Matrix Analysis

Cover......Page 1
Title......Page 4
Copyright......Page 5
Contents......Page 8
Preface to the Second Edition......Page 12
Preface to the First Edition......Page 16
0.1 Vector spaces......Page 20
0.2 Matrices......Page 24
0.3 Determinants......Page 27
0.4 Rank......Page 31
0.5 Nonsingularity......Page 33
0.6 The Euclidean inner product and norm......Page 34
0.7 Partitioned sets and matrices......Page 35
0.8 Determinants again......Page 40
0.9 Special types of matrices......Page 49
0.10 Change of basis......Page 58
0.11 Equivalence relations......Page 59
1.0 Introduction......Page 62
1.1 The eigenvalue–eigenvector equation......Page 63
1.2 The characteristic polynomial and algebraic multiplicity......Page 68
1.3 Similarity......Page 76
1.4 Left and right eigenvectors and geometric multiplicity......Page 94
2.1 Unitary matrices and the QR factorization......Page 102
2.2 Unitary similarity......Page 113
2.3 Unitary and real orthogonal triangularizations......Page 120
2.4 Consequences of Schur’s triangularization theorem......Page 127
2.5 Normal matrices......Page 150
2.6 Unitary equivalence and the singular value decomposition......Page 168
2.7 The CS decomposition......Page 178
3.0 Introduction......Page 182
3.1 The Jordan canonical form theorem......Page 183
3.2 Consequences of the Jordan canonical form......Page 194
3.3 The minimal polynomial and the companion matrix......Page 210
3.4 The real Jordan and Weyr canonical forms......Page 220
3.5 Triangular factorizations and canonical forms......Page 235
4.0 Introduction......Page 244
4.1 Properties and characterizations of Hermitian matrices......Page 246
4.2 Variational characterizations and subspace intersections......Page 253
4.3 Eigenvalue inequalities for Hermitian matrices......Page 258
4.4 Unitary congruence and complex symmetric matrices......Page 279
4.5 Congruences and diagonalizations......Page 298
4.6 Consimilarity and condiagonalization......Page 319
5.0 Introduction......Page 332
5.1 Definitions of norms and inner products......Page 333
5.2 Examples of norms and inner products......Page 339
5.4 Analytic properties of norms......Page 343
5.5 Duality and geometric properties of norms......Page 354
5.6 Matrix norms......Page 359
5.7 Vector norms on matrices......Page 390
5.8 Condition numbers: inverses and linear systems......Page 400
6.1 Gersgorin discs......Page 406
6.2 Gersgorin discs – a closer look......Page 415
6.3 Eigenvalue perturbation theorems......Page 424
6.4 Other eigenvalue inclusion sets......Page 432
7.0 Introduction......Page 444
7.1 Definitions and properties......Page 448
7.2 Characterizations and properties......Page 457
7.3 The polar and singular value decompositions......Page 467
7.4 Consequences of the polar and singular value decompositions......Page 477
7.5 The Schur product theorem......Page 496
7.6 Simultaneous diagonalizations......Page 504
7.7 The Loewner partial order and block matrices......Page 512
7.8 Inequalities involving positive definite matrices......Page 524
8.0 Introduction......Page 536
8.1 Inequalities and generalities......Page 538
8.2 Positive matrices......Page 543
8.3 Nonnegative matrices......Page 548
8.4 Irreducible nonnegative matrices......Page 552
8.5 Primitive matrices......Page 559
8.6 A general limit theorem......Page 564
8.7 Stochastic and doubly stochastic matrices......Page 566
Appendix A Complex Numbers......Page 574
Appendix B Convex Sets and Functions......Page 576
Appendix C The Fundamental Theorem of Algebra......Page 580
Appendix D Continuity of Polynomial Zeroes and Matrix Eigenvalues......Page 582
Appendix E Continuity......Page 584
Appendix F Canonical Pairs......Page 586
References......Page 590
Notation......Page 594
Hints for Problems......Page 598
Index......Page 626