Totally Positive Matrices

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Foreword......Page 11
1.1 Preliminaries......Page 15
1.2 Building (strictly) totally positive matrices......Page 19
1.3 Nonsingularity and rank......Page 26
1.4 Determinantal inequalities......Page 38
1.5 Remarks......Page 47
2 Criteria for total positivity and strict total positivity......Page 50
2.1 Criteria for strict total positivity......Page 51
2.2 Density and some further applications......Page 55
2.3 Triangular total positivity......Page 61
2.4 LDU factorizations......Page 64
2.5 Criteria for total positivity......Page 69
2.6 “Simple” criteria for strict total positivity......Page 74
2.7 Remarks......Page 88
3.1 Main equivalence theorems......Page 90
3.2 Intervals of strict total positivity......Page 97
3.3 Remarks......Page 99
4.1 Totally positive kernels and Descartes systems......Page 101
4.2 Exponentials and powers......Page 102
4.3 Cauchy matrix......Page 106
4.4 Green’s matrices......Page 108
4.5 Jacobi matrices......Page 111
4.6 Hankel matrices......Page 115
4.7 Toeplitz matrices......Page 118
4.8 Generalized Hurwitz matrices......Page 125
4.9 More on Toeplitz matrices......Page 131
4.10 Hadamard products of totally positive matrices......Page 133
4.11 Remarks......Page 139
5.1 Oscillation matrices......Page 141
5.2 The Gantmacher–Krein theorem......Page 144
5.3 Eigenvalues of principal submatrices......Page 154
5.4 Eigenvectors......Page 158
5.5 Eigenvalues as functions of matrix elements......Page 163
5.6 Remarks......Page 166
6.1 Preliminaries......Page 168
6.2 Factorizations of strictly totally positive matrices......Page 170
6.3 Factorizations of totally positive matrices......Page 178
6.4 Remarks......Page 181
Afterword......Page 183
References......Page 188
Author index......Page 194
Subject index......Page 196