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 second edition of this 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. This thoroughly revised and updated second edition is a text for a second course on linear algebra and has more than 1,100 problems and exercises, new sections on the singular value and CS decompositions and the Weyr canonical form, expanded treatments of inverse problems and of block matrices, and much more. Cover 1 Matrix Analysis 3 Title 5 Copyright 6 Dedication 7 Contents 9 Preface to the Second Edition 13 Preface to the First Edition 17 CHAPTER 0 Review and Miscellanea 21 0.0 Introduction 21 0.1 Vector spaces 21 0.2 Matrices 25 0.3 Determinants 28 0.4 Rank 32 0.5 Nonsingularity 34 0.6 The Euclidean inner product and norm 35 0.7 Partitioned sets and matrices 36 0.8 Determinants again 41 0.9 Special types of matrices 50 0.10 Change of basis 59 0.11 Equivalence relations 60 CHAPTER 1 Eigenvalues, Eigenvectors, and Similarity 63 1.0 Introduction 63 1.1 The eigenvalue--eigenvector equation 64 1.2 The characteristic polynomial and algebraic multiplicity 69 1.3 Similarity 77 1.4 Left and right eigenvectors and geometric multiplicity 95 CHAPTER 2 Unitary Similarity and Unitary Equivalence 103 2.0 Introduction 103 2.1 Unitary matrices and the QR factorization 103 2.2 Unitary similarity 114 2.3 Unitary and real orthogonal triangularizations 121 2.4 Consequences of Schur's triangularization theorem 128 2.5 Normal matrices 151 2.6 Unitary equivalence and the singular value decomposition 169 2.7 The CS decomposition 179 CHAPTER 3 Canonical Forms For Similarity and Triangular Factorizations 183 3.0 Introduction 183 3.1 The Jordan canonical form theorem 184 3.2 Consequences of the Jordan canonical form 195 3.3 The minimal polynomial and the companion matrix 211 3.4 The real Jordan and Weyr canonical forms 221 3.5 Triangular factorizations and canonical forms 236 CHAPTER 4 Hermitian Matrices, Symmetric Matrices, and Congruences 245 4.0 Introduction 245 4.1 Properties and characterizations of Hermitian matrices 247 4.2 Variational characterizations and subspace intersections 254 4.3 Eigenvalue inequalities for Hermitian matrices 259 4.4 Unitary congruence and complex symmetric matrices 280 4.5 Congruences and diagonalizations 299 4.6 Consimilarity and condiagonalization 320 CHAPTER 5 Norms for Vectors and Matrices 333 5.0 Introduction 333 5.1 Definitions of norms and inner products 334 5.2 Examples of norms and inner products 340 5.3 Algebraic properties of norms 344 5.4 Analytic properties of norms 344 5.5 Duality and geometric properties of norms 355 5.6 Matrix norms 360 5.7 Vector norms on matrices 391 5.8 Condition numbers: inverses and linear systems 401 CHAPTER 6 Location and Perturbation of Eigenvalues 407 6.0 Introduction 407 6.1 Gersgorin discs 407 6.2 Gersgorin discs -- a closer look 416 6.3 Eigenvalue perturbation theorems 425 6.4 Other eigenvalue inclusion sets 433 CHAPTER 7 Positive Definite and Semidefinite Matrices 445 7.0 Introduction 445 7.1 Definitions and properties 449 7.2 Characterizations and properties 458 7.3 The polar and singular value decompositions 468 7.4 Consequences of the polar and singularvalue decompositions 478 7.5 The Schur product theorem 497 7.6 Simultaneous diagonalizations, products, and convexity 505 7.7 The Loewner partial order and block matrices 513 7.8 Inequalities involving positive definite matrices 525 CHAPTER 8 Positive and Nonnegative Matrices 537 8.0 Introduction 537 8.1 Inequalities and generalities 539 8.2 Positive matrices 544 8.3 Nonnegative matrices 549 8.4 Irreducible nonnegative matrices 553 8.5 Primitive matrices 560 8.6 A general limit theorem 565 8.7 Stochastic and doubly stochastic matrices 567 APPENDIX A Complex Numbers 575 APPENDIX B Convex Sets and Functions 577 APPENDIX C The Fundamental Theorem of Algebra 581 APPENDIX D Continuity of Polynomial Zeroes and Matrix Eigenvalues 583 APPENDIX E Continuity, Compactness, and Weierstrass's Theorem 585 APPENDIX F Canonical Pairs 587 References 591 Notation 595 Hints for Problems 599 Index 627 The Thoroughly Revised And Updated Second Edition Of This Acclaimed Text Has Several New And Expanded Sections And More Than 1,100 Exercises-- 0. Review And Miscellanea -- 1. Eigenvalues, Eigenvectors, And Similarity -- 2. Unitary Similarity And Unitary Equivalence -- 3. Canonical Forms For Similarity And Triangular Factorizations -- 4. Hermitian Matrices, Symmetric Matrices, And Congruences -- 5. Norms For Vectors And Matrices -- 6. Location And Perturbation Of Eigenvalues -- 7. Positive Definite And Semidefinite Matrices -- 8. Positive And Nonnegative 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, Compactness, And Weierstrass's Theorem -- Appendix F. Canonical Pairs. Roger A. Horn, Charles R. Johnson. The Thoroughly Revised And Updated Second Edition Of This Acclaimed Text Has Several New And Expanded Sections And More Than 1,100 Exercises-- Provided By Publisher. Erscheint: Oktober 2012. Includes Bibliographical References And Index. Machine generated contents note: 1. Eigenvalues, eigenvectors, and similarity; 2. Unitary similarity and unitary equivalence; 3. Canonical forms for similarity, and triangular factorizations; 4. Hermitian matrices, symmetric matrices, and congruences; 5. Norms for vectors and matrices; 6. Location and perturbation of eigenvalues; 7. Positive definite and semi-definite matrices; 8. Positive and nonnegative matrices; Appendix A. Complex numbers; Appendix B. Convex sets and functions; Appendix C. The fundamental theorem of algebra; Appendix D. Continuous dependence of the zeroes of a polynomial on its coefficients; Appendix E. Continuity, compactness, and Weierstrass's theorem; Appendix F. Canonical Pairs. The thoroughly revised and updated second edition of this acclaimed text for a second course on linear algebra has more than 1,100 problems and exercises, along with new sections on the singular value and CS decompositions and the Weyr canonical form, expanded treatments of inverse problems and of block matrices and much more.