Totally Nonnegative Matrices
1st Edition
Published on 11. April 2011
264 pages
978-1-4008-3901-8 (ISBN)
Description
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.
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.
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Totally Nonnegative Matrices
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Shaun M. Fallat is professor of mathematics and statistics at the University of Regina. Charles R. Johnson is the Class of 1961 Professor of Mathematics at the College of William & Mary.
Content
List of Figures xi
Preface xiii
Chapter 0. Introduction 1
0.0 Definitions and Notation 1
0.1 Jacobi Matrices and Other Examples of TN matrices 3
0.2 Applications and Motivation 15
0.3 Organization and Particularities 24
Chapter 1. Preliminary Results and Discussion 27
1.0 Introduction 27
1.1 The Cauchy-Binet Determinantal Formula 27
1.2 Other Important Determinantal Identities 28
1.3 Some Basic Facts 33
1.4 TN and TP Preserving Linear Transformations 34
1.5 Schur Complements 35
1.6 Zero-Nonzero Patterns of TN Matrices 37
Chapter 2. Bidiagonal Factorization 43
2.0 Introduction 43
2.1 Notation and Terms 45
2.2 Standard Elementary Bidiagonal Factorization: Invertible Case 47
2.3 Standard Elementary Bidiagonal Factorization: General Case 53
2.4 LU Factorization: A consequence 59
2.5 Applications 62
2.6 Planar Diagrams and EB factorization 64
Chapter 3. Recognition 73
3.0 Introduction 73
3.1 Sets of Positive Minors Sufficient for Total Positivity 74
3.2 Application: TP Intervals 80
3.3 Efficient Algorithm for testing for TN 82
Chapter 4. Sign Variation of Vectors and TN Linear Transformations 87
4.0 Introduction 87
4.1 Notation and Terms 87
4.2 Variation Diminution Results and EB Factorization 88
4.3 Strong Variation Diminution for TP Matrices 91
4.4 Converses to Variation Diminution 94
Chapter 5. The Spectral Structure of TN Matrices 97
5.0 Introduction 97
5.1 Notation and Terms 98
5.2 The Spectra of IITN Matrices 99
5.3 Eigenvector Properties 100
5.4 The Irreducible Case 106
5.5 Other Spectral Results 118
Chapter 6. Determinantal Inequalities for TN Matrices 129
6.0 Introduction 129
6.1 Definitions and Notation 131
6.2 Sylvester Implies Koteljanski?I 132
6.3 Multiplicative Principal Minor Inequalities 134
6.4 Some Non-principal Minor Inequalities 146
Chapter 7. Row and Column Inclusion and the Distribution of Rank 153
7.0 Introduction 153
7.1 Row and Column Inclusion Results for TN Matrices 153
7.2 Shadows and the Extension of Rank Deficiency in Submatrices of
TN Matrices 159
7.3 The Contiguous Rank Property 165
Chapter 8. Hadamard Products and Powers of TN Matrices 167
8.0 Definitions 167
8.1 Conditions under which the Hadamard
Product is TP/TN 168
8.2 The Hadamard Core 169
8.3 Oppenheim's Inequality 177
8.4 Hadamard Powers of TP2 179
Chapter 9. Extensions and Completions 185
9.0 Line Insertion 185
9.1 Completions and Partial TN Matrices 186
9.2 Chordal Case--MLBC Graphs 189
9.3 TN Completions: Adjacent Edge Conditions 191
9.4 TN Completions: Single Entry Case 195
9.5 TN Perturbations: The Case of Retractions 198
Chapter 10. Other Related Topics on TN Matrices 205
10.0 Introduction and Topics 205
10.1 Powers and Roots of TP/TN Matrices 205
10.2 Subdirect Sums of TN Matrices 207
10.3 TP/TN Polynomial Matrices 212
10.4 Perron Complements of TN Matrices 213
Bibliography 219
List of Symbols 239
Index 245
Preface xiii
Chapter 0. Introduction 1
0.0 Definitions and Notation 1
0.1 Jacobi Matrices and Other Examples of TN matrices 3
0.2 Applications and Motivation 15
0.3 Organization and Particularities 24
Chapter 1. Preliminary Results and Discussion 27
1.0 Introduction 27
1.1 The Cauchy-Binet Determinantal Formula 27
1.2 Other Important Determinantal Identities 28
1.3 Some Basic Facts 33
1.4 TN and TP Preserving Linear Transformations 34
1.5 Schur Complements 35
1.6 Zero-Nonzero Patterns of TN Matrices 37
Chapter 2. Bidiagonal Factorization 43
2.0 Introduction 43
2.1 Notation and Terms 45
2.2 Standard Elementary Bidiagonal Factorization: Invertible Case 47
2.3 Standard Elementary Bidiagonal Factorization: General Case 53
2.4 LU Factorization: A consequence 59
2.5 Applications 62
2.6 Planar Diagrams and EB factorization 64
Chapter 3. Recognition 73
3.0 Introduction 73
3.1 Sets of Positive Minors Sufficient for Total Positivity 74
3.2 Application: TP Intervals 80
3.3 Efficient Algorithm for testing for TN 82
Chapter 4. Sign Variation of Vectors and TN Linear Transformations 87
4.0 Introduction 87
4.1 Notation and Terms 87
4.2 Variation Diminution Results and EB Factorization 88
4.3 Strong Variation Diminution for TP Matrices 91
4.4 Converses to Variation Diminution 94
Chapter 5. The Spectral Structure of TN Matrices 97
5.0 Introduction 97
5.1 Notation and Terms 98
5.2 The Spectra of IITN Matrices 99
5.3 Eigenvector Properties 100
5.4 The Irreducible Case 106
5.5 Other Spectral Results 118
Chapter 6. Determinantal Inequalities for TN Matrices 129
6.0 Introduction 129
6.1 Definitions and Notation 131
6.2 Sylvester Implies Koteljanski?I 132
6.3 Multiplicative Principal Minor Inequalities 134
6.4 Some Non-principal Minor Inequalities 146
Chapter 7. Row and Column Inclusion and the Distribution of Rank 153
7.0 Introduction 153
7.1 Row and Column Inclusion Results for TN Matrices 153
7.2 Shadows and the Extension of Rank Deficiency in Submatrices of
TN Matrices 159
7.3 The Contiguous Rank Property 165
Chapter 8. Hadamard Products and Powers of TN Matrices 167
8.0 Definitions 167
8.1 Conditions under which the Hadamard
Product is TP/TN 168
8.2 The Hadamard Core 169
8.3 Oppenheim's Inequality 177
8.4 Hadamard Powers of TP2 179
Chapter 9. Extensions and Completions 185
9.0 Line Insertion 185
9.1 Completions and Partial TN Matrices 186
9.2 Chordal Case--MLBC Graphs 189
9.3 TN Completions: Adjacent Edge Conditions 191
9.4 TN Completions: Single Entry Case 195
9.5 TN Perturbations: The Case of Retractions 198
Chapter 10. Other Related Topics on TN Matrices 205
10.0 Introduction and Topics 205
10.1 Powers and Roots of TP/TN Matrices 205
10.2 Subdirect Sums of TN Matrices 207
10.3 TP/TN Polynomial Matrices 212
10.4 Perron Complements of TN Matrices 213
Bibliography 219
List of Symbols 239
Index 245
