Totally nonnegative matrices

✍ Scribed by Fallat S., Johnson C.

• Publisher: PUP
• Year: 2011
• Tongue: English
• Leaves: 265
• Series: Princeton Series in Applied Mathematics
• Category: Library

✦ Synopsis

Totally nonnegative matrices arise in a remarkable variety of mathematical applications. This book is a comprehensive and self-contained study of the essential theory of totally nonnegative matrices, defined by the nonnegativity of all subdeterminants. It explores methodological background, historical highlights of key ideas, and specialized topics.The book uses classical and ad hoc tools, but a unifying theme is the elementary bidiagonal factorization, which has emerged as the single most important tool for this particular class of matrices. Recent work has shown that bidiagonal factorizations may be viewed in a succinct combinatorial way, leading to many deep insights. Despite slow development, bidiagonal factorizations, along with determinants, now provide the dominant methodology for understanding total nonnegativity. The remainder of the book treats important topics, such as recognition of totally nonnegative or totally positive matrices, variation diminution, spectral properties, determinantal inequalities, Hadamard products, and completion problems associated with totally nonnegative or totally positive matrices. The book also contains sample applications, an up-to-date bibliography, a glossary of all symbols used, an index, and related references.

✦ Table of Contents

Totally Nonnegative Matrices......Page 4
Contents......Page 8
List of Figures......Page 12
Preface......Page 14
0.0 Definitions and Notation......Page 18
0.1 Jacobi Matrices and Other Examples of TN matrices......Page 20
0.2 Applications and Motivation......Page 32
0.3 Organization and Particularities......Page 41
1.1 The Cauchy-Binet Determinantal Formula......Page 44
1.2 Other Important Determinantal Identities......Page 45
1.3 Some Basic Facts......Page 50
1.4 TN and TP Preserving Linear Transformations......Page 51
1.5 Schur Complements......Page 52
1.6 Zero-Nonzero Patterns of TN Matrices......Page 54
2.0 Introduction......Page 60
2.1 Notation and Terms......Page 62
2.2 Standard Elementary Bidiagonal Factorization: Invertible Case......Page 64
2.3 Standard Elementary Bidiagonal Factorization: General Case......Page 70
2.4 LU Factorization: A consequence......Page 76
2.5 Applications......Page 79
2.6 Planar Diagrams and EB factorization......Page 81
3.0 Introduction......Page 90
3.1 Sets of Positive Minors Sufficient for Total Positivity......Page 91
3.2 Application: TP Intervals......Page 97
3.3 Efficient Algorithm for testing for TN......Page 99
4.1 Notation and Terms......Page 104
4.2 Variation Diminution Results and EB Factorization......Page 105
4.3 Strong Variation Diminution for TP Matrices......Page 108
4.4 Converses to Variation Diminution......Page 111
5.0 Introduction......Page 114
5.1 Notation and Terms......Page 115
5.2 The Spectra of IITN Matrices......Page 116
5.3 Eigenvector Properties......Page 117
5.4 The Irreducible Case......Page 123
5.5 Other Spectral Results......Page 135
6.0 Introduction......Page 146
6.1 Definitions and Notation......Page 148
6.2 Sylvester Implies KoteljanskiI......Page 149
6.3 Multiplicative Principal Minor Inequalities......Page 151
6.4 Some Non-principal Minor Inequalities......Page 163
7.1 Row and Column Inclusion Results for TN Matrices......Page 170
7.2 Shadows and the Extension of Rank Deficiency in Submatrices of TN Matrices......Page 176
7.3 The Contiguous Rank Property......Page 182
8.0 Definitions......Page 184
8.1 Conditions under which the Hadamard Product is TP/TN......Page 185
8.2 The Hadamard Core......Page 186
8.3 Oppenheim’s Inequality......Page 194
8.4 Hadamard Powers of TP2......Page 196
9.0 Line Insertion......Page 202
9.1 Completions and Partial TN Matrices......Page 203
9.2 Chordal Case—MLBC Graphs......Page 206
9.3 TN Completions: Adjacent Edge Conditions......Page 208
9.4 TN Completions: Single Entry Case......Page 212
9.5 TN Perturbations: The Case of Retractions......Page 215
10.1 Powers and Roots of TP/TN Matrices......Page 222
10.2 Subdirect Sums of TN Matrices......Page 224
10.3 TP/TN Polynomial Matrices......Page 229
10.4 Perron Complements of TN Matrices......Page 230
Bibliography......Page 236
List of Symbols......Page 256
Index......Page 262