Generalized oscillatory matrices
type: Item
instance of: scholarly article
title: Generalized oscillatory matrices
zbMATH Open document ID: 1026.15008
DOI: 10.1016/S0024-3795(02)00436-6
author: Miroslav Fiedler
author: Thomas L. Markham
author: Shaun M. Fallat
published in: Linear Algebra and its Applications
publication date: 2003-03-23T00:00:00Z
review text: The paper is a continuation of the recent series of the authors' works [see for example \textit{M. Fiedler} and \textit{T. L. Markham}, Linear Algebra Appl. 345, 9-28 (2002; Zbl 0995.15016) and references therein] on the theory of totally nonnegative matrices over noncommutative rings. Let \(R\) be a noncommutative ring with the identity 1 and positive part \(R^+\) which is defined as follows: 1. if \(a,b\in R^+\) then \(a+b,ab\in R^+\); 2. if \(a\in R^+\) then \(a\) is invertible and \(a^{-1}\in R^+\). An LB-matrix (resp., UB-matrix) is a square bidiagonal matrix whose diagonal entries are either \(1\) or \(0\), and the subdiagonal (resp., superdiagonal) entries are either zero or positive elements of \(R\). A D-matrix is a diagonal matrix whose diagonal entries are either zero or positive. A B-matrix is either a \(k\times (k+1)\) matrix the diagonal entries of which are either \(1\) or \(0\) and the entries in the parallel second diagonal are either positive or zero, or a transpose of such matrix. A generalized totally nonnegative (GTN) matrix over \(R\) is a product of LB-, UB-, B-, and D-matrices in any order. A generalized invertible totally nonnegative (GITN) matrix over \(R\) is any finite product of invertible LB-, UB-, and D-matrices in any order. A square matrix over \(R\) is called generalized oscillatory (GO) if it is GTN and some power of it is generalized totally positive. Basic generalized oscillatory (BGO) matrices are defined as GITN-matrices of a special canonical form. It is proved that BGO-matrices are GO. The authors show that each BGO-matrix has both subdiagonal and superdiagonal rank one. A characterization theorem for an \(n\times n\)-matrix to be BGO is provided which implies that an \(n\times n\) BGO-matrix is uniquely determined by \(3n-2\) positive parameters and two increasing sequences of positive integers ending by \(n-1\). Some characterizations of GO matrices are obtained. In particular, it is proved that a matrix \(A\in M_n(R)\) is …
Mathematics Subject Classification ID: 15A23
Mathematics Subject Classification ID: 15B33
zbMATH DE Number: 1886828
zbMATH Keywords: totally nonnegative matrices
zbMATH Keywords: ring with identity
zbMATH Keywords: factorization
zbMATH Keywords: bidiagonal matrix
zbMATH Keywords: oscillatory matrix
zbMATH Keywords: exponent of positivity
reviewed by: Alexander Guterman
MaRDI profile type: Publication
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full work available at URL: s0024-3795(02)00436-6
OpenAlex ID: W2024282500
description: scientific article; zbMATH DE number 1886828