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Totally Nonnegative Matrices

Book 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.

Table of Contents

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