Totally positive matrices

✍ Scribed by Allan Pinkus

• Publisher: CUP
• Year: 2009
• Tongue: English
• Leaves: 195
• Series: Cambridge Tracts in Mathematics
• Category: Library

✦ Synopsis

Totally positive matrices constitute a particular class of matrices, the study of which was initiated by analysts because of its many applications in diverse areas. This modern account of the subject is comprehensive and thorough, with careful treatment of the central properties of totally positive matrices, full proofs and a complete bibliography. The history of the subject is also described: in particular, the book ends with a tribute to the four people who have made the most notable contributions to the history of total positivity: I. J. Schoenberg, M. G. Krein, F. R. Gantmacher and S. Karlin. This monograph will appeal to those with an interest in matrix theory, to those who use or have used total positivity, and to anyone who wishes to learn about this rich and interesting subject.

✦ Table of Contents

Cover......Page 1
Title Page......Page 4
Copyright......Page 5
Dedication......Page 6
Contents......Page 8
Foreword......Page 10
1.1 Preliminaries......Page 14
1.2 Building (strictly) totally positive matrices......Page 18
1.3 Nonsingularity and rank......Page 25
1.4 Determinantal inequalities......Page 37
1.5 Remarks......Page 46
2 Criteria for total positivity and strict total positivity......Page 49
2.1 Criteria for strict total positivity......Page 50
2.2 Density and some further applications......Page 54
2.3 Triangular total positivity......Page 60
2.4 LDU factorizations......Page 63
2.5 Criteria for total positivity......Page 68
2.6 ?°Simple?± criteria for strict total positivity......Page 73
2.7 Remarks......Page 87
3.1 Main equivalence theorems......Page 89
3.2 Intervals of strict total positivity......Page 96
3.3 Remarks......Page 98
4.1 Totally positive kernels and Descartes systems......Page 100
4.2 Exponentials and powers......Page 101
4.3 Cauchy matrix......Page 105
4.4 Green?ˉs matrices......Page 107
4.5 Jacobi matrices......Page 110
4.6 Hankel matrices......Page 114
4.7 Toeplitz matrices......Page 117
4.8 Generalized Hurwitz matrices......Page 124
4.9 More on Toeplitz matrices......Page 130
4.10 Hadamard products of totally positive matrices......Page 132
4.11 Remarks......Page 138
5.1 Oscillation matrices......Page 140
5.2 The Gantmacher¨CKrein theorem......Page 143
5.3 Eigenvalues of principal submatrices......Page 153
5.4 Eigenvectors......Page 157
5.5 Eigenvalues as functions of matrix elements......Page 162
5.6 Remarks......Page 165
6.1 Preliminaries......Page 167
6.2 Factorizations of strictly totally positive matrices......Page 169
6.3 Factorizations of totally positive matrices......Page 177
6.4 Remarks......Page 180
Afterword......Page 182
References......Page 187
Author index......Page 193
Subject index......Page 195