Totally Nonnegative Matrices

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