Characterization of \(Q\)-property for multiplicative transformations in semidefinite linear complementarity problems
type: Item
instance of: scholarly article
title: Characterization of \(Q\)-property for multiplicative transformations in semidefinite linear complementarity problems
zbMATH Open document ID: 1200.90161
author: R. Balaji
published in: Linear Algebra and its Applications
publication date: 2010-04-29T00:00:00Z
review text: Let \(V\overset.{}={\mathcal S}^{n\times n}\) be the vector space of real symmetric matrices of order \(n\) and \(\Sigma\) be the set of all posiive semidefinite matrices in \(V\). If \(X\in\Sigma\), we will use the notation \(X\succeq 0\). Suppose that \(L: V\to V\) is a linear transformation. Given an element \(Q\in V\), the semidefinite linear complementarity problem \(\text{SDLCP}(L, Q)\) is to find a matrix \(X\in V\) such that \[ X\succeq 0,\quad Y:= L(X)+ Q\succeq 0\quad\text{and}\quad XY= 0. \] Let \(A\) be a square matrix of order \(n\), the multiplicative transformation \(M_A: V\to V\) is defined by \(M_A(X)= AXA^\top\), and it is said to have the \(Q\)-property is \(\text{SDLCP}(M_A, Q)\) has solution for all \(Q\in V\). The following result is proved. Given \(A\in\mathbb{R}^{n\times n}\) the following are equivalent (a) \(A+ A^\top\) is either positive definite or negative definite; (b) For all \(Q\in V\), \(\text{SDLCP}(M_A, Q)\) has a unique solution; (c) \(\text{SDLCP}(M_A, 0)\) has a unique solution; (d) \(M_A\) has the \(Q\)-property.
Mathematics Subject Classification ID: 90C33
zbMATH DE Number: 5702818
zbMATH Keywords: multiplicative transformations
zbMATH Keywords: complementarity
zbMATH Keywords: inertia
reviewed by: Fabián Flores-Bazan
MaRDI profile type: Publication
full work available at URL: j.laa.2009.12.012
OpenAlex ID: W2024857089
cites work: Q3497902
cites work: Finite-Dimensional Variational Inequalities and Complementarity Problems
cites work: On semidefinite linear complementarity problems
cites work: Relationship Between Strong Monotonicity Property, P2-Property, and the Gus-Property in Semidefinite Linear Complementarity Problems
cites work: Positive operators and an inertia theorem
DOI: 10.1016/J.LAA.2009.12.012
description: scientific article; zbMATH DE number 5702818