynomial has the same total degree, each of the A, having degree 1.

LEMMA8.3.1. Let be homogeneous polynomials, of various positive degrees. Then there are only the following two possibilities.

(i) The polynomials have only a ,finite number of common zeros and there exists ajirst-degree polynomial that has no common zero with them. (ii) The polynomials have an injiiiity of common zeros and have a common zero with any other polynomial of positive degree. Here by "zero" we mean a set of values of the A,, ... , &, not all zero, which on substitution reduce the polynomials in question to zero. The zeros are to be considered projectively; two zeros of the form A,, ... , and c d l , . . . , d k are not considered distinct. It is easily seen that if the polynomials (8.3.1) have only a finite number of common zeros, we can find a linear form that does not vanish at any of them; the case of a finite field is excluded by our hypothesis of characteristic zero. Thus what must be proved is that if they have an infinity of common zeros, then any other homogeneous polynomial, not a constant, will have a common zero with them. Suppose then that (8.3.1) have an infinity of common zeros Aim'

, ... ,Aim) ,

m = 1,2, ... ,

(8.3.2)

which are distinct in that no set is proportional to any other set. We then note that the expressions u p

=

k

( C I~")xS}", s= 1

n = 0, 1,

... ,

(8.3.3)

8.3

139

AN ALGEBRAIC LEMMA

all satisfy the formal partial differential equations

p,(d/dX,

7

1.1

7

r

d/dxk)u = 0,

= 1,

... ,k

- 1.

(8.3.4)

Here differentiation is to be carried out according to the usual rules for polynomials; this procedure does not depend on the possibility of defining differentiation by a limiting process. Of course, u will satisfy (8.3.4) trivially if it is a polynomial in the x, of degree less than that of pr ; we are, however, concerned with the polynomials (8.3.3) when n is large. We write N for the set of all homogeneous polynomials u in x 1 , ... ,xk satisfying (8.3.4), and N , for the subset of N formed by homogeneous polynomials of degree n. This set N , will be a linear space. We claim that dim Nn --f

00,

n + 00.

(8.3.5)

We note that N, contains an infinity of elements (8.3.3). We prove (8.3.5) by showing that as n increases, the number of (8.3.3) that are linearly independent increases indefinitely. We choose any integer M , and wish to show that there exists an n' such that all m = 1, ... ,M ,

ui"'),

(8.3.6)

are linearly independent for n > n'. For this purpose, we construct a set of homogeneous polynomials Pt(A1,

...

7

Ak),

f =

1, ... M , 7

(8.3.7)

which vanish at all but one of the points (8.3.2) for rn = 1, ... , M , P , not vanishing at the point (8.3.2) for which rn = r. For example, let us construct M linear forms that vanish, respectively, at the lst, 2nd, ... , Mth of the points (8.3.2), but not at any other of the first M points (8.3.2). Then the P , can be products of M - 1 of these forms, and will be of degree M - 1. With the latter choice, for definiteness, we observe then that, if n 2 M - 1 we have P,(d/aXl, ... 8/8Xk)uim)= 0

( t # rn),

P,(a/axl,... ,a/dx,)u~"') # 0

( t = m), 1 I t , m IM ,

7

(8.3.8)

in view of our hypothesis concerning the zeros of P, as a polynomial. This shows that the polynomials (8.3.6) are linearly independent for n 2 M - 1, and so proves (8.3.5). Consider now any other homogeneous polynomial pk(Al , ... ,&), of positive degree q, say. We wish to show that it has a common zero with

140

8 THE SINGULARITYOF SQUARE ARRAYS

(8.3.1). We consider the corresponding partial differential operator acting

on elements of N according to

p,(a/axl , ... , a/ax,): N,,,,

+ N,,,

n = 0, 1,

... ;

(8.3.9)

it follows from (8.3.5) that for an infinity of n this map has nonzero kernel and consequently, by applying differentiations, for all n. In other words, the augmented system of differential equations

p,(a/axl , ... , a/ax,)u = 0,

r = 1, ... ,k,

(8.3.10)

has nontrivial homogeneous polynomial solutions in all degrees. We remark now that the solutions of (8.3.10) are annihilated by all homogeneous partial differential operators in the ideal generated by pl, ... ,pk in the graded algebra of homogeneous partial differential operators, and so this ideal must be a proper homogeneous ideal, being a strict subset of the algebra in every degree. This statement may be made equally in terms of the polynomials p r ( l l , ... ,Ak), r = 1, ... , k ; they generate an ideal that is proper in the graded algebra of homogeneous polynomials in 1,, ... , being a strict subset in every degree. It follows from a theorem of Hilbert that the polynomials must have a common zero, as was to be proved. 8.4 The Inductive Argument

We must show that the singularity of A, as given by the right of (6.2.1), implies the existence of 1,, ... , A,, not all zero, such that A,,I,, r = 1, . .. ,k, are all singular. This is tautological if k = 1, and has been proved in Section 6.3 if k = 2. Here we suppose it proved for k = n - 1, where n 2 2, and deduce it for k = n. We are dealing with an array of maps All

.-*

4" (8.4.1)

[Anl

A],

those in any one row being effectively square matrices of the same order. We form the determinants, homogeneous polynomials of various positive degrees, detsi=A1 r S A s ,

r = 1, ... ,n.

(8.4.2)

If the first n - 1 of these have an infinity of common zeros, then by the lemma they will have a common zero with the nth polynomial in (8.4.2), and

8.4

141

THE INDUCTIVE ARGUMENT

the result will be proved; here we follow the conventions of the last section in counting zeros. If, on the other hand, the first n - 1 of (8.4.2) have only a finite number of common zeros, we can find a linear form (8.4.3)

1

that does not have a common zero with them. We then claim that the deterniinantal map

A'

=

det

(8.4.4)

is nonsingular; this is a map of GI0 .-. 0 Gn-l, and the induced operators in this array act on this space or on other tensor products as in (4.6.8). To prove this, we note first that the ps are not all zero. We treat explicitly the case pl # 0, and in fact will take p l = 1; other cases may then be reduced to this by simple transformations. Since column manipulations are permitted, we may replace (8.4.4) by

A' = det

i

...

0 Ai2

An-1.1

AA-1.2

- pZAf1

0

Afn - pnA+

* a *

- ~zAnt-1,1 *..

t

An-l,n

- P n A nt- 1 . 1

1

(8.4.5)

and so by the minor given by the last n - 1 rows and columns of (8.4.5). We appeal now to the case k = n - 1. If the minor formed by the last n - 1 rows and columns of (8.4.5) were singular, there would exist p z , ... , p n , not all zero, such that the operators

are all singular. Thus if n

A1

= -CpSps, 2

A,

= pS, s = 2,

... , n,

(8.4.7)

we shall have that n

CArsA,,

s= 1

r = 1,

... , n - 1

(8.4.8)

142

8

THE SINGULARITY OF SQUARE ARRAYS

are all singular, while (8.4.9) and the 1, are not all zero. This contradicts our assumption concerning the linear form (8.4.3). We may therefore take it that (8.4.4) is nonsingular. We now form the array

i:l 0

A1 1

I

(8.4.10)

... An-1.n

An-1.1

An 1

Ann

where the map An, has similar action to A , ] , ... ,A,,, and is chosen to be nonsingular. From this n by (n + 1) array we form the determinantal maps As,

s=O

,..., n,

(8.4.1 1)

in the same way that (6.4.4) were formed from the array (6.4.1). We write n

(8.4.12)

A" = CpsAs. 1

Just as for (6.5.2), we can write this as

Pl

.*.

& = det

,

(8.4.13)

0 AL-1.1 An-1,n A;, A!, A!, where the induced operators A!s now act on G18 -.-0 G,.On comparison with (8.4.4), we see that (8.4.13) is compounded of two isomorphisms, arising from A,, acting on the factor G,,and A' acting on G10 ... 8 Gn-l, together with a sign-factor. Thus A" is nonsingular. By Theorem 6.7.2, it follows that the operators a * *

A"-lAS,

s = 0,

... , n,

(8.4.14)

all commute. We now suppose that A, = A is singular. If Go,a subspace of G, is the kernel of A,, we have that the operators (8.4.14) define endomorphisms of Go,which, of course, likewise commute. They therefore admit a common nonzero eigenvector g E G,so that for some set I , , ... ,Ak we have dog = 0,

A"-'A,g = I,g, s = 1,

... ,n.

(8.4.15)

8.5 Here the

SINGULARITY AND DECOMPOSABLE TENSORS

As cannot all be zero, since n

by (8.4.12).

143

c A A s g = g, n

,usAsg = A''-'

1

1

We extend the set (A, , ... , A,) by writing 1,

=

0, and then have

s = 0,... ,n.

A"-'ASg = &g,

It then follows from Theorem 6.6.1 that

c A!sAsg = 0, n

s=o

r = 1, ... ,n,

and so, since A. = 0, n

s=l

A!J,g

= 0,

r = 1,

... ,n.

By Theorem 4.7.2 it follows now that the operators

c A,1,, n

r = 1, ... , n

s= 1

are all singular, so that we have arrived at Case ( i ) of Theorem 8.2.1, as a consequence of (oi). This shows that the result is true for k = n, and so the inductive argument is complete, and therewith the proof of Theorem 8.2.1. 8.5 Singularity and Decomposable Tensors

The following corollary will be used in an application to the topic of positive definiteness. We have THEOREM 8.5.1. Let A be the determinantal homomorphism associated with the array (8.1.1), where A,, , ... ,A,, are linear maps between spaces of the sameJinite dimension, r = 1, .. . ,k . Then fi A annihilates a nonzero tensor, it annihilates some nonzero decomposable tensor. By Theorem 8.2.1, if A is singular, there exist nonzero g, E C,,

r = 1,

... , k ,

and scalars 1,, ... ,Ak , not all zero, such that

c A;,A,g, = 0, k

s= 1

r = 1, ... , k.

144

8

THE SINGULARITY OF SQUARE ARRAYS

It then follows that if g = g , @ ... @ gk, we have k

1A!sAsg = 0,

s= 1

r = 1,

... , k ,

and so, by the proof of Theorem 6.2.3, in particular from (6.2.6), AAtg

=

0,

t = 1,

... ,k .

Since the At are not all zero, and g is decomposable, we have that A annihilates a nonzero decomposable tensor. 8.6 The Positive Definiteness Theorem Reproved The theorem in question is Theorem 7.8.2, relating to the case that the A , in (8.1.1) are hermitian matrices, the field being the complex field. The assertion is that if g*Ag > 0 for nonzero decomposable g , then this is so for all nonzero g E G . The present proof makes use of the connectedness property of Section 7.7. As in the proof of Theorem 7.8.2 given in Section 7.8, we consider a path A(T),0 IT 5 1, joining A(0) = I with A(1) = A, where each A(T) is a determinantal map of the form in question, which is positive definite on decomposable tensors. As before, we define as the largest number in (0, I] such that for 0 It < T ’ ,A(.) is positive definite for general arguments. We now argue that A(T‘) is the limit, as T + T ’ from below, of positive definite maps, and is therefore either positive definite or positive semidefinite. In the latter event, it would have nonzero kernel. However, by Theorem 8.5.1, this would imply that it annihilated a nonzero decomposable tensor, which is excluded, since A(T) is positive definite on decomposable arguments. We therefore conclude that A(T’) > 0. If T ’ = 1, we have the desired result. If T’ < 1, we have that A(T) would be positive definite for T in some rightneighborhood of T ’ , which contradicts the definition of T ’ . This completes the proof.

8.7

Eigenvalues and Singularity

The following result, which will be needed in Chapter 10, relates to eigenvalues, as defined in Section 6.8, for a rectangular array (6.4.1) of maps between pairs of linear spaces, each pair having the same dimension. The field is to be algebraically closed but need not be the complex field. We use the notation (6.4.4) for the associated determinantal maps. We have then

8.7

EIGENVALUES AND SINGULARITY

145

THEOREM 8.7.1. The following two statements are equivalent.

c PSAS

( i ) Themap

k

(8.7.1)

0

is singular.

(ii) For some eigenvalue I , , ... , I k of the array (6.4. l),

(8.7.2) This result is related to Theorem 6.8.1. However, we do not make any definiteness assumption, such as that there should be an expression of the form (8.7.1) which is nonsingular. Suppose that I . , . . . , 1, is an eigenvalue satisfying (8.7.2), and that, say I p # 0. It then follows from (6.8.6) that for somef # 0, Aqf = ( l q l I p ) A p L q = 0,... , k, and so that k Pq Aq)f = P q l q )' 4 p f l J . p

(2

0,

b2 such that

c 2

%@zs

1

> 0,

(9.2.7)

(9.2.8) Geometrically, (9.2.7) states that there is a line through the origin such that C1,C2 lie strictly on one side of this line, and (9.2.8) that another such line has C1, C2 strictly on opposite sides; here “strictly” means that no point of these sets lies on either line. It is easily seen that the existence of two such lines implies that C, does not intersect either C2 or C2‘.In Section 9.4, we give a nongeometric proof of this property for a more general case. Conversely, suppose that the arcs C , , C2 are such that C, n (c, u c2’)=

4.

If the spaces H I , H2 are finite dimensional, we can say in addition that C,, C, and their antipodal sets are closed. Thus C, , C2 can be described in polar-coordinate terms by 0, I 0 I 0, , O3 I 0 5 O,, either respectively or otherwise, where O2 < 03,0, < 0, n. Then the line containing the ray 0 = t(0, 0,) will separate C1,C 2 , while that containing the ray 0 = f ( 0 , 0, TC)will have them both on the same side. The modification for the general case is given by

+ + +

+

THEOREM 9.2.2. I f in the last theorem we drop the requirement that H I , H 2 be finite dimensional, then in order to obtain conditions that are necessary as well as suficient, we mirst allow in (9.2.7) the additional possibilities 20,> O , or >0, 20, and in (9.2.8) the additional possibilities 20, O, 10. Geometrically, the extended (9.2.7) means that there is a line with C, , C2 both on the same side, with at most one of these sets having a boundary point on the line, and (9.2.8) as extended that there is a line separating CI, C 2 , with at most one of them having a point on the line. It is easy to show that this leads to the required conclusion. In the converse direction, the fact that H I , H2 need not be finite dimensional entails that C1, C2 need not be closed. We can construct lines satisfying the extended conditions by taking, for example, lines through the boundary points of C, .

150

9

ARRAYS OF HERMITIAN FORMS

9.3 Two by Three Arrays The simplest case of a k by ( k + 1) array, that for k = 1, having been considered in Chapter 2, we pass to the next case k = 2, in which we have two complex linear spaces H 1 , H 2 , and on each a set of three hermitian . are concerned forms qrs, s = 0, 1,2, with associated quadratic forms B r S We with three things: (i) the rank of the matrix

(

@lO(hl) @20(h2)

)

dhl) @12(hl) @21@2) @22(h2) @l

for nonzero arguments (9.2.5).

9

(9.3.1)

(ii) the existence of real scalars ,uo ,p l ,,uz such that

(9.3.2)

for all nonzero arguments. (iii) the existence of real scalars us,p,, s

c %@ls > 0, c Z

2

0

0

=

0, 1, 2, such that

%@zs

> 0,

(9.3.3)

and (9.3.4) In the case k = 2 these matters are simply related.

THEOREM 9.3.1. Let the complex linear spaces H 1 , H2 be finite dimensional. Then the following conditions are equivalent. (a) The matrix (9.3.1) has rank 2 for all nonzero arguments. (6) There exist real p,, s arguments.

=

0, 1,2, such that (9.3.2) holds for all nonzero

(c) There exist real scalars as,p,, s (9.3.4).

=

0, 1, 2, which satisfy (9.3.3) and

At this stage, we prove this only in part, in that the implication (a) => (c) will be left as a special case of a result for general k, to be proved later. The

9.4

GENERAL SQUARE ARRAYS

151

case k = 2 is special in that the implication (c) (b) fails in its natural generalization to higher values of k . We note first that (b) 3 (a) trivially; the relation between the two is essentially that between Definiteness Conditions I1 and I, in Sections 7.4 and 7.2. We prove next that (c) (b). Assuming (c), we choose the ps, not all zero, such that

c 2

0

c 2

USPS

= 0,

0

PSPS

(9.3.5)

= 0.

This will determine the ps, except for a scale factor, which is immaterial, and except for their signs, which we adjust so that the left of (9.3.2) has the correct sign. Since the sets of nonzero h, ,h2 are connected, it will be sufficient to show that the left of (9.3.2) does not vanish with the above choice of the PS.

Suppose then that we have equality in (9.3.2) for some nonzero arguments. Then for some real scalars ys , s = 0, 1,2, we have YOPS

+ Yl@)ls(hl)+ Y2@zs(hz)= 0,

s = 0, 1, 2,

(9.3.6)

the ys not being all zero. From (9.3.5) it then follows that

By (9.3.3) and (9.3.4), the coefficients of y , , y2 in the last equations are, respectively, both positive, and one positive and one negative. It follows that y, = yz = 0, and so yo # 0. By (9.3.6), this is impossible since the p s are not all zero. Thus (c) * (6). To complete the proof of Theorem 9.3.1, it will be sufficient to show that (a) (c). As forecast, we leave this as a special case of Theorem 9.8.1. 9.4 General Square Arrays

We now leave the special case k = 2, and suppose that we have k complex linear spaces Hr, r = 1, ... , k , on each of which is defined a set of k hermitian forms mrS(hr),s = 1, ... , k . In generalization of Theorem 9.2.1, THEOREM 9.4.1. Let the complex linear spaces H1, ... , Hk be finite dimensional. Then in order that the kth order determinant det@rs(hr) Z 0,

(9.4.1)

9

152

ARRAYS OF HERMITIAN FORMS

for all sets h,#O,

h,EH,,

r = l , ..., k,

it is necessary and suficient that f o r every set should exist a real set a l , , ak such that

...

k

E,

CI us@,, > 0,

r = 1,

s=

E,

=

(9.4.2)

=k 1, r = 1, ... ,k, there

... , k.

(9.4.3)

Starting with the sufficiency, we assume that (9.4.3) can be arranged, but that (9.4.1) fails, and derive a contradiction. If (9.4.1) fails for some set (9.4.2), then for some real set P I , ... , P k , not all zero, we have k

CP,@,,(h,)

=

r=l

s = 1,

0,

... , k.

For arbitrary real as we have k

k

(9.4.4)

We now choose the E, = f 1 such that E,P, 2 0, so that &,fir > 0 if P, # 0, and then choose the a, as in (9.4.3). Then the terms on the left of (9.4.4) are all nonnegative, some being positive. This gives a contradiction, and so proves the sufficiency. Turning to the necessity, we assume that (9.4.2) ensures (9.4.1). We define a set of real-valued functions xr(u,, ... ,ak),r = 1, ... , k of the real , , indefinite, we set xr = 0. variables a, as follows. If the form ~ ~ = l u , @ is If it is positive definite, we define k

k

(9.4.5)

the minimum being taken over h, E H,, h, # 0. If this same form is negative definite we define that xr is the maximum of the ratio appearing in (9.4.9, over the same set of h,. We note the following properties of the functions x,. They are continuous. They do not vanish together, except when u1 = ... = ctk = 0 ; this follows from (9.4.1). Finally, they have antipodal symmetry, in that xr('l,

...

3

ak)

=

-x ,(- U l ,... , -tLk),

r = 1, ... , k .

It follows that if we define the normalized functions k

confine the domain to 1 :a: = I, and consider the map (a1,... ,a k ) + ... ,(ik),we shall obtain a continuous map of the unit sphere into itself,

9.5

A PROPERTY OF CONVEX CONES

153

which takes antipodal points into antipodal points. By a theorem of Borsuk, such a map must be onto. This means that we can choose the a l , ... ,ak so that the functions I), have any set of values such that I): = 1. In particular, we can arrange that the I),, and so the x,, have any assigned collection of signs. This completes the proof. We have incidentally another proof of the implication (6) (c) in Theorem 9.3.1. If (9.3.2) holds, not only can we arrange (9.3.3) or (9.3.4), but we can also attach to ~ u , p ,or C/?,p, any assigned signs, or for that matter, demand that they vanish. It is clear from the above argument that the conditions (9.4.3) are sufficient for (9.4.1) even when the spaces H, are not all finite dimensional; however, as shown by Theorem 9.2.2 for the case k = 2, they are no longer necessary.

1:

9.5

A Property of Convex Cones

Our discussion of results (similar to those of the last section) for rectangular instead of square arrays, will not depend on antipodal mapping theorems, but instead on properties of convex cones. By a convex cone we shall mean a subset of a real linear space which is closed under addition, and which is closed under multiplication by nonnegative scalars. The general effect of our discussion will be that such a cone, if it does not contain two equal and opposite nonzero vectors, must lie in a half-space. We shall consider cones in finite dimensional real spaces, which we may suppose normed; we indicate the norm of a vector by I I. Our basic result is

THEOREM 9.5.1. Let V be a convex cone in R"', and let the closure vof V not contain a pair of' nonzero vectors whose sum is zero. Then there is a real linear junctional f on R"' such that for some c > 0.

fv 2

CIUI,

VE

v,

(9.5.1)

Since the conclusion will apply also to V, we may as well prove the result on the hypothesis that V itself is closed. We proceed by induction over m. If m = 1, V can only consist of the nonnegative, or the nonpositive scalars, and f may be the identity, or its negative. We therefore assume that m 2 2, and that the result has been established in lower dimensions. We remark that it suffices to show that there is a linear functional f such that fv > 0, V E v, u # 0, (9.5.2)

9 ARRAYS OF HERMITIAN FORMS

154

for it will then follow that fu > 0 on the subset of V such that 101 = 1 ; this subset is bounded, and closed since we assume V closed. Thus fv will attain its lower bound on this set, and denoting this by c, we have (9.5.1). We prove the result by a method of continuous variation. Let V(T), 0 IT I 1, denote the subset of V formed by elements of the form U{(l

- T)Uo f

U E

TU},

V,

IUI

< 1,

2 0,

(9.5.3)

where uo is some fixed element of V, with 1u01 = 1. In particular, V(0) consists of all nonnegative multiples of uo , while V(1) coincides with V. We first note that V(T)is a convex cone. It is clearly closed under multiplication by nonnegative scalars. It remains to show that it is closed under addition. Suppose that we have two elements of this form, say - T)Uo -k T U , } ,

LY,{(l

passing over the trivial case L Y ~= L in the form (a1

Y ~=

Y =

1, 2;

0, we note that their sun1 can be put

+ %){(l - T h o + Tug}

where U3

= (LYlUI

+ @ZUZ)/(% + u2).

Here, if lull I 1, IuZI I 1 , we have lu31 I 1, in accordance with (9.5.3). Hence V(T)is closed under addition. We must now prove that V(T)is closed. Suppose that we have a convergent sequence LY,{(l

- T ) U o -k ZU,},

n = 1, 2, ... ,

(9.5.4)

where LY, 2 0, Iu,I 1, n = 1,2, ... . We claim first that the sequence u, is bounded; here we take it that 0 I T < 1, since if T = 1, V(T)= V which is closed by hypothesis. Suppose that (9.5.4) converges, and that u, -+ co. It then follows that (1 - T)Uo -k T U , -+ 0. Here, by restriction to a subsequence, we may take it that the sequence U, converges, say to u, , which will be in V since V is closed. We shall then have (1 - T ) U ~ T V , = 0. By hypothesis, this is possible only if (1 - T ) U ~= 0, T U , = 0, and the first of these is impossible. Since the sequence LY, is bounded, we may choose a subsequence such that LY, + LY, , u, -+ u,. The limit of (9.5.4) will then be

+

Qm{(l

- T)Uo

Turn},

which has the form of an element of V(T).Thus V(T)is closed.

9.5 A PROPERTY OF CONVEX

CONES

155

I1.

(9.5.5)

Our final preliminary remark is that

0I 7 c

V(7) c V(7’),

7’

We must show that a typical element (9.5.3) can also be exhibited in the form of an element of V(7’). We write a{(l - 7)uo

where 0’

+

TO}

= 7‘-1{7u

Since IuI I I in (9.5.3), we have I7’-1{7lul

IU’I

+ r’u‘},

a{(l - 7‘)Oo

=

+

+

(7’

(7‘

- 7)ug).

- 7)luol>

II,

as required. Now write T for the set of r, 0 Iz I1, such that the theorem is true for V(7), or such that there exists a linear functionalf(7) with the property that f(7)U

> 0,

(9.5.6)

u E V(7), u # 0.

Clearly 0 E T, since V(0) consists of all nonnegative multiples of uo , and any linear functionalf(0) withf(0)uo > 0 will meet the requirement (9.5.6). Also, from (9.5.5), we have that if 7 ’ E T, then 7 E T for 7 in 0 I7 < 7 ’ . Thus T consists of some interval in [0, I], starting at 0. Suppose first that T = [0,7’1, with 0 I7 ’ c 1. We claim that there is an E > 0 such that f(z’)u > 0,

u # 0, u E V(7),

7’

< 7 < 7’

+ E.

(9.5.7)

In the contrary event there would be a decreasing sequence r l ,7 2 , ... , tending to z’, and a sequence u1 ,u 2 , ... , of elements of V, with Iu,I I 1, n = 1,2, ... , such that f(T’){(l

Here we make n urn, and obtain

+

- 7,)uo

+

5 0.

7,U,}

co,through a subsequence such that u, converges, say to f(~’)((l

- 7‘)Oo

+ 7‘Urn}

0.

This is contrary to the hypothesis 7 ’ E T, since (1 - 7’)Uo + T ’ U ~# 0, since (1 - 7’)Uo # 0. Thus T must extend beyond 7 ‘ , and we have a contradiction. Suppose next that T = [0,7 ’ ) , where 0 c 7 ’ I1. We will show that 7 ’ E T, which will eliminate this case. There will be an increasing sequence r l , 7 2 , ... , tending to T‘, and linear functionalsf(t,), such that f(Tn){(l

- 7,)Uo

+

7&}

> 0,

UE

v,

101

< 1.

(9.5.8)

156

9

ARRAYS OF HERMITIAN FORMS

Here we are at liberty to normalize the functionals f(z,,) so that they are bounded uniformly from above and below, in the sense that, for fixed positive C, C' we have

n = 1, 2,

c < sup If(z,)ul < C', IU141

... .

We can then choose a subsequence such that f ( z , ) converges as n -,co to a nonzero limit, say f l. On passage to the limit in (9.5.8) we then obtain flW

2 0,

w E V(z'), w # 0.

(9.5.9)

If here inequality holds, for all such w, the proof is finished. If not, we denote by W the subset of V(r') for which f,w = 0. Since V(z') is closed, W will also be closed, as a subset of R". Furthermore, W is closed under addition and under multiplication by nonnegative scalars, since this is true for V(z') and for the set of solutions of flw = 0. Finally, since f, # 0, we have that W is contained in an (rn - 1)-dimensional subspace of R" and also, as a subset of V, has the property that no two nonzero elements sum to zero. We now use the inductive hypothesis, and denote by f2 a linear functional, defined on the subspace fl = 0, such that f2w > 0,

W E

w, w # 0.

(9.5.10)

We suppose.f2 extended as a linear functional to R", by arbitrary definition on some element not in Kerf,. We then claim that for small q 0 the functional f 3 = f l !if2 has the property that

=-

+

f3U

> 0,

u E V(z'), u # 0.

(9.5.11)

Supposing the contrary, we have a sequence of positive yl,, n = 1,2, ... , tending to zero, and a sequence u, E V(t'), u, # 0, n = 1,2, ... , such that (fi

+ 4,f2)u, I 0,

n = 1,2, ... .

(9.5.12)

We normalize the u, by Iu,I = 1, and choose a subsequence so that they converge, say to u,, with (u,I = 1. On passing to the limit in (9.5.12) we have then f1Vm5

0,

and so flu, = 0, by (9.5.9), so that u, E W. Again, from (9.5.9) we have flu, 2 0, and this together with (9.5.12) shows that .f2u, 5 0, n = 1,2, ... , and a limiting transition shows that fiu, 5 0. Since u , E W, and Iu,I = I , this conflicts with (9.5.10). This disposes of the supposition that T = [0,z'), 0 c z' 5 1. We have thus shown that T,an interval in [0, 11 which includes 0, cannot have either of the forms [0,t'], where T' c 1, nor [0, z'), where z' 5 1. It

9.5

A PROPERTY OF CONVEX CONES

157

must therefore consist of the whole interval [O,11, which completes the proof of Theorem 9.5.1. We next treat the case in which no hypothesis regarding closure is made, and a slightly weaker result obtained.

THEOREM 9.5.2. Let V be a conuex cone in R"',such that no two elements, other than zero, haue zero sum. Then there is a nonzero linear functional f such that fu 2 0,

(9.5.13)

u E V.

It will be sufficient to prove this for the case that V is not contained in any proper subspace of R"';if it were, we could take the result for that subspace, and extend the linear functional in question to the whole of R"'. Let then wl,... , M', be a set of linearly independent elements of V , and define m

(9.5.14)

Vo = Y E1 W , ,

where y > 0 is chosen so that luol = 1. Since V contains all linear combinations of the w, with nonnegative coefficients, we see that uo will be an interior point of V. We define the subsets V(T),0 I T 5 1, of V by (9.5.3). As before, V(T) will be a convex cone. We cannot assert that it is closed. We claim, however, that the closure v(z) of V(T)is contained in V if 0 I T < 1. This is trivial if T = 0, and so we pass to the event that 0 < T < 1. Let (9.5.4) denote a convergent sequence of elements of V(T).As before, we prove first the boundedness of the sequence a,, . If this fails, we have that (1 - T ) U ~ TU,, + 0, and so TU,,/(T - 1) -+ uo . Since uo is an interior point of V, we have for large n that TU,,/(T - 1) E V. Since ~u,,/(l - T ) E V, we have two elements of V summing to zero, contrary to hypothesis. Thus the sequence a,,is bounded. The assertion to be proved, that the limit of (9.5.4) lies in V, is trivial if a,,+ 0, and so we assume that a,,tends to a positive limit. Then the sequence (1 - T ) U ~ TU,,, n = 1,2, ... , must also converge; it will be sufficient to prove that its limit is in V. The sequence u,, must converge, say to u, E we must show that ( I - T ) U ~ T U , E V. Let us define u,,' by

+

+

+

(1 -

T)U,,'

-k

TU,,

= (1

- T)Uo

+ TU,

;

(9.5.15)

this is possible since T < 1. Since u,, -+ u , , we have u,,' -+ uo as n --+ co. Since uo is an interior point of V , we have U , ' E V for large n. Then the left-hand side of (9.5.15) is a convex linear combination of elements of V,

158

9

ARRAYS OF HERMITIAN FORMS

and so is in V, the latter being a convex cone. Thus the right-hand side of (9.5.15) is in V , and so V(r) c V. Thus V(T),0 I z < 1, satisfies the conditions of Theorem 9.5.1, and we conclude that there is a linear functional f ( z ) such that f(T)V

> 0,

V E V(t),

U

# 0.

We choose an increasing sequence z,, z2, ... , tending to 1, normalize the functionals f(z,), and choose a subsequence converging to a nonzero limit f. For any v E V , IuI 5 1, we have then, as n --t 00 through this subsequence, fv

=

lim f(~,){(l

- zn)uO + z,u},

and here all terms on the right are nonnegative. Hence we obtain f u 2 0, V E V , if IuI 5 1, and so generally; this completes the proof of Theorem 9.5.2. We deduce a refinement of this result, also for the case that V need not be closed.

THEOREM 9.5.3. Let V be a convex cone in R", such that no two nonzero elements sum to zero. Then there is a linearly independent set of linear functionals f , , ... ,f , on R" sucly that

f * v 2 0, f 2 v 2 0,

VE

V,

u E Ker f , n V ,

(9.5.16) (9.5.17)

(9.5.18) v E Ker f , n -.. n Ker fm-l n V. f , v 2 0, Here inequality is to be possible whenever the indicated set of v contains a nonzero element. We suppose that V # 0, for otherwise the theorem is trivial. Let S , be the smallest subspace of R" containing V. By Theorem 9.5.2, there is a linear functional f , on S , that satisfies (9.5.16), and that is not zero. Heref, cannot vanish identically on V, for then S1 would not be the smallest subspace containing V. On supposing f , extended, if necessary, to the whole of R", we have the assertion concerning (9.5.16). Let next S2 be the smallest subspace of S , which contains Kerf, n V . It is certainly a proper subspace of S , since f , is not zero on S , , and S2 c Ker f , . If S2 is zero, there is nothing more to prove; as a formality, we can complete the linearly independent set f , , ... ,f , arbitrarily. If S2 is not zero, we consider the convex cone Ker f , n V , and in virtue of Theorem 9.5.2 take f 2 to be a linear functional on S,, not zero, which is nonnegative on Kerf, n V. Again we note that fi must be positive somewhere on

9.6

THE CASE OF SEVERAL CONES

159

Ker f , n V, since otherwise S, would not be the smallest subspace containing Ker fi n V. On extending.f, to R" we have the assertion concerning (9.5.17). We remark also that f , , f , are linearly independent. If S, = 0, this is so by construction. If S, # 0, we have that f, is constructed so as not to vanish everywhere on Ker f , n V, and so not on Ker f , ;thus f , cannot be a multiple off1 * We continue in this way. If fl, ... ,f,have been found, where I _< r c m, we define S,,, to be the smallest subspace containing Kerf, n ... n Kerf, n V. If S,,, = 0, the remaining linearly independent f,+,, ... ,f , are chosen arbitrarily. If not, we choosef,,, in virtue of Theorem 9.5.2 as a nonzero linear functional that is nonnegative on Ker f , n n Kerf, n V , and not identically zero thereon; it must therefore be linearly independent of fl, ... ,f,. The process must terminate with the construction off, , since we are dealing with a space of only m dimensions.

9.6 The Case of Several Cones We now take up the question of the linear independence of sets of vectors chosen one from each of several convex cones; it is in this way that we interpret questions such as that of the rank of the matrix (7.2.2), or the vanishing or nonvanishing of the determinant (9.4.1). We start with the simpler case that the cones are all closed. THEOREM 9.6.1. Let V , , ... , V,, t < m, be a set of' closed convex cones in R", each containing a nonzero element, and none containing a pair of nonzero elements which sum to zero. Then in order that every set of nonzero vectors u,

E

V , , ... , V , E V,,

(9.6.1)

be linearly independent, it is necessary and sufficient that for every set E , , . ,. ,E , , each equal to i-1, there should exist a linear functional f on R"'such that ESfU>O,

VEV,,

V # O ,

s=l,

..., t.

(9.6.2)

The sufficiency is easily checked; the proof follows the lines of a similar assertion in Theorem 9.4.1. We suppose that for some real p,, ... ,p, and some set (9.6.1) we have psv, = 0. We choose the E, so that Qs 2 0 s = 1, ... , f, and f accordingly. In the equation p,(fu,) = 0, all terms on the left are then nonnegative, and so all are zero. It then follows that for each s, either us = 0 or p, = 0, or both. Thus every nonzero set is linearly independent.

160

9

ARRAYS OF HERMITIAN FORMS

In the reverse direction, we postulate the linear independence of (9.6.l), and wish to establish the existence off for given sets E , , ... , E , . It will be = E , = 1 ; other cases may then be treated sufficient to do this for E , = by replacing V,, corresponding to E, = - 1 , by the corresponding reversed cones. We introduce the cone

v, + .*. + V,, (9.6.3) meaning by this the set of sums u1 + + u,, us E V,, s = 1 , ... , t. This is V

=

clearly a convex cone. We wish to show that it satisfies the other requirements of Theorem 9.5.1. We prove first that it is closed. It will be sufficient to prove that V, V2 is closed, the general case then following by induction. Suppose that we have a convergent sequence

+

unl

+ un2,

n

= 1,

2,

... ,

(9.6.4)

where unl E V , , un2 E V 2 . Suppose if possible that these sequences are not bounded. Then on division by lunll + 1un21 we obtain a sequence with the properties wnl

+

wn2

0,

lwnll

+

lwn2l

=

1.

Here the w,,,E V, , s = 1,2, are bounded, and so we can select an n-sequence such that they both converge, say to w1 ,w, . We shall have w, E V,, s = 1,2, since the V, are closed, and also w 1 + w2 = 0, lwll + Iw21 = 1. This contradicts our assumed linear independence property. Thus the summands in (9.6.4) are bounded, and will converge for some subsequence to elements of V , , V, . Hence V is closed. Finally, we claim that V has the property that no two nonzero elements sum to zero. Suppose that us, us' E V,, s = 1, ... , t , and that 1:us + us' = 0. We then have (0, + us') = 0, and so, by our linear independence hypothesis, us + us' = 0, s = 1, ... , t . Since we assume the property in question for each of the V,, we have us = 0, us' = 0, s = 1, ... , t, so that 1 :us, us' = 0. We can thus apply Theorem 9.5.1, and conclude that there is a functional f such that f u > 0, u E V , u # 0. This applies in particular to the subsets V, of V , and so have (9.6.2). This completes the proof. In the case that the V, are not necessarily closed, we have to use the more involved Theorem 9.5.3. THEOREM 9.6.2. Let the hypotheses of Theorem (9.6.1) hold, except that the V, need not be closed sets. Then for the linear independence of all nonzero sets (9.6.1) it is necessary and sufficient that, for every set of sign-factors

9.7 81 f l

SQUARE ARRAYS OF HERMITIAN FORM, CONTINUED

161

, ... , E , = 5 1, there should exist a set of linearly inaependent functionals , ... ,f,, such that E,fIV

2 0,

ESf2U

2 0,

v,, s = 1, ... , t, V E v,, f1V = 0, s = 1, ... , t,

E,f,V

2 0,

V E

(9.6.5)

V E

V,, f1v

=

* * a

= f m - l v = 0,

(9.6.6) s = 1,

... , t.

(9.6.7)

The functionals can be chosen so that inequality is possible in each case, whenever the indicated set of v contains a nonzero member.

For the sufficiency, we suppose again that psv, = 0, choose the E, so that E,B, 2 0, s = 1 , ... , t , and take the functionals fly ... ,f , to be linearly independent and to satisfy (9.6.5)-(9.6.7). We consider the equations t

CPs(f,v,) = 0,

s= 1

r = I, ... , m.

(9.6.8)

In the case r = 1, all terms on the left are nonnegative, and so all are zero. Thus (9.6.9) s = 1, ... , t. ps(flv,) = 0, We pass next to (9.6.8) for r = 2. Disregarding the s for which p, = 0, we have f l u , = 0, and so ~ , f ~2v ,0. Thus, by (9.6.6), the terms on the left of (9.6.8) for r = 2 will be nonnegative, and so all zero, so that p,(f2V,) = 0,

s = 1,

... , t.

(9.6.10)

Proceeding inductively, we thus conclude that

... ,m. Since thef, are linearly independent, we have either p, = 0 or v, ps(fiv,) = 0,

s = 1,

... , t , r

= 1,

(9.6.11)

= 0. This proves the sufficiency. For the necessity, we consider once more the cone (9.6.3), which need not be closed, but which has the properties demanded by Theorem 9.5.3. As in the proof of the last result, application of Theorem 9.5.3 yields the existence of a suitable set of functionals in the case .cl = = E , = 1 ; other cases may be dealt with by considering reversed cones.

9.7 Square Arrays of Hermitian Forms, Continued In Section 9.4, we derived a necessary and sufficient condition (9.4.3) for (9.4.1) in which the nontrivial part, the necessity, we established by use

162

9 ARRAYS OF HERMITIAN FORMS

of a theorem on mappings of spheres which preserved antipodes. Since we shall not prove this theorem here, we give an alternative treatment based on the results of the last two sections. In Section 9.4, we assumed that the complex linear spaces concerned were finite dimensional, which had the consequence that certain sets of vectors were closed. We shall now dispense with this assumption; the result will follow the pattern of the last theorem, rather than that of Section 9.4. We put the result in geometrical terms. For each of the complex linear spaces H,, r = 1 , ... , k, the set of values of the k-vector @r 1 (hr),

...

3

(9.7.1)

@rk(hr),

where the QrS are hermitian quadratic forms, as h, ranges over nonzero values in H,, will be denoted by U,. Thus U, may be considered as a set of vectors in R k ;it is appropriate to consider the case in which (9.7.1) do not vanish together, if h, # 0. The geometric principle is contained in THEOREM 9.7.1. Let U,, r = 1, ... ,k, be collections of nonzero vectors in Rk, each of which is a connected set. Then in order that every set U1E

u1, ... ,UkE

(9.7.2)

uk,

sliould be linearly independent, it is necessary and sufficient that for every set of sign-factors c l , ... ,E~ there should exist a set of linearly independent s = 1, ... , k, we have functionah f l , ... ,fk such that.for u E Us,

(9.7.3)

%flu 2 0, %f;U 2 0

&,fkU

20

( f l u = O), ( f l u = *..

(9.7.4) 'fk-lU

=

O),

(9.7.5)

where in each case inequality holds for some u, if the indicated set of u is nonempty.

The proof of sufficiency is the same as that of Theorem 9.6.2. In order to deduce the necessity from the same theorem, we introduce the cones V,=JU,,

r=l,

..., k,

(9.7.6)

where V, consists of all finite linear combinations of elements of U, with nonnegative coefficients. Then clearly V, is a convex cone containing Ur, and so containing nonzero elements. We must also show that no two nonzero elements of any V, sum to zero, and that a set of nonzero elements taken one from each of the V, is linearly independent.

9.8

RECTANGULAR ARRAYS OF HERMITIAN FORMS

163

The hypothesis of the linear independence of sets (9.7.2)can be written as det(ul , ... ,uk) # 0, where “det” indicates the determinant formed by writing the k-vectors u, in a square array. Since the U, are connected sets, this determinant must have a fixed sign; for definiteness, we assume it positive, and so det(u,,

... ,uk) > 0.

(9.7.7)

It follows from this that if n.

(9.7.8)

is a linear combination of elements of U, with nonnegative coefficients then, for any u p € U p , p = 1, ... , k , p # r, we have that det(u, ... , U, , ... ,uk) is positive, unless all the u,, u,’ E V,, we have det(u, ,... ,U, -k

U,’,

c P,, det(u, ,... , n,

=

s= 1

u,,

... ,uk)

Prs are zero, and so unless u, E V, is zero. Thus, for

... ,uk) = det(ul , ... ,U,, ... ,u k ) + det(ul ,... ,U,‘, ... ,uk),

the right being positive unless u, = u,‘ = 0; this shows that no two nonzero elements of V, sum to zero. Suppose now that we have a set u, E V,, r = 1, ... , k, which we may express in the form (9.7.8).We have then

and here the right is positive if, for every r = 1, ... ,k, one of the P,, is positive. Thus any nonzero set of u, E V,, r = 1, ... ,k, is linearly independent. This completes the proof that Theorem 9.6.2 is applicable. We deduce the existence of functionals f l, ... ,f k satisfying (9.7.3)-(9.7.5) for arguments u in V,, and so in particular for u E Us. 9.8 Rectangular Arrays of Hermitian Forms

We now take up from the geometrical viewpoint what in Section 7.2was termed “Definiteness Condition I,” and which forms a general basis for such properties as the reality of the eigenvalues, without however providing a positive definite scalar product. We put this in terms of hermitian quadratic forms rather than of operators or matrices. For k complex linear spaces Hl ... Hk , we suppose given on each a set of k 1 forms

+

@rdhr),

9

@rk(hh

Hr*

(9.8.1)

164

9

ARRAYS OF HERMITIAN FORMS

+

We denote by U, the range of values of the (k 1)-vector (9.8.1), as h, ranges over nonzero values. This definiteness condition is equivalent to the requirement that every set U,E U,, r = 1, ... ,k, (9.8.2) is linearly independent. In this case, it is not possible, as in the last section, to replace this by the requirement that a determinant should have constant sign; we have to make appeal to the fact that the u, do arise from hermitian forms. The result is simpler in the finite dimensional case.

THEOREM 9.8.1. Let the complex linear spaces H , , ... ,Hk be finite dimensional. Then for every set of k vectors (9.8.1), arising f7om nonzero arguments h, E H,, r = 1, ... ,k, to be linearly independent, it is necessary and suficient that for every set of sign-factors E , , ... ,E~ there should be a set of real ci, , ... ,ci& such that k

E,C~,@,, > 0, s=o

v = 1, ... ,k

(9.8.3)

The proof of the sufficiency is the same as in the case of a square array (Theorem 9.4.1). For the necessity, we embed the U, in cones V,, with a view to applying Theorem 9.6.1. We define now V, as the closure of the set of linear combinations (9.7.8) of elements of U, with nonnegative coefficients; it would be possible to argue that this closure operation is superfluous, but we do not need to discuss this. Thus the V, are automatically closed, as required by Theorem 9.6.1. To complete the proof, we must again show that no two elements of any V, sum to zero, unless both are zero, and that no nonzero set of elements of V,, ... , Vk are linearly dependent. We start by considering sets 01 E

V1,

u2 E u2,

...

9

UkE

u&.

(9.8.4)

By hypothesis, u2, ... , uk are linearly independent, and so generate a subspace of Rki which has dimension k - 1. Thus there exists a pair of linear functionals f l , f 2 on Rk+'which vanish together precisely on this subspace. Since U , does not intersect this subspace, we have that f l , f 2 do not vanish together on U,.Now f l , f 2 on U1 define two real linear combinations of the hermitian forms OlO(hl),... ,(Dlk(hl),and so themselves define hermitian forms on H , . Since these forms do not vanish together when h, # 0, and since H , is finite dimensional, we have by Section 2.8 that there is a linear combination of them that is positive definite on H 1 . Thus there is a linear functional f 3 = y1fl y 2 f i on Rkilsuch that

,

+

f3u

> 0,

UE

UlY

(9.8.5)

9.8

RECTANGULAR ARRAYS OF HERMITIAN FORMS

while for the fixed set u2 E U2, ... , uk E f3ur = 0,

r

u k , we

=

165

have

2, ... ,k.

(9.8.6)

Furthermore, since Hl is finite dimensional, the set of u E U, such that (u( = 1 is closed. Sincef3 is positive on this set, it is also bounded from zero on it, and so for some c > 0 we can strengthen (9.8.5) to f3u 2

CIUJ,

U E

(9.8.7)

Ul.

We now extend this to V,. Let

c n

v1

=

U l s E Ul

Psuls,

Pls

Y

2 0.

(9.8.8)

s= 1

Then

c n

.f3v1

=

s= 1

P~(f3~1s),

all terms on the right being nonnegative. Thus

This holds for all expressions (9.8.8), with any n, and so extends to the closure of the set of such v l . Thus f3Vl

2

clv11,

(9.8.9)

01 E V l .

We now prove that Vl has no pair of nonzero elements summing to zero. For if v l , ul'e V,, we have

+

f3(v1

+ ul')

= h V 1

+f3v1'

2

cIvll

+ clvl'l,

and so if v1 v l ' = 0, we must have v1 = ul' = 0. In the same way, we can show that V,, ... , Vk also have this property. It remains to show that a nonzero set v, E V,., r = 1, ... ,k, must be linearly independent. We do this inductively, and start by showing that a set (9.8.4) must be linearly independent. Here u 2 , ... ,uk are linearly independent by hypothesis, and so what we must show is that v 1 cannot be a linear combination of u 2 , ... , uk, unless it is zero. This follows at once from (9.8.6) and (9.8.9). In the next stage we consider sets of nonzero vectors 01 E v 1 y v2

E v2 y u3 E

u3y

...

y

uk E

uk

(9.8.10)

We have shown that v l , u3, ... , uk are linearly independent, so that there is a pair f4,f5of functionals that vanishes together on the subspace that it generates and nowhere else. In particular, f4,fs cannot vanish together on

166

9 ARRAYS

OF HERMITIAN FORMS

U2, since any set ul, u 2 , ... , u k is linearly independent. Thus there is a linear combinationf, of them such that

> 0,

fSU

while f6v1

= 0,

(9.8.11)

U E u2,

=

f6u3

"'

=f 6 u k =

0.

(9.8.12)

As before, we argue that, for some c2 > 0, h V 2

2

c21v219

v 2 E v2*

This shows that the set (9.8.10) of nonzero vectors is linearly independent. The argument continues in this way. In the final stage, we suppose that we have established the linear independence of nonzero sets of the form UkE u k . (9.8.13) ... V k - 1 E V k - 1 , For fixed nonzero vl, ... ,U k - 1 , which are thus known to be linearly 01 E v 1 ,

9

independent, we define a pair of linear functionals f j k - 2 , f 3 k - i that vanishes together only on the subspace that it generates, and so not on u k . This leads to a linear functional f 3 k which is positive on u k , and zero on v l , ... ,V k - l . Again we deduce that for some c k > 0 we have f 3 k V k 2 c k l o k l for V k E v k . Thus no nonzero v k can be a linear combination of u l , ... , V k - 1 . With this, we conclude the proof that Theorem 9.6.1 is applicable to the cones

vl, ...

vk.

It follows that there exists a linear functional f such that, for the signfactors E~ , . .. ,E ~ , U E V ~ v, # O ,

E,fv>O,

~ = l ..., , k.

(9.8.14)

It follows that this applies also to the subsets Us. By the linear independence hypothesis for (9.8.l), the elements of Usare nonzero, and so we have &,fi

> 0,

U E

Us, s

= 1,

... ,k.

(9.8.15)

This is equivalent to (9.8.3). In the general, not necessarily finite dimensional case, we must be content with a more involved formulation. THEOREM 9.8.2. In order that every set (9.8.1), arising from nonzero arguments h, E H,, i' = 1 , . .. ,k, be linearly independent, it is necessary and suficient that for every set of sign-factors e l , ... , Ek there should exist a nonsingular real matrix y s t , 0 I s, t I k, such that the following properties hold. (i) The inequality k

holdsfor t

=

0.

(9.8.16)

9.8

RECTANGULAR ARRAYS OF HERMITIAN FORMS

167

(ii) Inequality (9.8.16) holds for t = t' > 0 if for the hr in question equality holds for 0 I t < t'. (iii) For each h, Z 0, inequality holds for at least one t = 0, ... , k. For the sufficiency we suppose that for some set of nonzero hr and some real scalars P, we have k

IPr@Jhr) = 0, s= 1

We determine the sign-factors Then k

E~

k

C C Ysl@rs(hr)

r=l

so that

Pr

s=O

k

=

s = 0,

... , k.

E ~ 2P 0,~ and the k

C Y s l C Pr@rs(hr)

s=O

r=l

= 0,

(9.8.17) yst accordingly.

(9.8.18)

and so, by (9.8.16), for those r for which Pr # 0, k

For these r we have, by (ii), k

Pr

and this is true also if Pr

=

C Ysz@rs(k) s=o

2 0,

(9.8.19)

0. As in (9.8.18),

and so from (9.8.19) we conclude that

Continuing this argument, we obtain that for those r for which Pr # 0, we have equality in (9.8.16) for t = 1, ... ,k, which is excluded by (iii). In proving the necessity, we introduce the cones Vr, which are collections of linear combinations of the form (9.7.8); on this occasion we do not inquire whether they are closed, or introduce their closure. We have to show, for Theorem 9.6.2, that they do not contain a pair of nonzero elements summing to zero, and that a set of nonzero elements, taken one from each, is linearly independent. Starting from (9.8.4), we have, as before, that there is a pair of linear functionals which vanishes together on u2, ... , u k but not on U,. The conclusion now is that there is a linear combination f3 of them that is positive semidefinite on U , , and a second linear combination f3' that is positive where the first one vanishes.

168

9 ARRAY

OF HERMITIAN FORMS

We now consider u1 E Vl of the form (9.8.8). We have

c;

with equality only if for every s either /Is = 0 or f3uls = 0, or both. In this /IS(f3’u1,) > 0, unless all the /I, are zero, that is, event we have f 3 ’ ~ 1 = unless u1 = 0. We thus have that f3u1 2 0 if u1 E Vl , andf3‘u1 > 0 iff3ul = 0 and u1 # 0; also, f3, f3’ both vanish on u2 , ... , u,. From this we prove, much as before, that Vl does not contain a pair of nonzero elements summing to zero, and that a nonzero set (9.8.4) is linearly independent. Exactly the same argument proves, of course, that each of the V, contains no pair of nonzero elements summing to zero. The proof that no set of nonzero elements, taken one from each of the V,, is linearly dependent, follows the inductive lines of the proof of the previous theorem, in which we proceed from sets of the form (9.8.4) through (9.8.10) and (9.8.13) to sets of the required form. We then apply Theorem 9.6.2 with m = k + 1, t = k, and express the k 1 linearly independent functionals explicitly by means of the nonsingular matrix ys,, 0 Is, t Ik . Here (9.8.16) corresponds to (9.6.5); we have confined the assertion to elements of U,., as given by (9.8.1), which we may since U,. c V,. In a similar way, (ii) of the theorem corresponds to (9.6.6) and (9.6.7). If (iii) of the theorem were not true, one of the rows (9.8.1) would vanish, which is not the case if they are linearly independent. This completes our discussion of Theorem 9.8.2.

+

9.9 Relation between Definiteness Conditions I and 11 In the last section, we set up an equivalence, which was independent of k , between the requirement that k rows of k 1 hermitian forms be linearly independent, when applied to sets of nonzero arguments, and on the other hand the possibility of finding linear combinations of the elements of these rows, as in (9.8.3), which had assigned properties of sign-definiteness. We now consider the equivalence of the first of these situations with one that is formally more drastic, namely that there should be a fixed (k 1)-vector 1) quadratic forms, which is linearly independent of the given k rows of (k for any assigned nonzero arguments. We recall that, in Chapter 7, we commenced the study of eigenvalue problems under the first of these conditions, and then went over to the second. It turned out, in Secton 9.3, that these conditions were actually equivalent, in the finite dimensional case, if k = 2 ; part of the proof of this, Theorem 9.3.1, was left as a special case of Theorem 9.8.1. The two conditions are also equivalent, in the finite dimensional case, if k = 1 ; this was shown in Theorem 2.8.2.

+

+

+

9.9

WLATION BETWEEN DEFINITENESSCONDITIONS I AND 11

169

By means of an example, we now show that this equivalence fails if 3. In other words, there exist arrays of three sets of four hermitian forms,

k

=

@ro(hr),*.. @r3(hr),

hr E Hr

9

9

r = 1,2, 3 7

(9.9.1)

on three complex linear spaces, which are linearly independent for each set of three nonzero arguments, while there need not exist a nonzero real fourvector (p, , ... ,p3) which is linearly independent of all such sets. The example showing this will be based on the remarks at the end of Section 2.10, according to which we can choose the forms (9.9.1) for each r so that the range of the four-tuple (9.9.1) should be the convex set generated by any assigned finite set of four-vectors. We shall take the range U,., say, of the four-tuple (9.9.1), for each r and for hr # 0, to be the set generated by vectors of the form (x, , x l , x, ,x3), where x,

=

+1

or

x, = f ~ ,s

=

0, 1,2, 3,

(9.9.2)

for some positive E , and the signs of the x, in each Ur are fixed according to the following table: xo u 1

u, u

3

x1

x , x3

+ + - + - + +

/

+

-

(9.9.3)

-

We remark first that, whatever the choice of E > 0, vectors chosen one from each of the Ur must be linearly independent. In the first place, no vector in any of the Ur is zero. Secondly, no pair of vectors, chosen from two of U 1 ,U , , U3 can be linearly dependent; for example, a vector in U , and one in U, cannot be proportional, since the signs of xo in the two cases are the same, while those of x 1 are different. Suppose finally that there is a set of three vectors ( x ~ rxlrr , xzr, x3r) E ur,

r = 192, 3 9

(9.9.4)

which satisfy a homogeneous linear relation with nonzero coefficients; we suppose, that is, that for some real nonzero a l ,a,, a3 we have 3

1

U,X,

r= 1

= 0,

s = 0,

... , 3.

(9.9.5)

Since the signs of the xor are all positive, the czr cannot all have the same sign, and so two must be of one sign and the third of another. If for example

170

9

ARRAYS OF HERMITIAN FORMS

-=

a 1 > 0 and a2 < 0, a3 0, we obtain a contradiction with (9.9.5) when s = 1 in view of the signs of s1in (9.9.3). Other eventualities can be disposed

of similarly, and so we conclude that elements of Ul, U2, and U3 are linearly independent. It remains to show that for some E > 0, every nonzero vector p = ( p o , p , , p2,p 3 ) is a linear combination of some set of the form (9.9.4). In view of the linear independence of elements of the U,, it will be sufficient to show that the determinant

D =

Po

Pl

P2

P3

xO1

xll

x21

x31

x02

x12

x22

x32

x03

x13

x23

x33

(9.9.6)

can take the value zero, for gi :n p v , for some set (,.9.4). We first note that D willvanish at some point of the Cartesian product U = U1 x U2 x U 3 , with fixed p # 0, if it takes both signs on U. This is so since U is connected, as the Cartesian product of convex, and so of connected sets, in an obvious topology. It will thus be sufficient to show that for some fixed E > 0, and any p # 0, D takes both signs when the vectors (9.9.4) run through members of the extremal or generating sets described by (9.9.2) and (9.9.3). We can further reduce the task ahead of us by claiming that it is sufficient to show that for any p # 0 there is an E > 0 such that D takes both signs when (9.9.4) runs through generating sets (9.9.2) and (9.9.3). For this purpose, we can suppose p normalized by

&s” 3

(9.9.7)

= 1.

0

If for some such vector p we have an E > 0, such that D takes both signs on the set (9.9.2) and (9.9.3), the same E will, by continuity, have this property for any p ’ in some neighborhood of 11. By the Heine-Bore1 theorem, the unit sphere (9.9.7) can be covered by a finite number of such neighborhoods, and the least of the various &-valuesconcerned will then serve globally. In a further reduction, we note that if D takes both signs, for fixed 11 # 0, on the set (9.9.2) and (9.9.3) when E = 0, then this will be so also for some positive E ; this again follows by continuity. We have thus to show that for any LL # 0, D takes both signs when the arguments (9.9.4) range through the sets described by x, = + 1

or

x,

=

0,

s = 0, 1, 2, 3,

(9.9.8)

where in the first case the signs of x, are restricted by the table (9.9.3).

NOTES FOR CHAPTER

171

9

Suppose, for example, that p o # 0. We then note that D can take the values _+po, so that the result will be established for this case. Suitable choices of the lower three rows of D are given, respectively, by

b:j 0 1 0 0

E-:: 1 0 0

-1

It is easily seen that similar choices will attach to D the values & p l y + p 2 , i - p 3 , so that D must take both signs if p # 0. This completes the discussion of our example, to show that Definiteness Condition I does not imply Definiteness Condition 11, if k = 3. Notes for Chapter 9

For further information on cones and convex sets see Bonnesen and Fenchel (1934), Dines (1936), Grunbaum (1967) and Valentine (1964). The requirement that a cone with vertex, or “apex,” at the origin should not contain a pair of nonzero vectors with zero sum implies that the cone is “pointed.” I am indebted for discussions on these matters with Prof. C . Davis and Prof. P. Scherk. For the Borsuk theorem on maps of spheres, preserving antipodes, used in Section 9.4, see Borsuk (1933).

CHAPTER 10

COMPLETENESS OF EIGENVECTORS IN THE LOCALLY DEFINITE CASE

10.1 Introduction

In this chapter, we return to the topic of Chapter 7, and take up once more the case of a rectangular array (10.1.1)

of hermitian matrices, those in any one row being of the same size. In Chapter 7 we commenced the investigation of this assuming Definiteness Condition I, but established the completeness of the eigenvectors only under the formally stronger Condition 111, which was subsequently shown to be equivalent to the intermediate Condition 11. In Section 9.9 it was shown by an example that Condition I is actually weaker than the other two, if k 2 3. It is therefore desirable to extend to this case the completeness of the eigenvectors. 172

10.1

173

INTRODUCTION

The condition in question is that rank{gr*Arsgr} = k ,

gl # O,

...

Y

gk

# 0,

(10.1.2)

where the gr are column matrices of the appropriate sizes, and the matrix extends over r = 1, ... ,k , s = 0, ... ,k. In view of Theorem 9.8.1, the condition has the alternative expression that for every set of sign-factors E ~ ... , , &k there should exist real uo, ... , uk, such that k

E,

1u,A,, s=o

> 0,

r = 1, ... , k.

(10.1.3)

Denoting by G, the space of column matrices on which the Are, ... ,A,, act, we write again G = G1 0 0 c k . We use also the notation (6.4.4) for the determinantal endomorphisms formed from the array (10.1.1). As explained in Section 7.2, these define sesquilinear forms g*A,g‘ on G. Here we regard these as forming collectively a vector-valued scalar product. We write

...

d g ’ , g)

=

(g*AOg‘, * . .

3

g*dkg’).

(10.1.4)

Hypothesis (10.1.2) may then be written d g , g ) # 0,

g # 0, g decomposable.

(10.1.5)

If the values of q(g, g) lay in some open half-space, we would have the situation dealt with in Chapter 7; in this, some linear combination of the entries in (10.1.4) could serve as a positive definite scalar product in G , with respect to which the various eigensubspaces were orthogonal. We show here that the more limited assumption (10.1.5) makes possible the use of the vector-valued scalar product (10.1.4) to a similar effect. Although (10.1.4) does not, in general, yield a product that is positive definite on the whole of G, it will yield one for each eigensubspace, and indeed for eigensubspaces associated with eigenvalues lying in suitably small sets. It is for this reason that we use the term “locally definite” for this situation. 10.2 Eigenvalues and Eigensubspaces

Our definitions of these are as in Sections 7.2 and 7.3, where we proved various results, which were likewise based on our present assumption (10.1.2). Here we shall make some improvements, based on results proved in the interim. We take the eigenvalues to be real, on the basis of Theorem 7.2.1. As a sharpening of Theorem 7.2.2 we prove, subject to (10.1.2),

1 74

10

COMPLETENESS OF EIGENVECTORS

THEOREM 10.2.1. If l o ,... ,2,' is an eigenvalue, then for all nonzero associated eigenvectors g the form cp(g, g ) is a nonzero multiple of l o ,... , A,', with proportionality -factor of constant sign. The fact that cp(g, g ) is some multiple of l o ,... ,2,' follows from the fact that the vectors ..' (10.2.1) are themselves proportional to A,, . . . , ,Ik, by (6.8.7). It then follows from (10.1.2) that cp(g, g ) is not a zero multiple of l o ,... , A,' if g is a nonzero decomposable eigenvector and so, since the set of such g is connected, that the proportionality factor has constant sign for such g. We then claim that this is also true for general nonzero eigenvectors g, decomposable or not, by Theorem 7.8.2. In particular, if 1, # 0, then g*A,g is either positive definite or negative definite on the eigensubspace. On the issue of decomposability we note THEOREM 10.2.2. Every eigensubspace admits a linearly independent basis of decomposable elements that are porthogonal. We say that g, g' are cp-orthogonal if q(g', g) = 0, that is, if g*A,g' = 0, t = 0, .. . , k. In the case of elements of an eigensubspace, with eigenvalue Ao, ... , A k , it is sufficient for this, by Theorem 10.2.1, that g*A,g' = 0 for some t with 1, # 0. Jn the latter case, this product will be either positive definite or negative definite on this subspace. By Theorem 7.6.2, we can therefore say that there will be a linearly independent basis for this subspace formed by decomposable eigenvectors that are orthogonal with respect to A,, and so 9-orthogonal. Next we have THEOREM 10.2.3. For each eigenvalue, let there be chosen a cp-orthogonal basis of decomposable tensors. The resulting collection of decomposable eigenvectors is then linearly independent. The number of eigenvalues, counted according to multiplicity, does not exceed v = dim G . The last statement was proved in Theorem 7.3.3, without regard to multiplicity. The proof of the linear independence is very similar. The decomposable eigenvectors (7.3.6) are not now necessarily all associated with distinct eigenvalues. The orthogonality leading to (7.3.9), and to the proof of the assertion, holds by Theorem 7.3.1 if the eigenvectors are associated with distinct eigenvalues, and by construction, on the basis of Theorem 7.6.2, otherwise.

10.3

EIGENPROJECTORS

175

10.3 Eigenprojectors We wish to express a general h E G as the sum of components lying in the various eigensubspaces. As a first step, we separate out its component with respect to a single eigensubspace. Always subject to (10.1.2), we have THEOREM 10.3.1. Let A0, ... ,1, be an eigenvalue, and G' the associated eigensubspace. Tlien f o r each h E G there is a unique expression h = h'

+ h",

h'

E

G',

(10.3.1)

where h" is porthogonal to G', or cp(h",g)r= 0,

g E G'.

(10.3.2)

Let G" denote the subspace of h" with the property (10.3.2), an orthogonal complement of G' with respect to the vector-valued scalar product 40. For the uniqueness of (10.3.1) we must show that G' n G" = 0. Suppose that go E G' n G". It then follows from the definition of G" that cp(go,go) = 0. By Theorem 10.2.1 this shows that go = 0. To show that (10.3.1) is possible at all, we must show that for given h there exists h' E G' such that g*Ath

= g*Ath',

g E G',

(1 0.3.3)

for some t such that 1, # 0. It will be sufficient to ensure that this holds for each g in some set of elements forming a basis of G'. By orthogonalization, we can find a basis of G' of elements g'S'

,

s = 1,

... , m,

(10.3.4)

which are orthonormal in the sense that g(S)*A,g(S')=

I

0

(s # s'),

6

(s = s'),

where 6 = 1 or - 1 according to whether A, is positive or negative definite on G', respectively. We then choose m

h' = 6 C g's'(g's)*A,h) s= 1

and have then (10.3.3) for every g in the set (10.3.4). This completes the proof. The map that takes h into h' is thus well defined, and must be a linear operator and endomorphism of G , taking G into G'. We denote it by P,and

176

10

COMPLETENESS OF EIGENVECTORS

term it the “eigenprojector” associated with the eigenvalue in question. In justification of the name we note that

P2

(10.3.5)

= P,

and that P acts on G‘ as the identity map. Since Im P c G’ it is sufficient to prove the last statement. This consists in the remark that if g E G‘, its decomposition in the form (10.3.1) and (10.3.2) takes the form g 0, since g E G and h“ = 0 is cp-orthogonal to G’. We also have an orthogonality property.

+

10.3.2. r f P, Pt ure eigenprojectors associated with distinct THEOREM eigenvalues, then PPt = PtP

=

(10.3.6)

0.

In particulur, the various eigenprojectors commute. As before, let G‘ denote the eigensubspace associated with P, so that G’ is its image. Then the second equation (10.3.6) is equivalent to the statement that Ptg = 0 for g E G’. In fact, the decomposition of g E G‘ with respect to the eigenvalue associated with Pt takes the form g = 0 g, since 0 lies in the eigensubspace which is the image of Pt, and g is cp-orthogonal to this eigensubspace, by Theorem 7.3.1. For the case of several eigenvalues we have now

+

THEOREM 10.3.3. Let Aos,

*..

,A h ,

s = 1, - * . , p ,

(10.3.7)

be a set of distinct eigenvalues, and let P,,

s = 1)..., p .

(10.3.8)

be the associated eigenprojectors. For h E G write (10.3.9) Then (10.3.10) or any g in any of the eigensubspaces associated with (10.3.7).

By the last theorem we, have from (10.3.9) that ho = ( I - P i )

( I - Pp)h,

(10.3.11)

10.4

EXISTENCE OF A NONSINGULAR DETERMINANTAL MAP

177

so that ho is in the range of I - P I , and so satisfies (10.3.10) for any g in the eigensubspace associated with (10.3.7) for s = 1. The argument applies generally, since the factors on the right of (10.3.11) can be written in any order. We aim to show that ho = 0 if (10.3.7) includes all the eigenvalues. This needs an extra argument. 10.4 Existence of a Nonsingular Determinantal Map While our hypothesis does not call for the existence of a linear combination of the maps A, which should be positive definite, there will nevertheless be linear combinations that are nonsingular. We need one of these to complete the proof of the completeness of the eigenvectors. THEOREM 10.4.1. such that

If (10.1.2) holds, k

A =C

0

is nonsingular.

there exists a set of real p,, . .. , pk

P A

(10.4.1)

If this were not so, then, by Theorem 8.7.2, for every real set p o ,... ,pk there would be an eigenvalue A,, ... , 2, such that ~ k o p s A s= 0. However, by Theorem 10.2.3, or Theorem 7.3.3, there are only a finite number of eigenvalues. If they are denoted by (10.3.7), we claim that it is possible to choose the p, so that

in other words, given a finite number of points, in Rk+',none zero, we can find a k-dimensional subspace not containing any of them. This can be seen in various ways. For example, reasoning inductively, suppose that we have a subspace not containing p - 1 given points. This property persists under small perturbations of the coefficients defining the subspace. This perturbation may then be chosen, if necessary, so as to make the subspace avoid an additional pth point, the points concerned being nonzero. Thus a nonsingular (10.4.1) exists. With such a choice for A we have THEOREM 10.4.2. Let P be an eigenprojector. Then the operators A-IA,, map PG, ( I - P)G into themselves.

s = 0, ... ,k,

(10.4.3)

178

10

COMPLETENESS OF EIGENVECTORS

Here PG is simply an eigensubspace. If ,lo ... ,,2, is the associated eigenvaIue, and g E G, we have from (6.8.8) that k

A - l Asg = isCCP q 4 J - ' g ,

(10.4.4)

0

to that the operators (10.4.3) act on PG as various scalar multipliers, and so define endomorphisms of PG. In considering the case of ( I P)G, let us (as in Theorem 10.3.1) write C' for PG, G" for ( I - P)G. We must show that if q(h,g) = 0 for all g E G', so that / I E G", then cp(A-'A,h, g ) = 0 for all such g, so that A-'A,h E G". Explicitly, we postulate that h is such that

-

g*A,h

=

0,

t

=

0,... , k , g E C',

(10.4.5)

t = 0, ... , k , g E G'.

(10.4.6)

and wish to show that

g*A,A-'A,h = 0,

S,

Let us write ctSt for the left of (10.4.6). We show first, independently of (10.4.5), that for some p we have N,, =

BAJ,,

(10.4.7)

0 IS, t 5 k .

We then show that (10.4.5) implies that p = 0, which will prove (10.4.6). Let ii E G be arbitrary. We note that if 7, = g*A,w,

then

7,

t = 0,... ,k,

(10.4.8)

= u*A,g, and so

A,?, - A,?, = u*(A,A, - 1,A,)g by (6.8.7). Since the A, are real, we have AS7, - Atys = 0,

S,

=

0,

t = 0, ... , k ,

and so, since the 1, are not all zero, 7, = 1,q7

t = 0, ... , k ,

for some q. We apply this result to the left of (10.4.6), with u and obtain us, = ArqS,

0

s S, t -< k.

(10.4.9) =

A-'Ash, (1(

10)

We note now that a,, =

ctt,:,

0 5 s, t 5 k,

by (6.7.4), and this, together with (10.4.10), gives (10.4.7).

(10.4.1 1)

10.5

COMPLETENESS OF THE EIGENVECTORS

179

We now use the fact that k

k

s=o

0

C p p S t = g*AtA - ' ( C

/A,

A,)h = g* Ath = 0,

by (10.4.1) and (10.4.5). Inserting this in (10.4.7) we have k

pA,(C0 psAs) = 0,

0 I t S k.

Since ~ ~ p #J 0,, by (10.4.2) for all eigenvalues, we have PA, = 0. Since the At are not all zero, we have p = 0, which proves (10.4.6), and completes the proof of Theorem 10.4.2. 10.5 Completeness of the Eigenvectors

We have previously established the completeness of the eigenvectors for an array (10.1.1) in Section 7.9, under the condition (7.4.1), in extension of the result of Section 7.4, where we needed the formally stronger requirement that (7.4.1) extended to nondecomposable arguments, as in Definiteness Condition 111. We now assert this result under the weaker condition (10.1.2).

THEOREM 10.5.1. Let (10.1.2) hold f o r the hermitian maps A,, of the finite dimensional complex linear spaces G,, r = 1 , ... ,k . Let (10.3.7) denote all the distinct eigenvalues of the array (lO.l.l), and (10.3.8) the corresponding eigenprojectors. Then I

P

= I1P S .

(10.5.1)

Let us write P

GO = (I - C P,)G 1

=

(I D

= s= 1

- Pl)

(I

- Pp)G

( I - P,)G.

(10.5.2)

We must prove that Go = 0. Since the operators (10.4.3) all define endomorphisms of ( I - P J G , t = 1, ... , p , they define endomorphisms of their intersection Go. Since they commute, by Theorem 6.7.2, they must admit a simultaneous eigenvalue, so that we have A-lAsg = psg,

s = 0, ... ,k,

(10.5.3)

for some nonzero g E Go. It follows from Theorem 6.8.1 that po, ... ,pk is

180

10

COMPLETENESS OF EIGENVECTORS

an eigenvalue, with g an eigenvector. The subspace ( I - P,)G consists of elements that are 9-orthogonal to all eigenvectors associated with the eigenvalue corresponding to P,. Hence their intersection will consist of elements that are 9-orthogonal to all eigenvectors. It will follow that the above element g E Go will be porthogonal to itself, and this is excluded by Theorem (10.2.1).Hence Go = 0, and the proof is complete.

10.6 The Eigenvector Expansion We express Theorem 10.5.1. in more detail in THEOREM 10.6.1. Let (10.1.2) hold. Let the eigenualues of (10.1.1) be written, with repeats according to niultiplicity, in the form (7.6.1). There is a set (7.6.2) of corresponding decomposable eigenvectors, which form a basis of G, and which arc orthogonal in the sense (7.6.3). Here we have appealed to Theorem 10.2.2 for the existence of a decomposable set of 9-orthogonal eigenvectors, associated with any multiple eigenvalue. More explicitly, we have the orthogonality relations

qo(h'",

P')) =0

q(h'", A'f')) # 0

(t

# t')

(10.6.1)

( t = t').

We are not now in a position to assert any form of positivity here in the case t = t'. Nevertheless, we can proceed to evaluate Fourier coefficients. For any g E G we have the expansion g

c k

=

CIA''),

1

(10.6.2)

and now have ~ ( gh'") , = c1cp(h''),h@)).

(10.6.3)

Here both sides are vectors with k + 1 entries that are, however, proportional. By abuse of notation, we shall write C, =

q(g, h(r))/9(h''), A")).

(10.6.4)

By further abuse of notation we then deduce that (10.6.5)

CHAPTER 11

ARRAYS OF COMPACT OPERATORS

11.1 Introduction The completeness of the eigenvectors of a hermitian matrix can be used to establish the completeness of the eigenvectors of certain more general types of operator. In a slight generalization, one considers a hermitian map A of a hilbert space H such that A H is finite dimensional; since A induces a hermitian endomorphism of A H , the result for the matrix case shows that the eigenvectors of A generate A H . In a more substantial and highly important generalization, one uses a limiting process in combination with this last result to establish the completeness, in its range, of a hermitian compact operator on a hilbert space. This latter result includes as a special case completeness properties for eigenfunctions of differential and integral equations. Our purpose in this chapter is to carry out a similar program for multiparameter eigenvalue problems involving hermitian compact operators. Again, we shall have in view applications to similar problems concerning differential and integral equations. As in Chapters 6 and 7, we shall start by considering eigenvalue problems posed in separate spaces, which we shall do in Sections 11.1 and 11.3; in

181

11

182

ARRAYS OF COMPACT OPERATORS

Sections 11.8-1 1.10 we go over to a tensor product setting, which is necessary in order to give any sense to the question of the completeness of the eigenvectors. The intervening sections (1 1.4-1 1.7) are devoted to “truncation,” or to finite dimensional approximation and sequences of such approximations. In this section we note some definitions and properties, closely related to those of Chapter 7, which involve only minimal hypotheses ;the hypotheses will be strengthened in Section 11.3. We make, to start with, the following six assumptions.

(i) The spaces H I , ... , Hk are complex linear, none zero, and not all finite dimensional. (ii) In each H, there is a positive-definite hermitian scalar product ( 9 )r* (iii) G, is a nonzero subspace of H,, r = 1, ... ,k, not necessarily a

proper subspace.

(iv) A,, r = 1, ... ,k, s = 0, ... ,k are linear operators, acting, respectively, from G, to H,. (0)

(11.1.1)

The A,, are hermitian, in the sense that the forms (A,g,, g,),, g, E Gr, are real-valued.

(vi) There holds the “definiteness condition” (1 1.1.2)

subject to O#g,€G,,

r=l,

..., k.

(1 1.1.3)

We go over to an inhomogeneous notion of the term “eigenvalue”; this will be a k-tuple (A,, ... , A,) such that there exists a set (1 1.1.3) satisfying (1 1.1.4)

With the above assumptions we have THEOREM 1 1.1.1.

The eigenvalues are all real.

This is proved in the same way as Theorem 7.2.1. As in Section 7.3, we have a result which can be interpreted as an orthogonality of eigenvectors though, strictly speaking, the latter term can be interpreted only in terms of tensor products.

11.2 NOTIONS FROM HILBERT SPACE TEHORY

183

THEOREM 11.1.2. Let As, s = 1, ... , k, and As’, s = 1, . .. ,k, be distinct eigenvalues, in the sense that As, As’ are different for at least one s. In addition to (1 1.1.4), let k

(Aro Then the k by (k

+ C1& ‘ A r & ’ = 0,

r

s=

=

1,

... ,k.

(1 1.1.5)

+ 1) matrix

r = 1, * * * k, s = 0,

(Angry gr’)r,

9

* * a

9

k,

(11.1.6)

has rank less than k. In particular,

(1 1.1.7) The case (11.1.7) has special value, since it corresponds to a positive definite scalar product, in view of (1 1.1.2). The orthogonality (1 1.1.7) can be extended from distinct eigenvalues to multiple eigenvalues. As previously, the “multiplicity” of an eigenvalue will be the product of the dimensions of the solution-spaces of (11.1.4). We impose in Section 11.3 conditions which ensure that the multiplicity is finite. By Section 7.6, we can then find sets of solutions which are orthogonal in this sense. 11.2 Notions from Hilbert Space Theory

Here we recall some basic definitions and propositions. A complex linear space H is a hilbert space if it is endowed with a positive definite hermitian sesquilinear form or inner product ( , ), with associated norm llfll = ( f , f ) * 2 0, if it is “complete”; by the latter, we mean that every sequence of elements f l ,f 2 , . , which is “Cauchy-convergent” in the sense that llfm -&I1 -+ 0, mY n COY (1 1.2.1)

..

+

admits a limit f to which it converges, in the sense that

Ilf, - f l l

+

0,

n

-b

00.

(1 1.2.2)

We admit the possibility that H may be finite dimensional. If it is infinite dimensional, we distinguish the cases that it is, or is not “separable”; it is “separable” if there is a sequence of its elements which is everywhere dense. Equivalently, it is separable if and only if there exists a complete orthonormal system of elements e l , e 2 , .,. , with (1 1.2.3)

184

11

ARRAYS OF COMPACT OPERATORS

such that every f E H admits an expansion

f = 2 (f,en>en m

(11.2.4)

9

with convergence in the sense of (11.2.2). In addition to ordinary or “strong” convergence, as in (1 1.2.2), we need the notion of weak convergence. We say that f n tends to f weakly, and write f n -J if (fn - J g ) 0, n m, (11.2.5) +

+

for every g E H . It follows from the Cauchy-Schwarz inequality (Theorem 2.5.2) that a strongly convergent sequence is also weakly convergent, with the same limit. A weakly convergent sequence is necessarily bounded, in the sense that the normsfn form a bounded numerical sequence. In the converse direction, every bounded sequence contains a weakly convergent subsequence; bounded sequences are “weakly compact.” We next recall some notions concerning a linear operator A from a space G into H , where the domain G is a subspace of the hilbert space H . It is “bounded” if the ratio llAgllillgIl,

g e G,

is bounded above. If this ratio admits a A is “boundedly invertible.” Let A , B be two linear operators from relative to A” if from Agn-Ag, it follows that Bgn + Bg,

g

z 0,

(1 1.2.6)

positive lower bound, we say that G to H. We say that B is “compact a + a,

(1 1.2.7)

n

(11.2.8)

+

m,

weak convergence being replaced by strong convergence. 11.3 Discreteness of the Spectrum We revert to the operator problem of Section 11.1, where we assumed that the hermitian linear maps A , acted from G, to H , , and satisfied the definiteness requirement (1 1.1.2). We now make more drastic assumptions.

THEOREM 11.3.1. In addition to the above assumptions let ( i ) the A,, map G, onto H,, r = 1, ... , k , (ii) the A,, be boundedly invertible, from H, to G, , (iii) the A,,, s = 1, , k, be compact relative to A,, . Then the eigenvalues have 110finite point of accumulation.

...

11.3

185

DISCRETENESS OF THE SPECTRUM

It follows from (i) and (ii) that A,, : G, -,H, is an isomorphism; this can also be deduced from (iii) and (1 1.1.2). We deduce that any element of H, has a unique expression as Arogr, for some g, E G,. Thus any bounded sequence of elements of the form ArOgrn,n = 1,2, ... , will have a subsequence which converges weakly to some element of H,, and so to some element Ag,, where g, is uniquely determined by the subsequence in question. We note further that if we have two weakly convergent sequences Arogrn

then

(Arsgrn

For (Arsgrn

9

hrJr

9

-

hrJr

- (Arsgr > I?,),

Arogr,

+

=

Arohrn

Arohr

A

(Arsgr, h r h

(Arsgrn

- Arsgr

9

(1 1.3.1)

9

k.

1

s

hr)r

+ (Arsgrn > I7rn

(11.3.2) - hr)r *

Here, the first term on the right tends to zero, since A,,grn + A,,g,, by the compactness assumption. The remaining term can be written (grn, Arshrn - A,&,), ; this tends to zero since the second argument tends (strongly) to zero, and the first argument gr, is bounded uniformly in n, in view of (1 1.3.1) and the hypothesis (ii) of Theorem 11.3.1. We now proceed to the proof of the theorem, and suppose the conclusion to be false. There will then be a sequence of mutually distinct eigenvalues that tends to a finite limit; we shall thus have, with n = 1 ,2, ... , (A,,

k

+ C1As(")Ar,)hp)= 0,

r

=

... , k ,

1,

s=

A ~ ) - + ? L ~ , s =i,

..., k,

n+

03.

(1 1.3.3) (11.3.4)

while the hp' E G,, not being zero, may be normalized by IIArOh?)llr

( 11.3.5)

= 1.

In view of (11.3.5), the sequences A,,h,(") will be weakly compact, and we can choose an n-sequence such that they all converge weakly, with say A,,h?)

-

r = 1, ... , k .

A,,h,,

(1 1.3.6)

It then follows from assumption (iii) that A&''

-+ A,,h,,

s = 1,

... , k .

(11.3.7)

From (1 1.3.3) and (1 1.3.4), we then have that A,ohy' converges strongly, so that (1 1.3.6) holds with strong convergence. We must note that 17, # 0, since otherwise it would follow from (1 1.3.7) that A,&?' + 0, and so A,,h?' + 0, in contradiction to (1 1.3.5).

186

11 ARRAYS OF COMPACT OPERATORS

We now appeal to the orthogonality (1 1.1.7) of eigenvectors associated with distinct eigenvalues. We have det (Ar,h!"),h!"')),= 0,

(11.3.8)

n # n'.

1sr,ssk

By (1 1.3.1) and (1 1.3.2), we may make n, n' + co through pairs of distinct values, to obtain det (Arshr hr)r = 0, 9

1 sr,ssk

in contradiction to (1 1.1.2). This proves Theorem 11.3.1. In a closely related result we prove

THEOREM 11.3.2. Under the assumptions of Theorem 11.3.1, eigenvalues have finite multiplicity. Supposing the contrary, we take it that all of (11.1.4) have nontrivial solutions, whereas the solution space of one of them, or more, has infinite dimension. To simply the notation, we assume this so for r = 1. Let h, , ... ,hk denote nonzero solutions of (11.1.4), r = 2, ... ,k, and let F denote the solution space of (11.1.4) with r = 1. We consider the expression * * *

(Alkhl? hl')l

( A Z l h z ~h Z ) Z

.*.

(A2kh2, hZ)Z

( A k lhk

* **

(Akkhk,

hl')l

W l Y

hl')

= hk)k

hk)k

as a sesquilinear form on F, which is positive definite in view of (11.1.2). If F is infinite dimensional, we can form an infinite sequence of nonzero elements of it, say h?', n = 1, 2, ... , which are orthogonal with respect to $, and which are normalized as in (1 1.3.5). The remainder of the argument follows that of the proof of Theorem 11.3.1. We restrict n to a subsequence such that A,,hf" converges weakly, say to A , , h , , so that A,&') converges strongly to A , & , , 1 I s I k. It then follows that h , # 0. In the result $(h?), hy')) = 0,

n # n',

we then make n, n' + co through distinct pairs, and obtain a contradiction with (11.1.2).

1 1.4 TRUNCATED PROBLEMS

187

11.4 Truncated Problems

The scalar product ( , ), in H, induces, by restriction of the arguments, a scalar product in any subspace of H,, and so in any subspace of G,; this induced scalar product is, of course, positive definite. Similarly, the hermitian forms (Ars. , .),, which are defined at least on G,, induce similar forms in subspaces of the G,, and the definiteness condition (11.1.2) remains in force for these induced forms. We are particularly concerned with the case of finite dimensional subspaces of the G,. Here Theorem 2.6.1 ensures that the restricted forms (A,. , .), can be expressed by means of endomorphisms of the subspaces; we shall express these endomorphisms with the aid of projection operators. We are thus led to formulate a truncated version of the original eigenvalue problem, for which the results of Chapter 7 will be available. Let G,' be a finite dimensional subspace of G,. We will suppose G,' to be given as the range of a hermitan operator P,, which is a projection in the sense that Pr2 = P,. In the truncated eigenvalue problem, associated with these subspaces G,', r = 1, ... ,k, we ask for scalars ,I1, ... , j2k such that there exist gr=Prgr,

grZ0, r=l,.**,k,

(1 1.4.1)

such that k

P,(A,,

+ C ASA,Jg, = 0, s= 1

r = 1,

... , k.

(11.4.2)

Such a set of ,Is will be termed an eigenvalue of this truncated problem. By the theory of Chapter 7, the eigenvalues will be all real, and there will be a set of nonzero decomposable eigenvectors in G1' €3 ... €3 Gk', which are orthogonal in a certain sense. To be more precise, we denote by g;""'

€3

... €3 g;""),

rn = 1,2, ... ,

(11.4.3)

such a set of nonzero decomposable eigenvectors, orthogonal in the sense that det (Arsg;(*),g:'"'), = 0, rn # n. (11.4.4) 1 Sr,ssk

These will correspond to eigenvalues

,I;('"),

... ,

$"),

rn = 1,2, ... ,

(1 1.4.5)

repeated according to multiplicity. The completeness property can be expressed by means of the Parseval equality. For decomposable elements fl

€3

@h, LEG,',

r = 1,... , k ,

(1 1.4.6)

188

11 ARRAYS OF COMPACT OPERATORS

this will assert that

11.5 Sequences of Truncations

We set up a sequence of truncated problems, which approximate in a sense to the original eigenvalue problem (1 1.1.4). These will be determined by means of a sequence of sets of hermitian projectors of the type of the last section. Specifically, we assume the following five properties. (i) For each r

=

1, ... , k, there is a sequence P,,,

n = I, 2,

... ,

(11.5.1)

of hermitian projection operators on Hr. (1 1.5.2)

(ii) PmGr c Gr,

(iii) The Pr, have finite dimensional range. (io) For everyf, E H,, we have Prnf, +L, (0)

n

+

(1 1.5.3)

00,

for some positive c, we have IIPrnAr0Prn.L111 2

Cr IIAroPrnf, Ilr 9

(11.5.4)

for all f, E Hr; in particular, it is sufficient that P,, commute with Are, for all n. It follows from (11.5.3), and the hypothesis that at least one of the G, is infinite dimensional, that the dimension of PrnG, tends to co with n, for at least one r. We shall have a sequence of truncated eigenvalue problems, which call for nontrivial solution of

where h,, E PrnGr.We denote the eigenvalues, repeated according to multiplicity, by

A;:', ... , A,$",

m = I, 2,

... .

(11.5.6)

11.6

CONVERGENCE OF THE EIGENVALUES

189

There will be only a finite number of these, given by the dimension of the tensor product (11.5.7) PInGI @ ” ‘ @ PknGk. There will be a basis of this formed by tensor products of nontrivial solutions of (1 1.5.5), say

h!:)

.-. hi,),

m

=

I, 2, ... ,

(1 1.5.8)

and these may be taken to be orthogonal in the sense (11.1.7). Furthermore, we may normalize them by IIAroh(.;’llr = 1,

r = 1, ... , k.

(1 1.5.9)

We wish to arrange the numbering, both in m and in n, so that convergence obtains in these entities. 11.6 Convergence of the Eigenvalues This can be arranged rather simply. We suppose that for each n the eigenvalues (1 1.5.8) are ordered so that (11.6.1)

In:;)Jz s= 1

is nondecreasing in m. If (11.6.1) takes the same value for two or more distinct eigenvalues, we suppose these ordered lexicographically, for example. We then turn to considering the convergence properties of the sequences A(m) In,

... , Ai:),

n

+

co,

(1 1.6.2)

for each fixed rn. For any m, this sequence will be nonempty if n is sufficiently large, since at least one of the G, is infinite dimensional, and by (11.5.3). Starting with the case m = 1, we note that if the sequence is bounded, then there exists a convergent subsequence; if it is not bounded, we can select a subsequence tending to co,in the sense that (11.6.1) tends to 03. Thus we can select an n-sequence i t l l < n I 2 < ... such that, when rn = 1, (11.6.2) either converges or tends to co.We then select a subsequence n Z 1 < nZ2 < of n 1 n I 2 ,... such that the same holds for (1 1.6.2) when rn = 2, a subsubsequence such that this holds also when m = 3, and so on. Then the diagonal sequence I Z ~ nzz, ... will have the property that the sequence (11.6.2), for every m, either converges or tends to co, if n is restricted to this sequence. Here we remark that there are two possibilities. Either (11.6.2) tends to a finite limit for every in, as I Z runs through this diagonal sequence, or else this is so for some limited range of rn, say for 1 I rn Irn,,, while for

190

11

ARRAYS OF COMPACT OPERATORS

m > m, (1 1.6.2) tends to 03. This follows from our prescription that (I 1.6.1) is nondecreasing in m. To simplify the notation, we suppose the sequence of truncations reduced and renumbered, so that convergence holds for the eigenvalues (1 1.6.2), as n runs through the full sequence n = 1,2, ... , except conceivably for cases in which (11.6.2) tends to co. Where the limit of (11.6.2) is finite, we write

):1

+

A:'"),

s

1, ... , k.

=

(11.6.3)

11.7 Convergence of the Eigenvectors

Supposing that the eigenvalues converge, either to finite limits as in (11.6.3) or to co, we discuss the behavior of the nontrivial solutions of (11.5.5). We have

+

Pr,,(Aro where

k

=

P,,h!:'

E

r = 1, ... ,k,

(11.7.1)

~ ~ A r 0 h=~1.~ ) ~ ~ ,

(11.7.2)

= 0,

@&)hi:)

s= 1

G,,

In view of (1 1.7.2), the sequences A,,h:,"), n -+ coy will be weakly compact, and we can write, for some n-sequence, ~,,h!,") - ~ ~ , h ! " ) ,

-

n

+

co.

(11.7.3)

From our compactness assumption it then follows that, if 1 5 s 5 k,

Arsh!:'

A,,h!"'.

(11.7.4)

We wish to proceed to the limit in (11.7.1). We note the property that iff,,, -f, in H, as n + co then Pr,,f,, +f,. For P r L n - f, = (Prnf, - L)+ Prn(f,n - f,),

and the first term on the right tends to zero by (11.5.3), and the second since P,,, has norm 1. Suppose that m is such that convergence holds in (1 1.6.2) and (1 1.6.3). It then follows from (1 1.7.4) that (11.7.5) for the n-sequence in question. It then follows from (1 1.7.1) that the sequence

PrnAroh2)

(1 1.7.6)

converges strongly. However we have also that

P,,,A,,h!;)-

ArOh!").

(11.7.7)

11.7 For iff

CONVERGENCE OF THE EIGENVECTORS

191

E H,,

+ (Aroh!,"),P n f

-f ) r

Here the first two terms on the right give zero in the limit, by (1 1.7.3), and the last tends to zero by (11.53) and (11.7.2). Since (11.7.6) converges strongly, and has the weak limit given in (1 1.7.7), this is also its strong limit, and so (11.7.8) P,,A,,h!,") + Ar0lt!"). On proceeding to the limit in (11.7.1) we therefore have, for a suitable n-sequence, k

(Aro + C 1~"),4,,)h~'")= 0, s=l

r = 1, ... ,k.

( 11.7.9)

Furthermore, we note that h!") # 0. For it follows from (11.7.2) that lP,J,~h!,") 1, 2 c, > 0, and so the right of (1 1.7.8) cannot be zero. We have shown that for any rn such that the rnth eigenvalue converges to a finite limit (1 1.6.3), we can select an n-sequence such that the convergence properties (1 1.7.5) and (1 1.7.8) hold, with the consequence that the limit of this sequence of eigenvalues of the truncated problems is an eigenvalue of the original problem. As previously, we select an n-sequence such that this convergence holds when m = 1, a subsequence of this such that the convergence holds also when m = 2, and so on indefinitely, or at least so long as convergence to a finite limit holds for the mth eigenvalue. The diagonal sequence will then have the property that this convergence holds for all m if (1 1.6.3) holds for all rn; if it holds up to some point only, we can form a suitable n sequence by a finite number of selections only. Once again, we suppose the sequence of truncations reduced and renumbered, so that as n + co we have the convergence of eigenvalues (1 1.6.3), and the convergence of (11.7.1) to (11.7.9), either for all m, or for m up to some point only, after which the mth eigenvalue tends to infinity. Provisionally, we use the term "limiting eigenvalue" for an eigenvalue of the original problem which is given by (1 1.6.3), as a limit of eigenvalues of the truncated problems. In a sense yet to be made precise, we use the term "limiting eigenvector" for the expressions

hi"'

@

*.. @ hi").

(11.7.10)

yielded by the limiting processes (1 1.7.3). The eigenvectors of the truncated problems were chosen so as to be mutually orthogonal in the sense (11.1.7). One may proceed to the limit in

192

11

ARRAYS OF COMPACT OPERATORS

these orthogonality relationships, so as to establish a similar orthogonality in the limiting eigenvectors (1 1.7.lo). We have det (ArshLy),h!:')),

I 0 there is an M > 0 such that

196

11 ARRAYS OF COMPACT OPERATORS

if the sum is extended over those eigenvalues, of the nth truncated problem, for which

(11.8.18) It is here that we need the hypothesis (11.8.8). We extend the notation As from s = 0, given by (6.2.2), to general s as in (6.4.4). We have, as for (6.4.9),

Aous = A p ,

s = 1,

... ,k.

(1 1.8.19)

Also, we have from (11.7.1) and (11.8.8) that

+c k

(Aroh!,"',u ~ ) ~ A:;)(Arsh!;), s=1

r = 1,

= 0,

... ,k.

(11.8.20)

Using "Cramer's rule," we obtain a result that may be written

Ag;)(A

,u )

o '~( mn)

s = 1,

= (Asjim),u),

... , k .

(11.8.21)

Here the operators Ao, As may be transferred to the second arguments in the scalar products, or the arguments may be interchanged, without affecting the validity of the results. We deduce that, if A$') # 0,

(Aou, jp))= ( u , A0j!,"'lm,)= (l.:;))-'(u, Asjim)) =

(A;;))-'(Asu,

=

(A2;))-'(AOus, jLm)),

j:))

(11.8.22)

by (11.8.19). We note now that if (1 1.8.18) holds, then we have

p:?l

> (M/k)+,

cs

(11.8.23)

denotes the sum on for each n, m and at least one s = 1, ... , k. Thus if the left of (11.8.17) taken over all eigenvalues of the nth truncated problem for which (1 1.8.23) holds, then the sum over the range (1 1.8.18) does not , x k . Thus (11.8.17), subject to (11.8.18), will be ensured exceed 1 if we can choose M such that

+

.*+

+

c < Elk,

s = 1,

... , k .

S

To arrange this, we note that (1 1.8.22) implies that

c 5 ( k / M )c [(Aovs,~!?)~2/@0j~m)7 jLm% S

where the sum on the right may be taken over all m.

(11.8.24)

11.9

DISCUSSION OF THE EXPANSION

197

We now appeal to the Bessel inequality, a weaker form of the Parseval equality; the Bessel inequality is valid for orthogonal systems whether complete or not. Since the product (Ao . , .) is positive semidefinite, or definite, on ,'A and thej,!"', rn = 1,2, ... , are orthogonal with respect to it, we have

Thus there exists an M > 0 so that (11.8.24) holds, and so such that (I 1.8.17) holds for the sum over (11.8.18). On the basis of this result the procedure of passing to the limit as n -+ 00 in (1 1.8.13) is justified, and we obtain the desired result (1 1.8.10).

11.9 Discussion of the Expansion In Theorem 11.8.1, we obtained the Parseval equality for certain decomposable elements of the tensor-product space, with respect to a certain set of limiting eigenvectors. As is often possible in the theory of eigenfunction and eigenvector expansions, one can now appeal to general principles concerning the Parseval equality in order to obtain an improved result. One type of improvement relates to the circumstance that the limiting procedure of Sections 11.5-11.7 depended on the choice of the projection operators P,,,and these in view of (113.9) depend on the vector u being expanded. Thus the eigenvectors (1 1.8.11) are not independent of u. We must show that they may nevertheless be assumed independent of u, or that (1 1.8.10) is valid for some fixed set of /I("). Let us first note that the Parseval equality expresses an approximation property, and so indeed an "expansion." It will be convenient for this purpose to suppose the / I ( " ) of (1 1.8.11) normalized by

(Aolz'"), h'"')

=

1,

rn

=

1, 2,

... ;

(1 1.9.1)

this may be arranged by imposing a similar normalization on the eigenvectorsj:") of the truncated problems. For any integral p 2 1, we define (1 1.9.2)

198

11

ARRAYS OF COMPACT OPERATORS

where the Fourier coefficients a,,, are given by (1 1.9.3)

a, = (Aou, It'"").

It is then found that

-c P

@ow,, wp> = (Aou, u )

1

la,I2.

(11.9.4)

If u satisfies the requirements of Theorem 11.8.1, the Parseval equality ensures that

c m

(Aou, u ) =

1

(11.9.5)

lam12,

in view of the normalization (1 1.9.1). We deduce that (Aowp, wp)

0,

--f

P

+

a.

(1 1.9.6)

Thus u can be approximated arbitrarily closely by linear combinations of eigenvectors in the metric associated with the scalar product (Ao . , .); this may be a weaker form of approximation than that associated with Y

1.

Now let g(4)

,

q = 1, ... ,

(1 1.9.7)

be some fixed set of decomposable eigenvectors, tensor products of nonzero solutions of (1 1. 1.4), which are orthonormal in the sense (11.9.8) Let this set be maximal in the sense that the number of (11.9.7) associated with any eigenvalue be equal to its multiplicity; the existence of such sets is ensured by Chapter 7. Since wp can be expressed in the form xt = u

-

t

(11.9.9)

pqg(q),

1

in view of the fact that the h'") are finite linear combinations of the g ( 4 ) ,it follows that we can approximate to u arbitrarily closely by means of finite linear combinations of the g(4),in the metric given by (Ao . , .), provided that u satisfies the requirements of Theorem 11.8.1. Now write (1 1.9.10) Y g = (Aou, g'4'). With x , given by (1 1.9.9), a simple calculation gives t

(Aoxt, x t )

=

(Aou, u )

t

- C1 lY41' + C1 IPq - y4l2-

(11-9.11)

1 1.10

A SPECIAL CASE

199

Since A. is positive definite on finite linear combinations of decomposable elements, we have that the right is nonnegative, whatever the values of Bq, and so have the Bessel inequality

(1 1.9.12) On the other hand, we have from (1 1.9.6) that the right-hand side of (1 1.9.11) can be made arbitrarily small, by taking t large and choosing the pq suitably. co, Thus the left-hand side of (1 1.9.12) cannot tend to a positive limit as t and so must tend to zero; thus we have the Parseval equality (11.9.13)

for any u satisfying the conditions of Theorem 11.8.1, with respect to a fixed set of decomposable orthonormal eigenvectors, independent of u. In the preceding we have written sums over the set of eigenvalues and eigenvectors for the event that the number of eigenvalues is infinite; if this number is actually finite, the sums will, of course, be finite. Having fixed on some particular set (1 1.9.7) of orthonormal decomposable eigenvectors, we can turn to a different aspect-namely the extension of the class of elements for which the Parseval equality and associated approximation property holds. If dl), ... , d P )are elements satisfying the requirements of Theorem 11.8.1, they can all be approximated arbitrarily closely by finite linear combinations of elements (11.9.7). Since A. is positive definite on decomposable elements, it follows that any linear combination of dl),... ,d P ) can be arbitrarily closely approximated by linear combinations of (I 1.9.7), always in the sense of the metric given by (& . , .), and thence that the Parseval equality holds for finite linear combinations of dl), ... ,dP).One can then take the further step of extending this ParsevaI equality to the closure, with respect to (Ao , .), of this set of linear combinations of elements satisfying the requirements of Theorem 1 1 3 . 1 .

.

11.10 A Special Case In many cases, the operators A,, in the eigenvalue problem (11.1.4) are positive definite; for example, they may be differential operators of the Sturm-Liouville type, applied to spaces of functions subject to boundary conditions. The auxiliary inner products ( , ), may then be eliminated in favor of the products (Aro , .),. Here we achieve a similar effect by replacing the A,, by Z,, the identity on H, . The A,,, 1 I s I k, will then have to be compact with respect to I,, and so in fact just compact; the latter

.

200

11 ARRAYS OF COMPACT OPERATORS

will be interpreted to mean that for any weakly convergent sequence h,, one has A,,h,, -+ A,,h,, strongly.

-

h,

We bring together all necessary hypotheses in the following: ( i ) HI,... , Hk are separable hilbert spaces, not all finite dimensional, with scalar product written ( , ), . (ii) A,,, r , s = 1, ... , k, are bounded, symmetric, and compact endomorphisms of the H , , respectively. (iii) There holds the definiteness property det

15r.sSk

(Arshr

for all nonzero h, E H,, r = 1,

3

hr)r

(1 1.10.1)

> 0,

... ,k .

By X , we denote the hilbert space tensor product of the hilbert spaces H,, and by A,: the induced endomorphism corresponding to A,,; for the purposes of the following result, it will be permissible to replace X by a subspace of this hilbert space tensor product, so long as it includes all decomposable elements of the tensor product. As a special case of Theorem 11.8.1 we have THEOREM 11.10.1.

With the preceding assumptions, let Aim)

, ... ,Aim),

m = 1,2, ... ,

(1 1.10.2)

be a set of eigenvalues of the problem (1 1.10.3)

where 0#h,~H,,

r=l,

..., k,

(1 1.10.4)

each eigenvalue being repeated according to multiplicity. Let corresponding nontrivial solutions of (1 1.10.2) be

h!"',

r = l , ..., k , m = 1 , 2 ,...,

(1 1.10.5)

chosen so as to be orthonormal in the sense det (A,,h~"),h!'"')), = lSrs,Sk

Let 11

= u1 0

-.. 0 uk,

l110

u, E H , , r

(m = m'h (m

z

mf).

=

1,

... ,k ,

(1 I. 10.6)

(1 1.10.7)

NOTES FOR CHAPTER

11

20 1

be a decomposable element of X f o r which there exists other elements v l , . .. , v k of X such that k

r = l , ..., k.

u+xA$,=O, s= 1

(11.10.8)

Then there holds the Parseval equality

(A0u, U ) = 1I(A0u, h'"')1'. m

(11.10.9)

Here A, has the same meaning as in Section 11.8; the result differs from (1 1.8.10) only in that the eigenvectors are assumed normalized. The proof consists simply in checking that our assumptions guarantee that the various hypotheses of Sections 11.1, 11.3, and 11.8 are satisfied. In the present formulation, the subspaces G, coincide with the whole spaces H,. The conditions (i)-(ui) of Section 11.1 clearly hold. Conditions (i) and (ii) of Section 11.3 hold, since A,, = I,, the identity, while condition (iii) has been duly postulated with A,, = I,. In Section 11.8, the subspaces Y, Z of X now coincide, and X may be taken to be the hilbert space tensor product. As noted in Section 11.8, the bounded operators A , induce bounded maps of X , if the A,, are bounded, so that the requirements (iu)-(ui) are satisfied. Finally, we note that the requirement (6) of Theorem 11.8.1 is automatically satisfied; for any u, can be considered as the first of a complete orthogonal set of elements of H , ; we can then take P,, to be the hermitian projection whose range is the subspace generated by the first n members of this orthogonal set. The requirement (11.8.9) is then satisfied, as are also (i)-(0) of Section 11.5; in the case of the latter, (11.5.4), we can take c, = 1, since A,, = I,. This completes the proof.

Notes for Chapter 11 One of the principal assumptions for the expansion theorems of this chapter is that the operators A,, , ... , Ark should be compact relative to A,,, r = 1, ... , k . This may be realized in two ways. In one of these, the A,, correspond to differential operators, which will be unbounded, while the A , , , ... , Ark are well-behaved operators, such as those obtained by multiplying by bounded functions. In the second way, which arises when the differential operators are inverted by means of Green's functions, the A,, are identities, while the A r l , ... , Ark are integral operators associated with continuous kernels. It is the latter situation for which Theorem 11.10.1 is adapted. A version suitable for the previous situation has been given by Browne (1971).

202

11 ARRAYS OF COMPACT OPERATORS

-

Our definition of a compact operator V on a hilbert space, that for any h we should have Vh, -,Vh strongly, weakly convergent sequence h, may be shown in the case of hermitian V to be equivalent to another, perhaps more common definition; this is that for any bounded sequence g,, the sequence Vg, should contain a convergent subsequence. As a general reference on hilbert space theory, one may cite, for example, the text of Ahiezer and Glazman (1966). The tensor product of hilbert spaces was considered in Murray and von Neumann (1936). For a recent account see Prugovecki (1971). The motivation for the multiparameter completeness theorem of this chapter is to be found in the special case of multiparameter Sturm-Liouville theory, which will be the subject of Volume I1 of this work. Among early work on this subject, in the differential equation context, may be mentioned that of Bocher (189711898) and Carmichael (1921-1922); the latter author, as we have already mentioned, was moved to investigate the matrix analog, the subject of the present volume. Two-parameter problems for differential equations have been considered by Arscott (1964), and the k-parameter case by Faierman (1969). An early investigation of the completeness of eigenfunctions for a twoparameter differential equation problem was that of Hilbert (1912). A twoparameter integral equation problem was investigated by Anna J. Pel1 (1922).

Ahiezer, N. I. and Glazman, I. M. (1966). “Theory of linear operators in hilbert space.” 2nd ed. Moscow. Albert, A. A. (1938). A quadratic form problem in the calculus of variations, Bull. Amer. Math. SOC.44, 250-252. Anselone, P. M. (1963). Matrices of linear operators, Enseignement Math. 9 (2), 191-197. Arscott, F. M. (1964). Two-parameter eigenvalue problems in differential equations. Proc. London Math. SOC.14,459470. Atkinson, F. V. (1963). Boundary problems leading to orthogonal polynomials in several variables. Bull, Amer. Math. SOC.69, 345-351. Atkinson, F. V. (1964a). “Multivariate Spectral Theory: The Linked Eigenvalue Problem for Matrices.” Technical Summary Report No. 431, Math. Research Center (U.S. Army), Madison, Wisconsin. Atkinson, F. V. (1964b). “Discrete and Continuous Boundary Problems.” Academic Press, New York. Atkinson, F. V. (1965). Singularity of determinantal endomorphisms. Abstract No. 619-85, Notices Amer. Math. SOC.12, 81. Atkinson, F. V. (1968). Multiparameter spectral theory, Bull. Amer. Math. SOC.74, 1-27. Bellman, R. (1960). “Introduction to Matrix Analysis.” McGraw-Hill, New York. Berezanskii, Yu. M. (1965). “Expansions in Eigenfunctions of Self Adjoint Operators.” Naukova Dumka, Kiev. Engl. trans.: Translations of Mathematical Monographs 17 Amer. Math. SOC.,Providence, Rhode Island, 1968. Bocher, M. (1897-1898). The theorems of oscillation of Sturm and Klein, I. Bull. Amer. Math. SOC.4, 295-313. Bocher, M. (1897-1898). The theorems of oscillation of Sturm and Klein, 11. Bull. Amer. Math. SOC.4, 365-376. Bocher, M. (1897-1898). The theorems of oscillation of Sturm and Klein, 111, Bull. Amer. Math. SOC.5 , 2 2 4 3 . Bonnesen, T. and Fenchel, W. (1934). “Theorie der konvexen Korper,” Ergeb. der Math. 3. Springer, Berlin, 1934. de Boor, C. and Rice, J. R. (1964). Tensor products and commutative matrices. SIAM J. Appl. Math. 12, 892-896. Borsuk, K. (1933). Drei Satze uber die n-dimensionale euklidische Sphare, Fund. Math. 20, 177-190. Bourbaki, N. (1948). “Algebre,” Chap. 111, Paris, 1948. Brown, A. and Pearcy, C. (1966). Spectra of tensor products of operators. Proc. Amer. Mafh. SOC.17, 162-166. Browne, P. J. A multiparameter eigenvalue problem. J. Math. Anal. AppI. (to appear). Calabi, E. (1964). Linear systems of quadratic forms. Proc. Amer. Math. SOC.15, 844-846. Carmichael, R. D. (1921a). Boundary value and expansions problems: algebraic basis of the theory. Amer. J. Math. 43, 69-101. Carmichael, R. D. (1921b). Boundary value and expansion problems: formulation of various transcendental problems. Amer. J. Math. 43, 232-270. Carmichael, R. D. (1922). Boundary value and expansion problems: oscillatory, comparison and expansion problems. Amer. J. Math. 44, 129-152. Carroll, R. W. (1963). Problems in linked operators, I. Math. Ann. 151,272-282. Carroll, R. W. (1965). Problems in linked operators, 11. Math. Ann. 160, 233-256. Chambadal, L. and Ovaert, J. L. (1968). “Algebra lineaire et algebre tensorielle.” Dunod, Paris.

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Cordes, H. 0. (1953.) Separation der Variablen in Hilbertschen Raumen. Math. Ann. 125, 401-434. Cordes, H. 0. (1954-1955). Uber die Spektralzerlegung von hypermaximalen Operatoren, die durch Separation der Variablen zerfallen, I. Math. Ann. 128, 257-289. Cordes, H. 0. (1954-1955). Uber die Spektralzerlegung von hypermaximalen Operatoren, die durch Separation der Variablen zerfallen, 11. Math. Ann. 128, 373-411. Daleckii, Yu. L., Krein, M. G. (1970). “Stability of Solutions of Differential Equations in a Banach Space.” Izdatel’stvo Nauka, Moscow. Davis, C. (1970). Representing a commuting pair by tensor tensor products. Lin. Alg. Appl. 3, 355-357. Dines, L. L. (1936). Convex extensions and linear inequalities. Bull. Amer. Math. SOC.42, 353-365. Dines, L. L. (1941). On the mapping of quadratic forms. Bull. Amer. Math. SOC.47,494-498. Dines, L. L. (1942). On the mapping of n quadratic forms. Bull. Anier. Math. SOC.48, 467-471. Dines, L. L. (1943). On linear combinations of quadratic forms. Bull. Amer. Math. SOC.49, 388-393. Dunford, N. (1965-1966). A spectral theory for certain operators o n a direct sum of hilbert spaces. Math. Ann. 162, 294-300. Faierman, M. (1969). The completeness and expansion theorems associated with the multiparameter eigenvalue problem in ordinary differential equations. J. Differential Equations 5, 197-213. Finsler, P. (1937). Uber das Vorkommen definiter und semi-definiter Formen in Scharen quadratischer Formen, Comment Math. Helvet. 9, 188-192. Friedman, B. (1956). An abstract formulation of the method of separation of variables. In “Proceedings of the Conference on Differential Equations,” pp. 209-226. Univ. of Maryland, College Park, Maryland. Friedman, B. (1961). Eigenvalues of composite matrices. Proc. Cambridge Phil. SOC. 57. 37-49. Gantmaher, F. R. and Krein, M. G. (1960). “Oszillationsmatrizen, Oszillationskerne und kleine Schwingungen mechanischer Systeme,” Akademie-Verlag, Berlin. Gelbaum, B. R. (1962). Tensor products and related questions. Trans. Ainer. Math. SOC. 103, 525-548. Gel’fand, 1. M. (1948). “Lectures on Linear Algebra.” Moscow, 1948. Godement, R. (1968). “Algebra,” Hermann, Paris. Engl. trans.: Houghton-Mifflin, Boston, 1968. Greub, W. (1967a). “Linear Algebra,” 3rd ed. Springer-Verlag, Berlin. (Die Grundlehren der Math. Wiss. 97). Greub, W. (196713). “Multilinear Algebra.” Springer-Verlag, Berlin (Die Grundlehren der Math. Wiss., 136). Grobner, W. (1966). “Matrizenrechnung.” Bibliographisches Institut, Mannheim, 1966. Grothendieck, A. (1955.) Produits tensorielles topologiques et espaces nucleaires. Menioirs Amer. Math. SOC.16. Griinbaum, B. (1967). “Convex polytopes.” Wiley (Interscience), New York. Halmos, P. (1958). “Finite-Dimensional Vector Spaces,” 2nd ed. Van Nostrand-Keinhold, Princeton, New Jersey. Halmos, P. (1967). “A Hilbert Space Problem Book.” Van Nostrand-Reinhold, Princeton, New Jersey. Hausdorff, F. (1919). Der Wertvorrat einer bilinearform. Math. Z.3, 314-316. Hestenes, M. R. (1968). Pairs of quadratic forms. Linear Alg. Appl. 1, 397-407.

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Hestenes, M. R. and McShane, E. J. (1940). A theorem on quadratic forms and its application in the calculus of variations. Trans. Anier. Math. SOC.47,501-512. Heuser, H. (1969). Die Iteration finiter operatoren auf Raumen mit halbskalarprodukten. Math. Ann. 182,21 3-231. Hilbert, D. (1912). “Grundziige einer allgemeinen Theorie der linearen Integralglerchungen.” Teubner, Leipzig. Jacobson, N. (1953). “Lectures in Abstract Algebra,” Vol. 2. Van Nostrand-Reinhold, Princeton, New Jersey. Jaeger, A. and Kuo, Y. (1967). On tensor products of systems of linear equations. I n “Methods of Operations Research,” Vol. 111, pp. 249-269. Verlag, Anton Hain, Meisenheim am Clan. Karlin, S. (1968). “Total Positivity.” Stanford Univ. Press, Stanford, California. Lang, S. (1965). “Algebra.” Addison-Wesley, Reading, Massachusetts. Lynch, R. E., Rice, J. R. and Thomas, D. E. (1964a). Tensor product analysis of partial difference equations. Bull. Amer. Math. SOC.70, 378-384. Lynch, R . E., Rice, J. R. and Thomas, D. E. (1964b). Direct solutions of partial difference equations by tensor product methods. Numer. Math. 6, 185-199. Lynch, R. E., Rice, J. R . and Thomas, D. E. (1965). Tensor product analysis of alternating direction implicit methods. J . SOC.Znd. Appl. Math. 13,995-1006. Mal’cev A. I. ( I 948). “Foundations of Linear Algebra,” Moscow-Leningrad. Maclane, S. (1963). “Homology.” Springer-Verlag, Berlin (Die Grundlehren der Math. Wiss. 114). Marcus, M. (1960). Basic theorems in matrix theory, Nut. Bur. Stand. Appl. Matk. Ser., No. 57. Marcus, M. (1964). The use of multilinear algebra for proving matric inequalities. Zn “Recent Advances in Matrix Theory” (H. Schneider, ed.). Univ. of Wisconsin Press, Madison, Wisconsin. Marcus, M. and Minc, H. (1964). “A Survey of Matrix Theory and Matrix Analysis.” Allyn and Bacon, Boston. Markham, T. L. (1970). On oscillatory matrices. Lin. Alg. Appl. 3, 143-156. Morse, P. M. and Feshbach, H. (1953). “Methods of Mathematical Physics,” Vol. 1. McGraw-Hill, New York. Murray, F. J. and von Neumann, J. (1936). On rings of operators. Ann. Math. 37,116-229. Pell, Anna J. (1922). Linear equations with two parameters. Trans. Amer. Math. SOC.23, 198-211. Prugovecki, E. (1971). “Quantum Mechanics in Hilbert Space.” Academic Press, New York. Reid, W. T. (1938). A theorem on quadratic forms. Bull. Amer. Math. Soc. 44, 437-440. Schatten, R. (1950). A theory of cross-spaces. Annals Math. Studies, No. 26, Princeton Univ. Press, Princeton, New Jersey. Schatten, R. (1960). Norm ideals of completely continuous operators, Ergeb. der Math. Taussky, 0. (1967). Positive-definite matrices. In “Inequalities” (0. Shisha, ed.), pp. 309-319. Academic Press, New York. Taylor, A. E. (1966). Theorems on ascent, descent, nullity and defect of linear operators. Math. Ann. 163,18-49. Valentine, F. A. (1964). “Convex Sets,” New York. van der Waerden, B. L. (1966). “Algebra,” 1. Teil. Springer-Verlag, Berlin. van der Waerden, B. L. (1967). “Algebra.” 2. Teil. Springer-Verlag, Berlin. Westwick, T. R. (1967). Transformations on tensor spaces. Pacific J . Math. 23, 613-620. Williamson, S. G. (1969). Tensor contraction and hermitian forms. Linear Algebra and Its Apjdications 2, 335-347.

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INDEX

Convex cone, 153 Corank of eigenvalue, 46 Cordes, H. O., 92, 204

A Ahiezer, N. I., 202, 203 Albert, A. A., 39, 203 Algebraically closed, 2 Anselone, P. M., 114, 203 Arscott, F. M., 202, 203 Ascent, 40, 52, 59 Atkinson, F. V., 114, 135, 146, 203 Autornorphisrn, 4

D Davis, C., 78, 171, 204 de Boor, C., 78, 203 Definiteness, 107, 117, 121, 163, 168 local, 172 Dependence, linear, 10 Descent. 41, 59 Determinant, 92 operator-valued, 82, 101-105, 129, 143 Dimension, 10 of direct sum, 14 of dual space, 15 ofHom, 14 of tensor product, 65 Dines, L. L., 39, 171, 204 Direct product, 78 Direct sum, 8-10 Direct sum decomposition, 9,44 Dunford, N., 114, 204

B Basis, 11, 63 Bellman, R., 78, 92, 93, 203 Berezanskii, Yu, M., 78, 92, 203 Bilinear, 23 Bocher, M., ix, 202, 203 Bonnesen, T., 171, 203 Borsuk, K., 153, 171, 203 Bourbaki, N., 78, 203 Brown, A., 92,203 Browne, P. J., 201, 203

C Calabi, E., 39, 203 Carmichael, R. D., x, 113,202, 203 Carroll, R. W., 92, 203 Chambadal, L., 78,203 Characteristic equation, 48 Characteristic of field, 2 Coirnage, 5 Cokernel, 5 Compact operator, 184, 199-200 Connectedness, 18

E Eigenprojector, 175 Eigensubspace, 45, I12 Eigenvalue, 45 of array of operators, 111 , 117, 173 of commuting set of operators, 50 of induced endomorphism, 82 simple, 122, 126 207

208

INDEX

Eigenvector, 53-55 for array of operators, 112, 119 decomposable, 112, 119, 120, 127 of pair of forms, 57 Endomorphism, 4 hermitian, 30, 52 induced, 79 Epimorphism, 4

F Faierman, M., 135, 202, 204 Fenchel, W., 171, 203 Feshbach, H., ix, 205 Finder, P., 39, 204 Form, bilinear, 26, 74 sesquilinear, 27, 76 hermitian, 28, 77 Friedman, B., 92, 204 Functional, linear, 6 G

Gantmaher, F. R., 93, 204 Gelbauml B. R., 78 204 Gel’fand, I. M., 78,1204 Glazman, I. M., 202, 203 Godemont, R., 78,204 Greub, W., 19, 39, 59, 78, 92, 204 Grobner, W., 204 Grothendieck, A., 78, 204 Grunbaum. B., 171,204

H Halmos, P., 39, 78, 204 Hausdorff, F., 39, 204 Hermitian form, 28 arrays of, 147-153, 161-171 pairs of, 33-35, 57, 58 sets of, 35-39 zeros of, 31-33 Hestenes, M. R., 39, 204 Heuser, H., 39, 205 Hilbert, D., ix, 140, 202, 205 Hilbert space, 183, 202 tensor product, 200,202 Homomorphism, 4 induced, 6, 69

I Image, 4 Isomorphism,

4

J Jacobson, N., 78, 205 Jaeger, A., 92, 205

K Karlin, S., 93, 205 Kernel, 4 Klein, F., ix Krein, M. G., 93, 204 Kronecker product, 78, 84-86,92 Kronecker sum, 84-86 eigenvalues of, 86, 92, 93 Kunneth tensor formula, 78 Kuo, Y., 92, 205

L Lang, S., 39, 78, 205 Linear dependence, 10 Lynch, J. R., 92,205

M MacLane, S., 78,205 McShane, E. J., 39,204 Mal’cev, A. I., 39, 205 Map bilinear, 23 composite, 6 determinantal, 101, 177 induced, 6 linear, 4 multilinear, 20 semilinear, 19 Marcus, 78,205 Markham, T. L., 93, 205 Minc, H., 78,205 Minimal polynomial, 50 Monomorphism, 4 Morse, P. M., ix, 205 Multilinear, 20 Multiparameter Sturm-Liouville theory, ix, x, 121, 135, 201, 202 Multiplicity of eigenvalue, 122, 186 Murray, F. J., 202, 205

209

INDEX

Norm,

N

17

0 Orthogonality, 53, 119, 174 Ovaert, J. L., 78, 203 P Pearcy, C., 92, 203 Pell, Anna J., 202, 205 Positive-definite, 27, 29 Prugovecki, E., 202,205 Quadratic,

25

Q R

Range, 4 numerical, 35, 39 of quadratic form, 32 Rank of eigenvalue, 45 Reid, W. T., 39, 205 Rice, J. R., 78, 203, 205 Root subspace, 45, 59, 113 S Schatten, R., 78, 205 Scherk, P., 171

Separation of variables, ix, 88, 92 Sesquilinear, 27 Skewsymmetric, 26 Space dual, 6 finite-dimensional, 10 linear, 1 quotient, 3 zero, 2 Subspace, 3 T Taussky, O., 39, 205 Taylor, A. E., 59, 205 Tensor, 62 decomposable, 62, 63, 65, 66, 130, 143 product, 60, 192 Thomas, D. E., 205 Topology, 16

V Valentine, F. A., 171, 205 van der Waerden, B. L., 39, 78, 146, 205 von Neumann J., 202, 205 W

Westwick, T. R., 78, 205

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