Topics in matrix analysis

Contents......Page 2
Preface......Page 4
1.0 Introduction......Page 6
1.1 Definitions......Page 10
1.2 Basic properties of the field of values......Page 13
1.3 Convexity......Page 22
1.4 Axiomatization......Page 33
1.5 Location of the field of values......Page 35
1.6 Geometry......Page 53
1.7 Products of matrices......Page 70
1.8 Generalizations of the field of values......Page 82
2.0 Motivation......Page 94
2.1 Definitions and elementary observations......Page 96
2.2 Lyapunov's theorem......Page 100
2.3 The Routh-Hurwitz conditions......Page 106
2.4 Generalizations of Lyapunov's theorem......Page 107
2.5 M-matrices, P-matrices, and related topics......Page 117
3.0 Introduction and historical remarks......Page 139
3.1 The singular value decomposition......Page 149
3.2 Weak majorization and doubly substochastic matrices......Page 168
3.3 Basic inequalities for singular values and eigenvalues......Page 175
3.4 Sums of singular values: the Ky Fan k-norms......Page 200
3.5 Singular values and unitarily invariant norms......Page 208
3.6 Sufficiency of Weyl's product inequalities......Page 222
3.7 Inclusion intervals for singular values......Page 228
3.8 Singular value weak majorization for bilinear products......Page 236
4.0 Motivation......Page 244
4.1 Matrix equations......Page 246
4.2 The Kronecker product......Page 247
4.3 Linear matrix equations and Kronecker products......Page 259
4.4 Kronecker sums and the equation AX+XB=C......Page 273
4.5 Additive and multiplicative commutators and linear preservers......Page 293
5.O Introduction......Page 303
5.1 Some basic observations......Page 309
5.2 The Schur product theorem......Page 313
5.3 Generalizations of the Schur product theorem......Page 317
5.4 The matrices A o (A-1)1, and A o A-1......Page 327
5.5 Inequalities for Hadamard products of general matrices: an overview......Page 337
5.6 Singular values of a Hadamard product: a fundamental inequality......Page 354
5.7 Hadamard products involving nonnegative matrices and M-matrices......Page 361
6.0 Introduction......Page 387
6.1 Polynomial matrix functions and interpolation......Page 388
6.2 Nonpolynomial matrix functions......Page 412
6.3 Hadamard matrix functions......Page 454
6.4 Square roots, logarithms, nonlinear matrix equations......Page 464
6.5 Matrices of functions......Page 495
6.6 A chain rule for functions of a matrix......Page 525
Hints for problems......Page 566
References......Page 589
Notation......Page 595
Index......Page 600