**Matrix Algebra and Applications**

3 Matrix Algebra and Applications ... We used matrices in Chapter 2 simply to organize our ... We use these operations to study game theory and input-output models in ...

Graduate Texts in Mathematics 216,Editorial Board,S Axler F W Gehring K A Ribet. This page intentionally left blank,Denis Serre,Theory and Applications. Denis Serre,Ecole Normale Supe rieure de Lyon,Lyon Cedex 07 F 69364. Denis SERRE umpa ens lyon fr,Editorial Board,S Axler F W Gehring K A Ribet. Mathematics Department Mathematics Department Mathematics Department. San Francisco State East Hall University of California. University University of Michigan Berkeley, San Francisco CA 94132 Ann Arbor MI 48109 Berkeley CA 94720 3840.

USA USA USA, axler sfsu edu fgehring math lsa umich edu ribet math berkeley edu. Mathematics Subject Classification 2000 15 01, Library of Congress Cataloging in Publication Data. Serre D Denis,Matrices English,Matrices theory and applications Denis Serre. p cm Graduate texts in mathematics 216,Includes bibliographical references and index. ISBN 0 387 95460 0 alk paper,1 Matrices I Title II Series.

QA188 S4713 2002,512 9 434 dc21 2002022926,ISBN 0 387 95460 0 Printed on acid free paper. Translated from Les Matrices The orie et pratique published by Dunod Paris 2001. 2002 Springer Verlag New York Inc, All rights reserved This work may not be translated or copied in whole or in part without the. written permission of the publisher Springer Verlag New York Inc 175 Fifth Avenue New York. NY 10010 USA except for brief excerpts in connection with reviews or scholarly analysis Use. in connection with any form of information storage and retrieval electronic adaptation computer. software or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names trademarks service marks and similar terms even if. they are not identified as such is not to be taken as an expression of opinion as to whether or not. they are subject to proprietary rights,Printed in the United States of America. 9 8 7 6 5 4 3 2 1 SPIN 10869456, Typesetting Pages created by the author in LaTeX2e. www springer ny com,Springer Verlag New York Berlin Heidelberg.

A member of BertelsmannSpringer Science Business Media GmbH. To Pascale and Joachim,This page intentionally left blank. The study of matrices occupies a singular place within mathematics It. is still an area of active research and it is used by every mathematician. and by many scientists working in various specialities Several examples. illustrate its versatility, Scienti c computing libraries began growing around matrix calculus. As a matter of fact the discretization of partial di erential operators. is an endless source of linear nite dimensional problems. At a discrete level the maximum principle is related to nonnegative. Control theory and stabilization of systems with nitely many degrees. of freedom involve spectral analysis of matrices, The discrete Fourier transform including the fast Fourier transform. makes use of Toeplitz matrices, Statistics is widely based on correlation matrices. The generalized inverse is involved in least squares approximation. Symmetric matrices are inertia deformation or viscous tensors in. continuum mechanics, Markov processes involve stochastic or bistochastic matrices.

Graphs can be described in a useful way by square matrices. viii Preface, Quantum chemistry is intimately related to matrix groups and their. representations, The case of quantum mechanics is especially interesting Observables. are Hermitian operators their eigenvalues are energy levels In the. early years quantum mechanics was called mechanics of matrices. and it has now given rise to the development of the theory of large. random matrices See 23 for a thorough account of this fashionable. This text was conceived during the years 1998 2001 on the occasion of. a course that I taught at the E cole Normale Supe rieure de Lyon As such. every result is accompanied by a detailed proof During this course I tried. to investigate all the principal mathematical aspects of matrices algebraic. geometric and analytic, In some sense this is not a specialized book For instance it is not as. detailed as 19 concerning numerics or as 35 on eigenvalue problems. or as 21 about Weyl type inequalities But it covers at a slightly higher. than basic level all these aspects and is therefore well suited for a gradu. ate program Students attracted by more advanced material will nd one. or two deeper results in each chapter but the rst one given with full. proofs They will also nd further information in about the half of the. 170 exercises The solutions for exercises are available on the author s site. http www umpa ens lyon fr serre exercises pdf, This book is organized into ten chapters The rst three contain the. basics of matrix theory and should be known by almost every graduate. student in any mathematical eld The other parts can be read more or. less independently of each other However exercises in a given chapter. sometimes refer to the material introduced in another one. This text was rst published in French by Masson Paris in 2000 under. the title Les Matrices the orie et pratique I have taken the opportunity. during the translation process to correct typos and errors to index a list. of symbols to rewrite some unclear paragraphs and to add a modest. amount of material and exercises In particular I added three sections. concerning alternate matrices the singular value decomposition and the. Moore Penrose generalized inverse Therefore this edition di ers from the. French one by about 10 percent of the contents, Acknowledgments Many thanks to the Ecole Normale Supe rieure de Lyon.

and to my colleagues who have had to put up with my talking to them. so often about matrices Special thanks to Sylvie Benzoni for her constant. interest and useful comments,Lyon France Denis Serre. December 2001,Preface vii,List of Symbols xiii,1 Elementary Theory 1. 1 1 Basics 1,1 2 Change of Basis 8,1 3 Exercises 13. 2 Square Matrices 15,2 1 Determinants and Minors 15. 2 2 Invertibility 19,2 3 Alternate Matrices and the Pfa an 21.

2 4 Eigenvalues and Eigenvectors 23,2 5 The Characteristic Polynomial 24. 2 6 Diagonalization 28,2 7 Trigonalization 29,2 8 Irreducibility 30. 2 9 Exercises 31,3 Matrices with Real or Complex Entries 40. 3 1 Eigenvalues of Real and Complex Valued Matrices 43. 3 2 Spectral Decomposition of Normal Matrices 45,3 3 Normal and Symmetric Real Valued Matrices 47. x Contents, 3 4 The Spectrum and the Diagonal of Hermitian Matrices 51.

3 5 Exercises 55,4 Norms 61,4 1 A Brief Review 61,4 2 Householder s Theorem 66. 4 3 An Interpolation Inequality 67,4 4 A Lemma about Banach Algebras 70. 4 5 The Gershgorin Domain 71,4 6 Exercises 73,5 Nonnegative Matrices 80. 5 1 Nonnegative Vectors and Matrices 80,5 2 The Perron Frobenius Theorem Weak Form 81. 5 3 The Perron Frobenius Theorem Strong Form 82,5 4 Cyclic Matrices 85.

5 5 Stochastic Matrices 87,5 6 Exercises 91, 6 Matrices with Entries in a Principal Ideal Domain. Jordan Reduction 97,6 1 Rings Principal Ideal Domains 97. 6 2 Invariant Factors of a Matrix 101, 6 3 Similarity Invariants and Jordan Reduction 104. 6 4 Exercises 111,7 Exponential of a Matrix Polar Decomposition and. Classical Groups 114,7 1 The Polar Decomposition 114.

7 2 Exponential of a Matrix 116,7 3 Structure of Classical Groups 120. 7 4 The Groups U p q 122,7 5 The Orthogonal Groups O p q 123. 7 6 The Symplectic Group Spn 127,7 7 Singular Value Decomposition 128. 7 8 Exercises 130,8 Matrix Factorizations 136,8 1 The LU Factorization 137. 8 2 Choleski Factorization 142,8 3 The QR Factorization 143.

8 4 The Moore Penrose Generalized Inverse 145,8 5 Exercises 147. 9 Iterative Methods for Linear Problems 149,Contents xi. 9 1 A Convergence Criterion 150,9 2 Basic Methods 151. 9 3 Two Cases of Convergence 153,9 4 The Tridiagonal Case 155. 9 5 The Method of the Conjugate Gradient 159,9 6 Exercises 165.

10 Approximation of Eigenvalues 168,10 1 Hessenberg Matrices 169. 10 2 The QR Method 173,10 3 The Jacobi Method 180,10 4 The Power Methods 184. 10 5 Leverrier s Method 188,10 6 Exercises 190,References 195. This page intentionally left blank,List of Symbols. A 80 EK 28,a b 97 E 29,A B 59 End E 7,A 0 80 exp A 116.

a b 97 F G 2,A 15 97 F G 3,B C 13 F 11,BP 106 G 152. Cn 33 G A 71,Cr 83 G 125,n 87 gcd 98,ij 5 GLn A 20. det M 16 G0 126,diag d1 dn 5 H h 42,dim E 3 H 0n 42. dimK F 3 Hn 41,Dk N 102 HPD,ei 3 imaginary part 56. xiv List of Symbols,In 5 On K 20,J 151 O n 123,J a r 110 O p q 120.

Jik 100 A 160,J3 132 Pf 22,J4 132 PG 156,K A 162 PJ 156. ker M 7 P 156,ker u 7 p 62,KI 2 PSL2 IR 56,Kn 57 RA F 57. K X 15 rk M 5,k X Y 99 real part 63,k A 57 A 61,L E F 7 R M 8. L 152 r x 70 160,adj M 17 x y 11 41,M 40 S n 90,M 17 r 188. i1 i2 ip sj A 75,j1 j2 jp sk a 52,M k 6 SLn A 20,M 1 20 sm 189.

M k 20 S n 15,M T 20 SOn K 20,M N 6 S 1 86,Mn K 5 Sp M 24. Mn m K 5 SpK M 24,M 40 SPDn 42,M 40 Spm 120,M T 10 Spm 120. S 2 56 126,A 64 SUn 41,A p 65 Symn K 10,x A 154 151.

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Matrices : Theory & Applications Additional exercises Denis Serre Ecole Normale Sup erieure de Lyon Contents Topics 2 Themes of the exercises 4 Exercises 21

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