A remark on oscillatory matrices
type: Item
instance of: scholarly article
title: A remark on oscillatory matrices
zbMATH Open document ID: 1084.15012
author: Shaun M. Fallat
published in: Linear Algebra and its Applications
publication date: 2005-01-11T00:00:00Z
review text: The class of oscillatory matrices introduced by \textit{F. P. Gantmacher} and \textit{M. G. Krein} [Oscillation matrices and kernels and small vibrations of mechanical systems (2002; Zbl 1002.74002)] is considered. An \(n\times n\) matrix \(A\) is called totally nonnegative (resp. totally positive) if any minor of \(A\) is nonnegative (resp. positive). A square totally nonnegative matrix \(A\) is called oscillatory if some positive integral power of \(A\) is totally positive. The classical criterion due to Gantmacher and Krein states that an \(n\times n\) totally nonnegative matrix \(A=[a_{ij}]\) is oscillatory if and only if \(A\) is nonsingular and \(a_{i,i+1}>0,\;a_{i+1,i}>0\) for \(i=1,2,\dots,n-1\). The aim of this paper is to shed new light on this criterion by a new proof which incorporates bidiagonal factorizations of totally nonnegative matrices and some planar diagrams associated with such factorizations. Some key ideas are emphasized, which underline the combinatorial character of this new proof.
reviewed by: Valeriu Prepeliţă
Mathematics Subject Classification ID: 15A23
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 05C50
Mathematics Subject Classification ID: 05C38
zbMATH DE Number: 2126347
zbMATH Keywords: oscillatory matrices
zbMATH Keywords: planar diagrams
zbMATH Keywords: totally nonnegative matrices
zbMATH Keywords: bidiagonal factorizations
MaRDI profile type: Publication
full work available at URL: j.laa.2003.12.006
OpenAlex ID: W2026752664
cites work: Totally positive matrices
cites work: Combinatorics and total positivity
cites work: Some properties of totally positive matrices
cites work: Bidiagonal Factorizations of Totally Nonnegative Matrices
cites work: Generalized oscillatory matrices
cites work: Generalized totally positive matrices
cites work: Generalized totally nonnegative matrices
cites work: Total positivity: tests and parametrizations.
cites work: Q3261425
cites work: Q4375491
cites work: Total positivity and Neville elimination
cites work: Q4381523
cites work: Q5624248
cites work: On the boundary of totally positive upper triangular matrices
cites work: A reduction theorem for totally positive matrices
DOI: 10.1016/J.LAA.2003.12.006
description: scientific article; zbMATH DE number 2126347