Some properties for the existence of a positive definite solution of matrix equation \(X+A^*X^{-2^m}A=I\)
type: Item
instance of: scholarly article
title: Some properties for the existence of a positive definite solution of matrix equation \(X+A^*X^{-2^m}A=I\)
zbMATH Open document ID: 1031.15015
DOI: 10.1016/S0096-3003(01)00019-4
author: Salah M. El-Sayed
author: Mahmoud M. El-Alem
published in: Applied Mathematics and Computation
publication date: 2003-01-28T00:00:00Z
review text: Let \(P(n)\) be the set of all positive definite \((n\times n)\) matrices; \(I\) is the \((n\times n)\) identity matrix; and \(m, n\) \(\in \mathbb{N}\). The authors studies the matrix equation \[ X+A^*X^{-2^m}A=I \tag{1} \] with given matrix \(A\in P(n)\) and unknown matrix \(X\in P(n)\). They give necessary and sufficient conditions for the solvability of equation (1). The main results of this paper are: Theorem 3. If there exist numbers \(a, b\) satisfying \(0<a<b<1\) and the inequalities \[ a^{2m}(1-a)I<AA^*<b^{2m}(1-b)I \] hold, then (1) has a positive definite solution. Theorem 6. If (1) has a positive definite solution \(X\), then \[ A^*A+ (AA^*)^{\frac{1}{2m}}<I. \]
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 15B48
zbMATH DE Number: 1861063
zbMATH Keywords: matrix equation
zbMATH Keywords: positive definite solution
reviewed by: V. M. Prokip
MaRDI profile type: Publication
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full work available at URL: s0096-3003(01)00019-4
OpenAlex ID: W2002965757
description: scientific article; zbMATH DE number 1861063