Totally Nonnegative Matrices

by Fallat, Shaun M.; Johnson, Charles R.

• ISBN13: 9780691121574
• ISBN10: 0691121575
• Format: Hardcover
• Copyright: 2011-04-11
• Publisher: Princeton Univ Pr

Summary

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 carefully 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.

Author Biography

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.

Table of Contents

List of Figures	p. xi
Preface	p. xiii
Introduction	p. 1
Definitions and Notation	p. 1
Jacobi Matrices and Other Examples of TN matrices	p. 3
Applications and Motivation	p. 15
Organization and Particularities	p. 24
Preliminary Results and Discussion	p. 27
Introduction	p. 27
The Cauchy-Binet Determinantal Formula	p. 27
Other Important Determinantal Identities	p. 28
Some Basic Facts	p. 33
TN and TP Preserving Linear Transformations	p. 34
Schur Complements	p. 35
Zero-Nonzero Patterns of TN Matrices	p. 37
Bidiagonal Factorization	p. 43
Introduction	p. 43
Notation and Terms	p. 45
Standard Elementary Bidiagonal Factorization: Invertible Case	p. 47
Standard Elementary Bidiagonal Factorization: General Case	p. 53
LU Factorization: A consequence	p. 59
Applications	p. 62
Planar Diagrams and EB factorization	p. 64
Recognition	p. 73
Introduction	p. 73
Sets of Positive Minors Sufficient for Total Positivity	p. 74
Application: TP Intervals	p. 80
Efficient Algorithm for testing for TN	p. 82
Sign Variation of Vectors and TN Linear Transformations	p. 87
Introduction	p. 87
Notation and Terms	p. 87
Variation Diminution Results and EB Factorization	p. 88
Strong Variation Diminution for TP Matrices	p. 91
Converses to Variation Diminution	p. 94
The Spectral Structure of TN Matrices	p. 97
Introduction	p. 97
Notation and Terms	p. 98
The Spectra of IITN Matrices	p. 99
Eigenvector Properties	p. 100
The Irreducible Case	p. 106
Other Spectral Results	p. 118
Determinantal Inequalities for TN Matrices	p. 129
Introduction	p. 129
Definitions and Notation	p. 131
Sylvester Implies KoteljanskiI	p. 132
Multiplicative Principal Minor Inequalities	p. 134
Some Non-principal Minor Inequalities	p. 146
Row and Column Inclusion and the Distribution of Rank	p. 153
Introduction	p. 153
Row and Column Inclusion Results for TN Matrices	p. 153
Shadows and the Extension of Rank Deficiency in Submatrices of TN Matrices	p. 159
The Contiguous Rank Property	p. 165
Hadamard Products and Powers of TN Matrices	p. 167
Definitions	p. 167
Conditions under which the Hadamard Product is TP/TN	p. 168
The Hadamard Core	p. 169
Oppenheim's Inequality	p. 177
Hadamard Powers of TP2	p. 179
Extensions and Completions	p. 185
Line Insertion	p. 185
Completions and Partial TN Matrices	p. 186
Chordal Case-MLBC Graphs	p. 189
TN Completions: Adjacent Edge Conditions	p. 191
TN Completions: Single Entry Case	p. 195
TN Perturbations: The Case of Retractions	p. 198
Other Related Topics on TN Matrices	p. 205
Introduction and Topics	p. 205
Powers and Roots of TP/TN Matrices	p. 205
Subdirect Sums of TN Matrices	p. 207
TP/TN Polynomial Matrices	p. 212
Perron Complements of TN Matrices	p. 213
Bibliography	p. 219
List of Symbols	p. 239
Index	p. 245
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