Totally Nonnegative Matrices ：Totally Nonnegative Matrices ( Princeton Series in Applied Mathematics )

Publication subTitle ：Totally Nonnegative Matrices

Publication series ：Princeton Series in Applied Mathematics

Author： Fallat Shaun M.;Johnson Charles R.  

Publisher： Princeton University Press‎

Publication year： 2011

E-ISBN: 9781400839018

P-ISBN(Paperback): 9780691121574

Subject： O151.21 矩阵论

Keyword： 数学,数理科学和化学

Language： ENG

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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.

Chapter

Totally Nonnegative Matrices

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 KoteljanskiI

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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