On some properties of \(\mathbf P\)-matrix sets
type: Item
instance of: scholarly article
title: On some properties of \(\mathbf P\)-matrix sets
zbMATH Open document ID: 0949.15027
DOI: 10.1016/S0024-3795(98)10228-8
author: M. Seetharama Gowda
author: Yoon J. Song
author: G. Ravindran
published in: Linear Algebra and its Applications
publication date: 1999-09-30T00:00:00Z
review text: Let \({\mathcal C}\) be a non-empty set of matrices in \(\mathbb{R}^{n\times n}\). The paper introduces the concept of row-P-property for the set of matrices in \({\mathcal C}\). It is shown that this property is equivalent to the fact that for any \(x\neq 0\) in \(\mathbb{R}^n\), there exists an index \(i\) with \(x_i(M x)_i > 0\) for all \(M\in{\mathcal C}\). It is also shown that a compact convex set \({\mathcal C}\) in \(\mathbb{R}^{n\times n}\) has the row-P-property if and only if the set of extreme point of \({\mathcal C}\) has the row-P-property. It is also proved that a compact set \({\mathcal C}\) in \(\mathbb{R}^{n\times n}\) has the row-P-property if and only if the complementarity problem \(CP({\mathcal C},q)\) has a unique solution for every compact \(q\) in \(\mathbb{R}^n\). As a corollary, the existence of a unique \(u>0\) such that \(Mu>0\) for all \(M\in{\mathcal C}\) is obtained which is shown to fail without the row-P-property.
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 90C33
Mathematics Subject Classification ID: 15A60
zbMATH DE Number: 1331376
zbMATH Keywords: P-matrix
zbMATH Keywords: Z-matrix
zbMATH Keywords: linear complementarity problem
zbMATH Keywords: convex set
zbMATH Keywords: compact set
reviewed by: Arbind Kumar Lal
MaRDI profile type: Publication
cites work: Q5340130
cites work: Q4003375
cites work: Interval P-Matrices
cites work: The Generalized Order Linear Complementarity Problem
cites work: Q5638112
cites work: Q4143788
cites work: Generalizations of \(\mathbf P_ 0\)- and \(\mathbf P\)-properties; extended vertical and horizontal linear complementarity problems
description: scientific article; zbMATH DE number 1331376