Matrix Analysis

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Matrix Analysis

Second Edition

Linear algebra and matrix theory are fundamental tools in mathematical and physical

science, as well as fertile ﬁelds for research. This new edition of the acclaimed text presents

results of both classic and recent matrix analysis using canonical forms as a unifying theme,

and demonstrates their importance in a variety of applications.

The authors have thoroughly revised, updated, and expanded on the ﬁrst edition. The

book opens with an extended summary of useful concepts and facts and includes numerous

new topics and features, such as:

rNew sections on the singular value and CS decompositions

rNew applications of the Jordan canonical form

rA new section on the Weyr canonical form

rExpanded treatments of inverse problems and of block matrices

rA central role for the von Neumann trace theorem

rA new appendix with a modern list of canonical forms for a pair of Hermitian

matrices and for a symmetric–skew symmetric pair

rExpanded index with more than 3,500 entries for easy reference

rMore than 1,100 problems and exercises, many with hints, to reinforce understand-

ing and develop auxiliary themes such as ﬁnite-dimensional quantum systems, the

compound and adjugate matrices, and the Loewner ellipsoid

rA new appendix provides a collection of problem-solving hints.

Roger A. Horn is a Research Professor in the Department of Mathematics at the University

of Utah. He is the author of Topics in Matrix Analysis (Cambridge University Press 1994).

Charles R. Johnson is the author of Topics in Matrix Analysis (Cambridge University Press

1994).

Matrix Analysis

Second Edition

Roger A. Horn

University of Utah

Charles R. Johnson

cambridge university press

Cambridge, New York, Melbourne, Madrid, Cape Town,

Singapore, S˜

ao Paulo, Delhi, Mexico City

Cambridge University Press

32 Avenue of the Americas, New York, NY 10013-2473, USA

www.cambridge.org

Information on this title: www.cambridge.org/9780521548236

Roger A. Horn and Charles R. Johnson 1985, 2013

This publication is in copyright. Subject to statutory exception

and to the provisions of relevant collective licensing agreements,

no reproduction of any part may take place without the written

permission of Cambridge University Press.

First published 1985

First paperback edition 1990

Second edition ﬁrst published 2013

Printed in the United States of America

A catalog record for this publication is available from the British Library.

Library of Congress Cataloging in Publication Data

Horn, Roger A.

Matrix analysis / Roger A. Horn, Charles R. Johnson. – 2nd ed.

p. cm.

Includes bibliographical references and index.

ISBN 978-0-521-83940-2 (hardback)

1. Matrices. I. Johnson, Charles R. II. Title.

QA188.H66 2012

512.9434–dc23 2012012300

ISBN 978-0-521-83940-2 Hardback

ISBN 978-0-521-54823-6 Paperback

Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external

or third-party Internet Web sites referred to in this publication and does not guarantee that any content

on such Web sites is, or will remain, accurate or appropriate.

To the matrix theory community

Contents

Preface to the Second Edition page xi

Preface to the First Edition xv

0 Review and Miscellanea 1

0.0 Introduction 1

0.1 Vector spaces 1

0.2 Matrices 5

0.3 Determinants 8

0.4 Rank 12

0.5 Nonsingularity 14

0.6 The Euclidean inner product and norm 15

0.7 Partitioned sets and matrices 16

0.8 Determinants again 21

0.9 Special types of matrices 30

0.10 Change of basis 39

0.11 Equivalence relations 40

1 Eigenvalues, Eigenvectors, and Similarity 43

1.0 Introduction 43

1.1 The eigenvalue–eigenvector equation 44

1.2 The characteristic polynomial and algebraic multiplicity 49

1.3 Similarity 57

1.4 Left and right eigenvectors and geometric multiplicity 75

2 Unitary Similarity and Unitary Equivalence 83

2.0 Introduction 83

2.1 Unitary matrices and the QR factorization 83

2.2 Unitary similarity 94

2.3 Unitary and real orthogonal triangularizations 101

2.4 Consequences of Schur’s triangularization theorem 108

2.5 Normal matrices 131

vii

viii Contents

2.6 Unitary equivalence and the singular value decomposition 149

2.7 The CS decomposition 159

3 Canonical Forms for Similarity and Triangular Factorizations 163

3.0 Introduction 163

3.1 The Jordan canonical form theorem 164

3.2 Consequences of the Jordan canonical form 175

3.3 The minimal polynomial and the companion matrix 191

3.4 The real Jordan and Weyr canonical forms 201

3.5 Triangular factorizations and canonical forms 216

4 Hermitian Matrices, Symmetric Matrices, and Congruences 225

4.0 Introduction 225

4.1 Properties and characterizations of Hermitian matrices 227

4.2 Variational characterizations and subspace intersections 234

4.3 Eigenvalue inequalities for Hermitian matrices 239

4.4 Unitary congruence and complex symmetric matrices 260

4.5 Congruences and diagonalizations 279

4.6 Consimilarity and condiagonalization 300

5 Norms for Vectors and Matrices 313

5.0 Introduction 313

5.1 Deﬁnitions of norms and inner products 314

5.2 Examples of norms and inner products 320

5.3 Algebraic properties of norms 324

5.4 Analytic properties of norms 324

5.5 Duality and geometric properties of norms 335

5.6 Matrix norms 340

5.7 Vector norms on matrices 371

5.8 Condition numbers: inverses and linear systems 381

6 Location and Perturbation of Eigenvalues 387

6.0 Introduction 387

6.1 Gerˇ

sgorin discs 387

6.2 Gerˇ

sgorin discs – a closer look 396

6.3 Eigenvalue perturbation theorems 405

6.4 Other eigenvalue inclusion sets 413

7 Positive Deﬁnite and Semideﬁnite Matrices 425

7.0 Introduction 425

7.1 Deﬁnitions and properties 429

7.2 Characterizations and properties 438

7.3 The polar and singular value decompositions 448

7.4 Consequences of the polar and singular value decompositions 458

7.5 The Schur product theorem 477

7.6 Simultaneous diagonalizations, products, and convexity 485

7.7 The Loewner partial order and block matrices 493

7.8 Inequalities involving positive deﬁnite matrices 505

Contents ix

8 Positive and Nonnegative Matrices 517

8.0 Introduction 517

8.1 Inequalities and generalities 519

8.2 Positive matrices 524

8.3 Nonnegative matrices 529

8.4 Irreducible nonnegative matrices 533

8.5 Primitive matrices 540

8.6 A general limit theorem 545

8.7 Stochastic and doubly stochastic matrices 547

Appendix A Complex Numbers 555

Appendix B Convex Sets and Functions 557

Appendix C The Fundamental Theorem of Algebra 561

Appendix D Continuity of Polynomial Zeroes and Matrix

Eigenvalues 563

Appendix E Continuity, Compactness, and Weierstrass’s Theorem 565

Appendix F Canonical Pairs 567

References 571

Notation 575

Hints for Problems 579

Index 607

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MatrixAnalysisSecondEditionLinearalgebraandmatrixtheoryarefundamentaltoolsinmathematicalandphysicalscience,aswellasfertileﬁeldsforresearch.Thisneweditionoftheacclaimedtextpresentsresultsofbothclassicandrecentmatrixanalysisusingcanonicalformsasaunifyingtheme,anddemonstratestheirimportanceinavarietyof...

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