Bidiagonal factorizations and quasi-oscillatory rectangular matrices
type: Item
instance of: scholarly article
title: Bidiagonal factorizations and quasi-oscillatory rectangular matrices
zbMATH Open document ID: 1154.15022
published in: Linear Algebra and its Applications
publication date: 2008-10-06T00:00:00Z
review text: An \(m\times n\) real matrix \(A\) is called totally nonnegative if all its minors are nonnegative, and totally positive if all its minors are positive. The authors define a totally nonnegative matrix \(A\) to be quasi-oscillatory if some positive integral power of \(AA^{\top}\) is totally positive. A square matrix \(E_{k}(\alpha)\) whose diagonal entries are \(1\), whose \((k,k-1)\)st entry is \(\alpha\) and all other entries are \(0\) is called an elementary bidiagonal matrix. The authors show how to factor a totally nonnegative matrix \(A\) in terms of a diagonal matrix, reduced identity matrices and elementary bidiagonal matrices. They then give necessary and sufficient conditions on these elementary bidiagonal matrices for \(A\) to be quasi-oscillatory. They also prove that some integral power of an \(n\times n\) matrix \(A\) is totally positive if and only if \(A=BC\) where \(B\) is a lower triangular matrix, \(C\) is upper triangular, and both \(B\) and \(C\) are quasi-oscillatory.
reviewed by: John D. Dixon
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 15A23
zbMATH DE Number: 5349156
zbMATH Keywords: totally nonnegative matrices
zbMATH Keywords: oscillatory matrix
zbMATH Keywords: Neville elimination
zbMATH Keywords: bidiagonal factorization
author: Maria T. Gassó
author: Juan Ramón Torregrosa Sánchez
MaRDI profile type: Publication
full work available at URL: j.laa.2008.05.030
OpenAlex ID: W2057375254
Wikidata QID: Q114152430
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DOI: 10.1016/J.LAA.2008.05.030
description: scientific article; zbMATH DE number 5349156