Matrix analysis.pdf(Second Edition)

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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 Gersgorin discs ˇ 387
6.2 Gersgorin discs – a closer look ˇ 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