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    Cover
    Title
    Copyright
    Contents
    Preface to the Second Edition
    Preface to the First Edition
    0. Review and Miscellanea
    0.0 Introduction
    0.1 Vector spaces
    0.2 Matrices
    0.3 Determinants
    0.4 Rank
    0.5 Nonsingularity
    0.6 The Euclidean inner product and norm
    0.7 Partitioned sets and matrices
    0.8 Determinants again
    0.9 Special types of matrices
    0.10 Change of basis
    0.11 Equivalence relations
    1. Eigenvalues, Eigenvectors, and Similarity
    1.0 Introduction
    1.1 The eigenvalue–eigenvector equation
    1.2 The characteristic polynomial and algebraic multiplicity
    1.3 Similarity
    1.4 Left and right eigenvectors and geometric multiplicity
    2. Unitary Similarity and Unitary Equivalence
    2.0 Introduction
    2.1 Unitary matrices and the QR factorization
    2.2 Unitary similarity
    2.3 Unitary and real orthogonal triangularizations
    2.4 Consequences of Schur’s triangularization theorem
    2.5 Normal matrices
    2.6 Unitary equivalence and the singular value decomposition
    2.7 The CS decomposition
    3. Canonical Forms for Similarity and Triangular Factorizations
    3.0 Introduction
    3.1 The Jordan canonical form theorem
    3.2 Consequences of the Jordan canonical form
    3.3 The minimal polynomial and the companion matrix
    3.4 The real Jordan and Weyr canonical forms
    3.5 Triangular factorizations and canonical forms
    4. Hermitian Matrices, Symmetric Matrices, and Congruences
    4.0 Introduction
    4.1 Properties and characterizations of Hermitian matrices
    4.2 Variational characterizations and subspace intersections
    4.3 Eigenvalue inequalities for Hermitian matrices
    4.4 Unitary congruence and complex symmetric matrices
    4.5 Congruences and diagonalizations
    4.6 Consimilarity and condiagonalization
    5. Norms for Vectors and Matrices
    5.0 Introduction
    5.1 Definitions of norms and inner products
    5.2 Examples of norms and inner products
    5.3 Algebraic properties of norms
    5.4 Analytic properties of norms
    5.5 Duality and geometric properties of norms
    5.6 Matrix norms
    5.7 Vector norms on matrices
    5.8 Condition numbers: inverses and linear systems
    6. Location and Perturbation of Eigenvalues
    6.0 Introduction
    6.1 Gersgorin discs
    6.2 Gersgorin discs – a closer look
    6.3 Eigenvalue perturbation theorems
    6.4 Other eigenvalue inclusion sets
    7. Positive Definite and Semidefinite Matrices
    7.0 Introduction
    7.1 Definitions and properties
    7.2 Characterizations and properties
    7.3 The polar and singular value decompositions
    7.4 Consequences of the polar and singular value decompositions
    7.5 The Schur product theorem
    7.6 Simultaneous diagonalizations
    7.7 The Loewner partial order and block matrices
    7.8 Inequalities involving positive definite matrices
    8. Positive and Nonnegative Matrices
    8.0 Introduction
    8.1 Inequalities and generalities
    8.2 Positive matrices
    8.3 Nonnegative matrices
    8.4 Irreducible nonnegative matrices
    8.5 Primitive matrices
    8.6 A general limit theorem
    8.7 Stochastic and doubly stochastic matrices
    Appendix A Complex Numbers
    Appendix B Convex Sets and Functions
    Appendix C The Fundamental Theorem of Algebra
    Appendix D Continuity of Polynomial Zeroes and Matrix Eigenvalues
    Appendix E Continuity
    Appendix F Canonical Pairs
    References
    Notation
    Hints for Problems
    Index
    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: r New sections on the singular value and CS decompositions r New applications of the Jordan canonical form r A new section on the Weyr canonical form r Expanded treatments of inverse problems and of block matrices r A central role for the von Neumann trace theorem r A new appendix with a modern list of canonical forms for a pair of Hermitian matrices and for a symmetric–skew symmetric pair r Expanded index with more than 3,500 entries for easy reference r More 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 r A 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 C 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) I. Johnson, Charles R. 1. Matrices. QA188.H66 2012 512.9 2012012300 II. Title. 434–dc23 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 Preface to the First Edition 0 Review and Miscellanea 0.0 Introduction 0.1 Vector spaces 0.2 Matrices 0.3 Determinants 0.4 Rank 0.5 Nonsingularity 0.6 The Euclidean inner product and norm 0.7 Partitioned sets and matrices 0.8 Determinants again 0.9 Special types of matrices 0.10 Change of basis 0.11 Equivalence relations 1 Eigenvalues, Eigenvectors, and Similarity Introduction 1.0 1.1 The eigenvalue–eigenvector equation 1.2 The characteristic polynomial and algebraic multiplicity 1.3 Similarity 1.4 Left and right eigenvectors and geometric multiplicity 2 Unitary Similarity and Unitary Equivalence Introduction 2.0 2.1 Unitary matrices and the QR factorization 2.2 Unitary similarity 2.3 Unitary and real orthogonal triangularizations 2.4 Consequences of Schur’s triangularization theorem 2.5 Normal matrices vii page xi xv 1 1 1 5 8 12 14 15 16 21 30 39 40 43 43 44 49 57 75 83 83 83 94 101 108 131
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