Totally Nonnegative Matrices

This book is a comprehensive study of totally nonnegative matrices,defined by the nonnegativity of all subdeterminants. It explores methodological background,historical highlights,and specialized topics,with a unifying theme of elementary bidiagonal factorization. Recent work has shown that bidiagonal factorizations provide deep insights into this class of matrices.

Format: Paperback / softback
Length: 264 pages
Publication date: 13 December 2022
Publisher: Princeton University Press

Totally nonnegative matrices find widespread application across diverse mathematical fields. This comprehensive and self-contained volume delves into the fundamental theory of totally nonnegative matrices, characterized by the nonnegativity of all subdeterminants. It encompasses a methodological background, highlights significant historical ideas, and explores specialized topics. The book employs both classical and ad hoc tools, with a unifying theme being the elementary bidiagonal factorization, which has emerged as the single most crucial tool for this particular class of matrices. Recent research has revealed that bidiagonal factorizations can be concisely described through combinatorial approaches, yielding profound insights. Despite a gradual pace of development, bidiagonal factorizations, alongside determinants, have become the dominant methodology for comprehending total nonnegativity. The subsequent chapters delve into essential topics such as recognizing 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. Additionally, the book includes sample applications, an up-to-date bibliography, a glossary of all symbols utilized, an index, and related references.

Dimension: 235 x 156 (mm)
ISBN-13: 9780691242415