\(M\)-matrices whose inverses are totally positive
type: Item
instance of: scholarly article
title: \(M\)-matrices whose inverses are totally positive
zbMATH Open document ID: 0827.15022
DOI: 10.1016/0024-3795(93)00244-T
author: J. M. Peña
published in: Linear Algebra and its Applications
publication date: 1995-07-03T00:00:00Z
review text: A real invertible \(n \times n\) matrix with nonpositive off-diagonal elements is an \(M\)-matrix provided \(Ax \geq 0\) implies \(x \geq 0\) for all \(x \in {\mathcal F}^n\). The author shows that if \(A\) is such a matrix, then \(A^{- 1}\) is totally positive (i.e., all minors are nonnegative) if and only if \(A\) is a tridiagonal matrix. As a kind of dual, if \(A\) is nonsingular and totally positive, then it is shown that \(A^{- 1}\) is an \(M\)-matrix if and only if \(A^{- 1}\) is tridiagonal if and only if \(\text{det} A (i |j) = 0\) whenever \(|i - j |= 2\).
reviewed by: Stanley E. Payne
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 15A09
zbMATH DE Number: 769058
zbMATH Keywords: totally positive matrix
zbMATH Keywords: \(M\)-matrix
zbMATH Keywords: tridiagonal matrix
MaRDI profile type: Publication
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Wikidata QID: Q126550725
description: scientific article; zbMATH DE number 769058