Matrix Theory Xingzhi Zhan +VEHYEXI7XYHMIW MR1EXLIQEXMGW :SPYQI %QIVMGER1EXLIQEXMGEP7SGMIX] Matrix Theory https://doi.org/10.1090//gsm/147 Matrix Theory Xingzhi Zhan Graduate Studies in Mathematics Volume 147 American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE David Cox (Chair) Daniel S. Freed Rafe Mazzeo Gigliola Staffilani 2010 Mathematics Subject Classification. Primary 15-01, 15A18, 15A21, 15A60, 15A83, 15A99, 15B35, 05B20, 47A63. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-147 Library of Congress Cataloging-in-Publication Data Zhan, Xingzhi, 1965– Matrix theory / Xingzhi Zhan. pages cm — (Graduate studies in mathematics ; volume 147) Includes bibliographical references and index. ISBN 978-0-8218-9491-0 (alk. paper) 1. Matrices. 2. Algebras, Linear. I. Title. QA188.Z43 2013 512.9434—dc23 2013001353 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2013 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 181716151413 Contents Preface ix Chapter 1. Preliminaries 1 §1.1. Classes of Special Matrices 2 §1.2. The Characteristic Polynomial 6 §1.3. The Spectral Mapping Theorem 8 §1.4. Eigenvalues and Diagonal Entries 8 §1.5. Norms 10 §1.6. Convergence of the Power Sequence of a Matrix 13 §1.7. Matrix Decompositions 14 §1.8. Numerical Range 18 §1.9. The Companion Matrix of a Polynomial 21 §1.10. Generalized Inverses 22 §1.11. Schur Complements 23 §1.12. Applications of Topological Ideas 24 §1.13. Gr¨obner Bases 25 §1.14. Systems of Linear Inequalities 27 §1.15. Orthogonal Projections and Reducing Subspaces 29 §1.16. Books and Journals about Matrices 31 Exercises 31 Chapter 2. Tensor Products and Compound Matrices 35 §2.1. Definitions and Basic Properties 35 §2.2. Linear Matrix Equations 40 v vi Contents §2.3. Frobenius-K¨onig Theorem 44 §2.4. Compound Matrices 46 Exercises 49 Chapter 3. Hermitian Matrices and Majorization 51 §3.1. Eigenvalues of Hermitian Matrices 51 §3.2. Majorization and Doubly Stochastic Matrices 56 §3.3. Inequalities for Positive Semidefinite Matrices 68 Exercises 74 Chapter 4. Singular Values and Unitarily Invariant Norms 77 §4.1. Singular Values 77 §4.2. Symmetric Gauge Functions 88 §4.3. Unitarily Invariant Norms 90 §4.4. The Cartesian Decomposition of Matrices 97 Exercises 100 Chapter 5. Perturbation of Matrices 103 §5.1. Eigenvalues 103 §5.2. The Polar Decomposition 112 §5.3. Norm Estimation of Band Parts 114 §5.4. Backward Perturbation Analysis 116 Exercises 118 Chapter 6. Nonnegative Matrices 119 §6.1. Perron-Frobenius Theory 120 §6.2. Matrices and Digraphs 132 §6.3. Primitive and Imprimitive Matrices 134 §6.4. Special Classes of Nonnegative Matrices 138 §6.5. Two Theorems about Positive Matrices 142 Exercises 147 Chapter 7. Completion of Partial Matrices 149 §7.1. Friedland’s Theorem about Diagonal Completions 150 §7.2. Farahat-Ledermann’s Theorem about Borderline Completions 153 §7.3. Parrott’s Theorem about Norm-Preserving Completions 157 §7.4. Positive Definite Completions 159 Chapter 8. Sign Patterns 165 Contents vii §8.1. Sign-Nonsingular Patterns 168 §8.2. Eigenvalues 169 §8.3. Sign Semi-Stable Patterns 173 §8.4. Sign Patterns Allowing a Positive Inverse 174 Exercises 179 Chapter 9. Miscellaneous Topics 181 §9.1. Similarity of Real Matrices via Complex Matrices 181 §9.2. Inverses of Band Matrices 182 §9.3. Norm Bounds for Commutators 184 §9.4. The Converse of the Diagonal Dominance Theorem 188 §9.5. The Shape of the Numerical Range 192 §9.6. An Inversion Algorithm 197 §9.7. Canonical Forms for Similarity 198 §9.8. Extremal Sparsity of the Jordan Canonical Form 207 Chapter 10. Applications of Matrices 213 §10.1. Combinatorics 214 §10.2. Number Theory 216 §10.3. Algebra 217 §10.4. Geometry 220 §10.5. Polynomials 222 Unsolved Problems 227 Bibliography 237 Notation 249 Index 251 Preface The systematic study of matrices began late in the history of mathematics, but matrix theory is an active area of research now and it has applications in numerical analysis, control and systems theory, optimization, combinatorics, mathematical physics, differential equations, probability and statistics, eco- nomics, information theory, and engineering. One attractive feature of matrix theory is that many matrix problems can be solved naturally by using tools or ideas from other branches of math- ematics such as analysis, algebra, graph theory, geometry and topology. The reverse situation also occurs, as shown in the last chapter. This book is intended for use as a text for graduate or advanced under- graduate level courses, or as a reference for research workers. It is based on lecture notes for graduate courses I have taught five times at East China Normal University and once at Peking University. My aim is to provide a concise treatment of matrix theory. I hope the book contains the basic knowledge and conveys the flavor of the subject. When I chose material for this book, I had the following criteria in mind: 1) important; 2) elegant; 3) ingenious; 4) interesting. Of course, a very small percentage of mathematics meets all of these criteria, but I hope the results and proofs here meet at least one of them. As a reader I feel that for clarity, the logical steps of a mathematical proof cannot be omitted, though routine calculations may be or should be. Whenever possible, I try to have a conceptual understanding of a result. I always emphasize methods and ideas. ix x Preface Most of the exercises are taken from research papers, and they have some depth. Thus if the reader has difficulty in solving the problems in these exercises, she or he should not feel frustrated. Parts of this book appeared in a book in Chinese with the same title published by the Higher Education Press in 2008. Thanks go to Professors Pei Yuan Wu and Wei Wu for discussions on the topic of numerical range and to Dr. Zejun Huang for discussions on Theorem 1.2 and Lemma 9.13. I am grateful to Professors Tsuyoshi Ando, Rajendra Bhatia, Richard Brualdi, Roger Horn, Erxiong Jiang, Chi-Kwong Li, Zhi- Guo Liu, Jianyu Pan, Jia-Yu Shao, Sheng-Li Tan, and Guang Yuan Zhang for their encouragement, friendship and help over the years. I wish to express my gratitude to my family for their kindness. This work was supported by the National Science Foundation of China under grant 10971070. Shanghai, December 2012 Xingzhi Zhan Bibliography [1] W.W. Adams and P. Loustaunau, An Introduction to Gr¨obner Bases, GSM 3, Amer. Math. Soc., Providence, RI, 1994. [2] S.S. 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Barrett and P.J. Feinsilver, Inverses of banded matrices, Linear Algebra Appl., 41(1981), 111–130. [20] A. Barvinok, A Course in Convexity, GSM 54, Amer. Math. Soc., Providence, RI, 2002. [21] L. Bassett, J. Maybee and J.