Table Of ContentGEOMETRY
Of:
MATRICES
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GEOMETRY
OF
MATRICES
In Memory of Professor L K Hua (1910 -1985)
Zhe-Xian Wan
Chinese Academy of Sciences, China
Lund University, Sweden
{World Scientific
SSiinnggaappoorree*'N Neeww JJeerrsseeyy •• LLo ndon •Hong Kong
Published by
World Scientific Publishing Co Pte Ltd
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UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data
Wan, Che-hsien.
Geometry of matrices / Zhe-xian Wan.
p. cm.
"In memory of Professor L. K. Hua (1910-1985)."
Includes bibliographical references and index.
ISBN 9810226381
1. Matrices. I. Title.
QA188.W36 1996
516.3'5--dc20 96-2179
CIP
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 1996 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
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Preface
The present monograph is a state of the art survey of the geometry of matri
ces. Professor L. K. Hua initiated the work in this area in the middle forties.
In this geometry, the points of the space are a certain kind of matrices of
a given size, and the four kinds of matrices studied by Hua are rectangular
matrices, symmetric matrices, skew-symmetric matrices and hermitian ma
trices. To each such space there is associated a group of motions, and the
aim of the study is then to characterize the group of motions in the space
by as few geometric invariants as possible. At first, Professor Hua, relating
to his study of the theory of functions of several complex variables, began
studying the geometry of matrices of various types over the complex field.
Later, he extended his results to the case when the basic field is not necessar
ily commutative, discovered that the invariant "adjacency" alone is sufficient
to characterize the group of motions of the space, and applied his results to
some problems in algebra and geometry. Professor Hua's pioneer work in the
area has been followed by many mathematicians, and more general results
have been obtained. I think it is now time to summarize all results obtained
so far, and this has been my motivation for the present work.
In order to be as self-contained as possible the book covers some material of
linear algebra over division rings in Chapter 1, which is necessary for later
chapters. This chapter can also be read independently as an introduction to
linear algebra over division rings. The fundamental theorems of the affine
geometry and of the projective geometry over any division ring constitute
the main contents of Chapter 2. In particular, Hua's beautiful theorem on
semi-automorphisms of a division ring and its application to the fundamental
theorem of the one-dimensional projective geometry over a division ring are
v
VI PREFACE
included. Following these chapters, the geometry of rectangular matrices
over any division ring, alternate matrices over any field, symmetric matrices
over any field, and the geometry of hermitian matrices over any division ring
which possesses an involution are discussed in detail in Chapters 3, 4, 5, and
6, respectively. Applications to problems in algebra, geometry, and graph
theory are included throughout.
Finally, the author is indebted to Yangxian Wang and Mulan Liu for their
helpful comments on the first draft of the book, to Rongquan Feng, Lei Hu,
Xinwen Wu, and Zhanfei Zhou for their laborious typewriting, and to Lena
Mansson for her beautiful improvement of the camera-ready copy.
Zhe-xian Wan
Contents
Preface v
1 Linear Algebra over Division Rings 1
1.1 Vector Spaces over Division Rings 1
1.2 Matrices over Division Rings 11
1.3 Matrix Representations of Subspaces 27
1.4 Systems of Linear Equations 29
1.5 Hermitian, Symmetric, and Alternate Matrices 35
1.6 Comments 44
2 Affine Geometry and Projective Geometry 45
2.1 Affine Spaces and Affine Groups 45
2.2 Fundamental Theorem of the Affine Geometry 54
2.3 Projective Spaces and Projective Groups 66
2.4 Fundamental Theorem of the Projective Geometry 76
2.5 One-dimensional Projective Geometry 80
2.6 Comments 87
3 Geometry of Rectangular Matrices 89
3.1 The Space of Rectangular Matrices 89
3.2 Maximal Sets of Rank 1 93
3.3 Maximal Sets of Rank 2 97
3.4 Proof of the Fundamental Theorem 106
3.5 Application to Algebra 118
3.6 Application to Geometry 123
3.7 Application to Geometry (Continued) 139
vn
viii CONTENTS
3.8 Application to Graph Theory 153
3.9 Comments 155
4 Geometry of Alternate Matrices 157
4.1 The Space of Alternate Matrices 157
4.2 Maximal Sets 159
4.3 Proof of the Fundamental Theorem 168
4.4 Application to Geometry 177
4.5 Application to Geometry (Continued) 199
4.6 Application to Graph Theory 211
4.7 Comments 215
5 Geometry of Symmetric Matrices 217
5.1 The Space of Symmetric Matrices 217
5.2 Maximal Sets of Rank 1 222
5.3 Maximal Sets of Rank 2 (Characteristic Not Two) 224
5.4 Proof of the Fundamental Theorem (I) 231
5.5 Maximal Sets of Rank 2 (Characteristic Two) 244
5.6 Proof of the Fundamental Theorem (II) 252
5.7 Proof of the Fundamental Theorem (III) 264
5.8 Application to Algebra 281
5.9 Application to Geometry 285
5.10 Application to Graph Theory 296
5.11 Comments 303
6 Geometry of Hermitian Matrices 305
6.1 The Space of Hermitian Matrices 305
6.2 Maximal Sets of Rank 1 308
6.3 Maximal Sets of Rank 2 311
6.4 Proof of the Fundamental Theorem (the Case n > 3) . . .. 323
6.5 Maximal Sets of Rank 2 (the Case n = 2) 341
6.6 Proof of the Fundamental Theorem (the Case n = 2) . . .. 348
6.7 Application to Algebra 355
6.8 Application to Geometry 356
6.9 Application to Graph Theory 363
CONTENTS ix
6.10 Comments 365
Bibliography 367
Index 371
Chapter 1
Linear Algebra over Division
Rings
1.1 Vector Spaces over Division Rings
Let D be any division ring and n a positive integer. We use
D{n) = {On, a? , • • •, i ) I Xi G D, i = 1, 2, • • •, n}
2 n
to denote the n-dimensional row vector space (or left vector space) over D
formed by the set of all n-tuples (or n-dimensional row vectors)
(x x , • • •, x ), Xi£ D, i = 1, 2, • • •, n,
u 2 n
over D with addition and scalar multiplication defined by
(x x , • • •, x ) + (2/1,2/2, * • •, 2/n) = («i + 2/1, x + 2/2, • • • x + y )
u 2 n 2 9 n n
and
X\Xiy X2, * " * , Xfij — y*LXi) aJX2, , d/Xjij^
respectively, where x, xi, ar , •••, x , ?/i, ?/ , •••, Vn € D. For any row
2 n 2
vectors u, u, iu G i^ and re, y G D we have the following manipulation
rules
U + V = V + U,
(u + v) + w = u + (v + w),
x(?i + v) = xu + xv,
(x + 2/)^ = a;w + yu,
(xy)u = x(yu),
lu = u,
1
