Multilinear Algebra

304 Pages·1978·4.04 MB·English

by  Werner Greub

Werner Greub
Multilinear
Igenra
2nd Edition
a
I
Springer-Verlag
New York Heidelberg Berlin
Un iversitext

Werner Greub
Multiinear Algebra
2nd Edition
Springer-Verlag
New York Heidelberg Berlin
Werner Greub
Department of Mathematics
University of Toronto
Toronto M5S 1A1
Canada
AMS Subject Classifications: 15-01, 15A75, 15A72
Library of Congress Cataloging in Publication Data
Greub, Werner Hildbert, 1925-
Multilinear algebra,
(Universitext)
Includes index.
1. Algebras, Linear. I. Title.
QA 184.G74 1978 512'.5 78-949
ISBN-13:978-0-387-90284-5 e-ISBN-13:978-1-4613-9425-9
DO!: 10.1007/ 978-1-4613-9425-9
All rights reserved.
No part of this book may be translated or reproduced in any
form without written permission from Springer-Verlag.
© 1967 by Springer-Verlag Berlin Heidelberg
© 1978 by Springer-Verlag New York Inc.
987654321
Preface
This book is a revised version of the first edition and is intended as a
sequel and companion volume to the fourth edition of Linear Algebra
(Graduate Texts in Mathematics 23).
As before, the terminology and basic results of Linear Algebra are
frequently used without reference. In particular, the reader should be
familiar with Chapters 1-5 and the first part of Chapter b of that book,
although other sections are occasionally used.
In this new version of Multilinear Algebra, Chapters 1-5 remain essen-
tially unchanged from the previous edition. Chapter b has been completely
rewritten and split into three (Chapters b, 7, and 8). Some of the proofs
have been simplified and a substantial amount of new material has been
added. This applies particularly to the study of characteristic coefficients
and the Pf of f ian.
The old Chapter 7 remains as it stood, except that it is now Chapter 9.
The old Chapter 8 has been suppressed and the material which it con-
tained (multilinear functions) has been relocated at the end of Chapters 3,
S, and 9.
The last two chapters on Clifford algebras and their representations are
completely new. In view of the growing importance of Clifford algebras
and the relatively few references available, it was felt that these chapters
would be useful to both mathematicians and physicists.
In Chapter 10 Clifford algebras are introduced via universal properties
and treated in a fashion analogous to exterior algebra. After the basic
isomorphism theorems for these algebras (over an arbitrary inner product
space) have been established the chapter proceeds to a discussion of
finite-dimensional Clifford algebras. The treatment culminates in the com-
plete classification of Clifford algebras over finite-dimensional complex
and real inner product spaces.
v
vi Preface
The book concludes with Chapter 11 on representations of Clifford
algebras. The twisted adjoint representation which leads to the definition
of the spin-groups is an important example. A version of Wedderburn's
theorem is the key to the classification of all representations of the Clifford
algebra over an 8-dimensional real vector space with a negative definite
inner product. The results are applied in the last section of this chapter to
study orthogonal multiplications between Euclidean spaces and the ex-
istence of orthonormal frames on the sphere. In particular, it is shown that
the (n -1)-sphere admits an orthonormal k-frame where k is the
Radon-Hurwitz number corresponding to n. A deep theorem of F. Adams
states that this result can not be improved.
The problems at the end of Chapter 11 include a basis-free definition of
the Cayley algebra via the complex cross-product analogous to the defini-
tion of quaternions in Section 7.23 of the fourth edition of Linear Algebra.
Finally, the Cayley multiplication is used to obtain concrete forms of some
of the isomorphisms in the table at the end of Chapter 10.
I should like to express my deep thanks to Professor J. R. Vanstone who
worked closely with me through each stage of this revision and who made
numerous and valuable contributions to both content and presentation. I
should also like to thank Mr. M. S. Swanson who assisted Professor
Vanstone and myself with the proof reading.
Toronto, April 1978 W. H. Greub
Table of Contents
Chapter 1 Tensor Products
1
Chapter 2 Tensor Products of Vector Spaces with
Additional Structure 41
Chapter 3 Tensor Algebra 60
Chapter 4 Skew-Symmetry and Symmetry in the Tensor Algebra 84
Chapter 5 Exterior Algebra 96
Chapter 6 Mixed Exterior Algebra 148
Chapter 7 Applications to Linear Transformations 174
Chapter 8 Skew and Skew-Hermitian Transformations 193
Chapter 9 Symmetric Tensor Algebra 209
Chapter 10 Clifford Algebras 227
Chapter 11 Representations of Clifford Algebras 260
Index 291
vii

Tensor Products
Throughout this chapter except where noted otherwise all vector spaces will be defined over a
fixed, but arbitrarily chosen, field T.
Multilinear Mappings
1.1. Bilinear Mappings
Suppose E, F and G are any three vector spaces, and consider a mapping
p : E x F -+ G.
'p is called bilinear if it satisfies the conditions
p(t x 1 + µx 2 , y) = p(x1,y) + µup(x2 , y) x1, x2EE, yeF,A,µeF,
xeE,y1,y2eF.
'p(x, Ay 1 + µY2) _ A p(x, Y 1) + µup(x, Y2)
Recall that if G = F, then 'p is called a bilinear function.
The set S of all vectors in G of the form '(x, y), x e E, y e F is not in
general a vector subspace of G. As an example, let E = F and G be re-
spectively 2- and 4-dimensional vector spaces. Select a basis a 1, a2 in E
and a basis c (v = 1, ... , 4) in G and define the bilinear mapping 'p by
co(x, Y) = 'i 'c1 + 1i2C2 + 2i 'c3 + 2ii2C4
where x = c 1a 1 + 2a2 and y = ri' a 1 + Then it is easy to see that
r12a2.
a vector
z =
VC
of G is contained in S if and only if the components satisfy the relation
-
= o.
1