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by Charles R. Johnson, Shaun M. Fallat
Totally Nonnegative Matrices
Cover
Title Page
Copyright
Dedication
Contents
List of Figures
Preface
Chapter 0. Introduction
0.0 Definitions and Notation
0.1 Jacobi Matrices and Other Examples of TN matrices
0.2 Applications and Motivation
0.3 Organization and Particularities
Chapter 1. Preliminary Results and Discussion
1.0 Introduction
1.1 The Cauchy-Binet Determinantal Formula
1.2 Other Important Determinantal Identities
1.3 Some Basic Facts
1.4 TN and TP Preserving Linear Transformations
1.5 Schur Complements
1.6 Zero-Nonzero Patterns of TN Matrices
Chapter 2. Bidiagonal Factorization
2.0 Introduction
2.1 Notation and Terms
2.2 Standard Elementary Bidiagonal Factorization: Invertible Case
2.3 Standard Elementary Bidiagonal Factorization: General Case
2.4 LU Factorization: A consequence
2.5 Applications
2.6 Planar Diagrams and EB factorization
Chapter 3. Recognition
3.0 Introduction
3.1 Sets of Positive Minors Sufficient for Total Positivity
3.2 Application: TP Intervals
3.3 Efficient Algorithm for testing for TN
Chapter 4. Sign Variation of Vectors and TN Linear Transformations
4.0 Introduction
4.1 Notation and Terms
4.2 Variation Diminution Results and EB Factorization
4.3 Strong Variation Diminution for TP Matrices
4.4 Converses to Variation Diminution
Chapter 5. The Spectral Structure of TN Matrices
5.0 Introduction
5.1 Notation and Terms
5.2 The Spectra of IITN Matrices
5.3 Eigenvector Properties
5.4 The Irreducible Case
5.5 Other Spectral Results
Chapter 6. Determinantal Inequalities for TN Matrices
6.0 Introduction
6.1 Definitions and Notation
6.2 Sylvester Implies Koteljanski?
6.3 Multiplicative Principal Minor Inequalities
6.4 Some Non-principal Minor Inequalities
Chapter 7. Row and Column Inclusion and the Distribution of Rank
7.0 Introduction
7.1 Row and Column Inclusion Results for TN Matrices
7.2 Shadows and the Extension of Rank Deficiency in Submatrices of TN Matrices
7.3 The Contiguous Rank Property
Chapter 8. Hadamard Products and Powers of TN Matrices
8.0 Definitions
8.1 Conditions under which the Hadamard Product is TP/TN
8.2 The Hadamard Core
8.3 Oppenheim’s Inequality
8.4 Hadamard Powers of TP2
Chapter 9. Extensions and Completions
9.0 Line Insertion
9.1 Completions and Partial TN Matrices
9.2 Chordal Case—MLBC Graphs
9.3 TN Completions: Adjacent Edge Conditions
9.4 TN Completions: Single Entry Case
9.5 TN Perturbations: The Case of Retractions
Chapter 10. Other Related Topics on TN Matrices
10.0 Introduction and Topics
10.1 Powers and Roots of TP/TN Matrices
10.2 Subdirect Sums of TN Matrices
10.3 TP/TN Polynomial Matrices
10.4 Perron Complements of TN Matrices
Bibliography
List of Symbols
Index
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