Ⅰ A Review of Linear Algebra 1
Ⅰ.1 Vector Spaces and Inner Product Spaces 1
Ⅰ.2 Linear Operators and Matrices 3
Ⅰ.3 Direct Sums 9
Ⅰ.4 Tensor Products 12
Ⅰ.5 Symmetry Classes 16
Ⅰ.6 Problems 20
Ⅰ.7 Notes and References 26
Ⅱ Majorisation and Doubly Stochastic Matrices 28
Ⅱ.1 Basic Notions 28
Ⅱ.2 Birkhoff's Theorem 36
Ⅱ.3 Convex and Monotone Functions 40
Ⅱ.4 Binary Algebraic Operations and Majorisation 48
Ⅱ.5 Problems 50
Ⅱ.6 Notes and References 54
Ⅲ Variational Principles for Eigenvalues 57
Ⅲ.1 The Minimax Principle for Eigenvalues 57
Ⅲ.2 Weyl's Inequalities 62
Ⅲ.3 Wielandt's Minimax Principle 65
Ⅲ.4 Lidskii's Theorems 68
Ⅲ.5 Eigenvalues of Real Parts and Singular Values 73
Ⅲ.6 Problems 75
Ⅲ.7 Notes and References 78
Ⅳ Symmetric Norms 84
Ⅳ.1 Norms on Cn 84
Ⅳ.2 Unitarily Invariant Norms on Operators on Cn 91
Ⅳ.3 Lidskii's Theorem (Third Proof) 98
Ⅳ.4 Weakly Unitarily Invariant Norms 101
Ⅳ.5 Problems 107
Ⅳ.6 Notes and References 109
Ⅴ Operator Monotone and Operator Convex Functions 112
Ⅴ.1 Definitions and Simple Examples 112
Ⅴ.2 Some Characterisations 117
Ⅴ.3 Smoothness Properties 123
Ⅴ.4 Loewner's Theorems 131
Ⅴ.5 Problems 147
Ⅴ.6 Notes and References 149
Ⅵ Spectral Variation of Normal Matrices 152
Ⅵ.1 Continuity of Roots of Polynomials 153
Ⅵ.2 Hermitian and Skew-Hermitian Matrices 155
Ⅵ.3 Estimates in the Operator Norm 159
Ⅵ.4 Estimates in the Frobenius Norm 165
Ⅵ.5 Geometry and Spectral Variation:the Operator Norm. 168
Ⅵ.6 Geometry and Spectral Variation:wui Norms 173
Ⅵ.7 Some Inequalities for the Determinant 181
Ⅵ.8 Problems 184
Ⅵ.9 Notes and References 190
Ⅶ Perturbation of Spectral Subspaces of Normal Matrices 194
Ⅶ.1 Pairs of Subspaces 195
Ⅶ.2 The Equation AX－XB=Y 203
Ⅶ.3 Perturbation of Eigenspaces 211
Ⅶ.4 A Perturbation Bound for Eigenvalues 212
Ⅶ.5 Perturbation of the Polar Factors 213
Ⅶ.6 Appendix: Evaluating the (Fourier) constants 216
Ⅶ.7 Problems 221
Ⅶ.8 Notes and References 223
Ⅷ Spectral Variation of Nonnormal Matrices 226
Ⅷ.1 General Spectral Variation Bounds 227
Ⅷ.4 Matrices with Real Eigenvalues 238
Ⅷ.5 Eigenvalues with Symmetries 240
Ⅷ.6 Problems 244
Ⅷ.7 Notes and References 249
Ⅸ A Selection of Matrix Inequalities 253
Ⅸ.1 Some Basic Lemmas 253
Ⅸ.2 Products of Positive Matrices 255
Ⅸ.3 Inequalities for the Exponential Function 258
Ⅸ.4 Arithmetic-Geometric Mean Inequalities 262
Ⅸ.5 Schwarz Inequalities 266
Ⅸ.6 The Lieb Concavity Theorem 271
Ⅸ.7 Operator Approximation 275
Ⅸ.8 Problems 279
Ⅸ.9 Notes and References 285
Ⅹ Perturbation of Matrix Functions 289
Ⅹ.1 Operator Monotone Functions 289
Ⅹ.2 The Absolute Value 296
Ⅹ.3 Local Perturbation Bounds 301
Ⅹ.4 Appendix: Differential Calculus 310
Ⅹ.5 Problems 317
Ⅹ.6 Notes and References 320
References 325
Index 339