Totally Nonnegative Matrices

Shaun M. Fallat; Charles R. Johnson

Shaun M. Fallat and Charles R. Johnson

Totally Nonnegative Matrices

In: Princeton Series in Applied Mathematics, 35

Princeton University Press | 2011

https://doi.org/10.1515/9781400839018

Overview
Contents

About this book

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.

Author information

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.

Reviews

This book is a very useful new reference on the subject of TN matrices and it will be of interest to researchers on matrix theory as well as to researchers of any field where total positivity has applications.---Juan Manuel Pefia, Mathematical Reviews

"This book is a valuable new reference on the subject of totally nonnegative matrices and its insights will be much appreciated by a broad community of readers interested in matrix theory and its applications."—Charles Micchelli, City University of Hong Kong and State University of New York, Albany

Frontmatter Free access PDF PDF	i
Contents Free access PDF PDF	vii
List of Figures Access restricted Content is available PDF PDF	xi
Preface Access restricted Content is available PDF PDF	xiii
Chapter 0. Introduction Access restricted Content is available PDF PDF	1
Chapter One. Preliminary Results and Discussion Access restricted Content is available PDF PDF	27
Chapter Two. Bidiagonal Factorization Access restricted Content is available PDF PDF	43
Chapter Three. Recognition Access restricted Content is available PDF PDF	73
Chapter Four. Sign Variation of Vectors and TN Linear Transformations Access restricted Content is available PDF PDF	87
Chapter Five. The Spectral Structure of TN Matrices Access restricted Content is available PDF PDF	97
Chapter Six. Determinantal Inequalities for TN Matrices Access restricted Content is available PDF PDF	129
Chapter Seven. Row and Column Inclusion and the Distribution of Rank Access restricted Content is available PDF PDF	153
Chapter Eight. Hadamard Products and Powers of TN Matrices Access restricted Content is available PDF PDF	167
Chapter Nine. Extensions and Completions Access restricted Content is available PDF PDF	185
Chapter Ten. Other Related Topics on TN Matrices Access restricted Content is available PDF PDF	205
Bibliography Access restricted Content is available PDF PDF	219
List of Symbols Access restricted Content is available PDF PDF	239
Index Access restricted Content is available PDF PDF	245

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Totally Nonnegative Matrices

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