On the nonlinear matrix equation \(X+A^*{\mathcal F}(X)A=Q\): solutions and perturbation theory
type: Item
instance of: scholarly article
title: On the nonlinear matrix equation \(X+A^*{\mathcal F}(X)A=Q\): solutions and perturbation theory
zbMATH Open document ID: 1086.15013
DOI: 10.1016/S0024-3795(01)00508-0
author: André C. M. Ran
author: Martine C. B. Reurings
published in: Linear Algebra and its Applications
publication date: 2002-06-24T00:00:00Z
review text: Denote by \({\mathcal M}(n)\), \({\mathcal P}(n)\) respectively the sets of all \(n\times n\) and of all \(n\times n\) Hermitian positive semidefinite matrices. The nonlinear matrix equation \(X+A^*{\mathcal F}(X)A=Q\) is considered where \({\mathcal F}:{\mathcal P}(n)\rightarrow {\mathcal M}(n)\), \(Q\) is Hermitian positive definite and \(A\in {\mathcal M}(n)\). Sufficient conditions for the existence and uniqueness of a positive semidefinite solution are derived. Particular attention is paid to the cases when \({\mathcal F}:{\mathcal P}(n)\rightarrow \pm {\mathcal P}(n)\) is monotone or antimonotone, i.e. when \(X\leq Y\) implies respectively \({\mathcal F}(X)\leq {\mathcal F}(Y)\) or \({\mathcal F}(X)\geq {\mathcal F}(Y)\). (The inequality \(X\leq Y\) means that \(Y-X\) is positive semidefinite.) Conditions are given for the continuous dependence of the solution on \(A\) and \(Q\). The perturbed equation is also considered.
Mathematics Subject Classification ID: 15A24
zbMATH DE Number: 1758606
zbMATH Keywords: nonlinear matrix equations
zbMATH Keywords: perturbation theory
zbMATH Keywords: semidefinite solution
reviewed by: Vladimir P. Kostov
MaRDI profile type: Publication
cites work: Q4002249
cites work: Positive solutions to \(X=A-BX^{-1}B^*\)
cites work: On an iteration method for solving a class of nonlinear matrix equations
cites work: Necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation \(X+A^*X^{-1}A=Q\)
cites work: Q2724520
cites work: Q3917065
cites work: Properties of positive definite solutions of the equation \(X+A^*X^{-2}A=I\)
cites work: On matrix equations \(X\pm A^*X^{-2}A=I\)
description: scientific article; zbMATH DE number 1758606