Orthogonal bases that leads to symmetric nonnegative matrices
type: Item
instance of: scholarly article
title: Orthogonal bases that leads to symmetric nonnegative matrices
zbMATH Open document ID: 0891.15019
DOI: 10.1016/S0024-3795(97)00302-9
author: M. Neumann
author: Ludwig Elsner
author: Reinhard Nabben
published in: Linear Algebra and its Applications
publication date: 1998-06-30T00:00:00Z
review text: Interpreting inequalities for matrices elementwise, then a matrix \(R\in{\mathbb{R}}^{n\times n}\) is called a Soules matrix if (1) it is orthogonal, (2) its first column \(r_1\geq0\), (3) for all diagonal matrices \(\Lambda\geq 0\) with nonincreasing diagonal elements, \(S=R\Lambda R^T\geq0\). A matrix \(A\) is called an M-matrix if it is of the form \(cI-G\) with \(G\geq0\) and \(c\geq\) the spectral radius of \(G\). If \(A^k\) is an irreducible M-matrix for all \(k\geq1\), then \(A\) is called an MMA-matrix. A matrix \(A\) is called an inverse MMA-matrix if its inverse is an MMA-matrix. This paper shows how to construct all Soules matrices, starting from its first column \(r_1\). Furthermore, it is proved that the class of matrices \(S\) of the above form coincides with the class of symmetric inverse MMA-matrices and that the class of nonsingular such matrices \(S\) can be mapped onto another class which is up to a conjugation by positive diagonal matrices the so called class of strictly ultrametric matrices. Finally, a special sign pattern for the elements of \(Q\) and \(R\) in the QR factorization of certain such matrices \(S\) is explained.
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 65F05
zbMATH DE Number: 1116058
zbMATH Keywords: Soules matrix
zbMATH Keywords: M-matrix
zbMATH Keywords: MMA-matrix
zbMATH Keywords: ultrametric matrix
zbMATH Keywords: QR factorization
zbMATH Keywords: orthogonal bases
zbMATH Keywords: symmetric nonnegative matrices
zbMATH Keywords: irreducible
reviewed by: Adhemar Bultheel
MaRDI profile type: Publication
cites work: Q4326384
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Wikidata QID: Q127771425
description: scientific article; zbMATH DE number 1116058