On the positive definite solutions of the matrix equations \(X^{s}\pm A^{\text T} X^{-t} A=I_{n}\)
type: Item
instance of: scholarly article
title: On the positive definite solutions of the matrix equations \(X^{s}\pm A^{\text T} X^{-t} A=I_{n}\)
zbMATH Open document ID: 1025.15018
DOI: 10.1016/S0024-3795(02)00661-4
author: Hua Gao
author: Liu Xinguo
published in: Linear Algebra and its Applications
publication date: 2003-07-30T00:00:00Z
review text: The paper deals with the matrix equations \(X^s+\sigma A^\top X^{-t}A=I_n\), where \(\sigma=\pm 1\), \(s,t\) are positive integers, \(I_n\) is the identity \(n\times n\) matrix and \(X\in \mathbb{R}^{n\times n}\) is the solution. The case \(s=1\) is well studied in the literature. Using the Brouwer fixed point principle the authors prove theorems for the existence of symmetric positive definite solutions. Condition numbers for the solutions of these equations are derived and other perturbation results are also obtained. Finally the authors consider the iterative schemes \(X_{k+1}=(I_n- \sigma A^\top X_k^{-t}A)^{1/s}\) for computing the solutions.
reviewed by: Mihail M. Konstantinov
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 65F30
Mathematics Subject Classification ID: 65H10
zbMATH DE Number: 1956793
zbMATH Keywords: Matrix equation
zbMATH Keywords: positive definite solution
zbMATH Keywords: sensitivity analysis
zbMATH Keywords: iterative methods
MaRDI profile type: Publication
cites work: Positive solutions to \(X=A-BX^{-1}B^*\)
cites work: On the existence of a positive definite solution of the matrix equation \(X+A^ T X^{-1} A=I\)
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: Iterative solution of two matrix equations
cites work: On matrix equations \(X\pm A^*X^{-2}A=I\)
cites work: Properties of positive definite solutions of the equation \(X+A^*X^{-2}A=I\)
cites work: Q3998482
cites work: On the matrix equation \(X+A^ TX^{-1}A=I\)
cites work: Computing the Extremal Positive Definite Solutions of a Matrix Equation
cites work: Q4244570
cites work: Q5652137
cites work: Computing real square roots of a real matrix
cites work: A Theory of Condition
cites work: Q3754747
description: scientific article; zbMATH DE number 1956793