Chapter 0 Review and miscellanea 1
0.0 Introduction 1
0.1 Vector spaces 1
0.2 Matrices 4
0.3 Determinants 7
0.4 Rank 12
0.5 Nonsingularity 14
0.6 The usual inner product 14
0.7 Partitioned matrices 17
0.8 Determinants again 19
0.9 Special types of matrices 23
0.10 Change of basis 30
Chapter 1 Eigenvalues,eigenvectors,and similarity 33
1.0 Introduction 33
1.1 The eigenvalue-eigenvector equation 34
1.2 The characteristic polynomial 38
1.3 Similarity 44
1.4 Eigenvectors 57
Chapter 2 Unitary equivalence and normal matrices 65
2.0 Introduction 65
2.1 Unitary matrices 66
2.2 Unitary equivalence 72
2.3 Schur's unitary triangularization theorem 79
2.4 Some implications of Schur's theorem 85
2.5 Normal matrices 100
2.6 QR factorization and algorithm 112
Chapter 3 Canonical forms 119
3.0 Introduction 119
3.1 The Jordan canonical form:a proof 121
3.2 The Jordan canonical form:some observations and applications 129
3.3 Polynomials and matrices:the minimal polynomial 142
3.4 Other canonical forms and factorizations 150
3.5 Triangular factorizations 158
Chapter 4 Hermitian and symmetric matrices 167
4.0 Introduction 167
4.1 Definitions,properties,and characterizations of Hermitian matrices 169
4.2 Variational characterizations of eigenvalues of Hermitian matrices 176
4.3 Some applications of the variational characterizations 181
4.4 Complex symmetric matrices 201
4.5 Congruence and simultaneous diagonalization of Hermitian and symmetric matrices 218
4.6 Consimilarity and condiagonalization 244
Chapter 5 Norms for vectors and matrices 257
5.0 Introduction 257
5.1 Defining properties of vector norms and inner products 259
5.2 Examples of vector norms 264
5.3 Algebraic properties of vector norms 268
5.4 Analytic properties of vector norms 269
5.5 Geometric properties of vector norms 281
5.6 Matrix norms 290
5.7 Vector norms on matrices 320
5.8 Errors in inverses and solutions of linear systems 335
Chapter 6 Location and perturbation of eigenvalues 343
6.0 Introduction 343
6.1 Ger？gorin discs 344
6.2 Ger？gorin discs-a closer look 353
6.3 Perturbation theorems 364
6.4 Other inclusion regions 378
Chapter 7 Positive definite matrices 391
7.0 Introduction 391
7.1 Definitions and properties 396
7.2 Characterizations 402
7.3 The polar form and the singular value decomposition 411
7.4 Examples and applications of the singular value decomposition 427
7.5 The Schur product theorem 455
7.6 Congruence:products and simultaneous diagonalization 464
7.7 The positive semidefinite ordering 469
7.8 Inequalities for positive definite matrices 476
Chapter 8 Nonnegative matrices 487
8.0 Introduction 487
8.1 Nonnegative matrices-inequalities and generalities 490
8.2 Positive matrices 495
8.3 Nonnegative matrices 503
8.4 Irreducible nonnegative matrices 507
8.5 Primitive matrices 515
8.6 A general limit theorem 524
8.7 Stochastic and doubly stochastic matrices 526
Appendices 531
A Complex numbers 531
B Convex sets and functions 533
C The fundamental theorem of algebra 537
D Continuous dependence of the zeroes of a polynomial on its coefficients 539
E Weierstrass's theorem 541
References 543
Notation 547
Index 549