On the existence of a positive definite solution of the matrix equation \(X+A^ T X^{-1} A=I\)
type: Item
instance of: scholarly article
title: On the existence of a positive definite solution of the matrix equation \(X+A^ T X^{-1} A=I\)
zbMATH Open document ID: 0798.15013
DOI: 10.1016/0024-3795(93)90115-5
published in: Linear Algebra and its Applications
publication date: 1994-04-28T00:00:00Z
review text: First, the author demonstrates that the problem described in the title is equivalent to the recursion problem \(\forall n \in \mathbb{N}\), is \(X_ n > AA^ T\) if \(X_{n+1} = I -A^ T X_ n^{-1}A\) with \(X_ 0= I\). From this follow not only necessary solvability conditions in terms of the spectral radius of \(A\), \(A+A^ T\) and \(A-A^ T\), respectively, but also an algorithm for the numerical computation of the solution. For normal matrices \(A\) one of these conditions is necessary and sufficient. Moreover, it is demonstrated that the matrix \(A\) need not be regular and how this equation can be used to solve an indefinite linear/quadratic, discrete-time optimal control problem. The paper is well-organized, and easy to read.
reviewed by: Inge Troch
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 93C55
Mathematics Subject Classification ID: 65F30
Mathematics Subject Classification ID: 49N10
zbMATH DE Number: 540059
zbMATH Keywords: positive definite solution
zbMATH Keywords: matrix equation
zbMATH Keywords: algorithm
zbMATH Keywords: normal matrices
zbMATH Keywords: indefinite linear/quadratic, discrete-time optimal control problem
author: J. C. Engwerda
MaRDI profile type: Publication
full work available at URL: 0024-3795(93)90115-5
OpenAlex ID: W2145933873
cites work: Matrix Analysis
cites work: Spectral theory of the linear-quadratic optimal control problem: Discrete-time single-input case
cites work: Fast projection methods for minimal design problems in linear system theory
cites work: The Stabilizing Solution of the Algebraic Riccati Equation
cites work: Existence and comparison theorems for algebraic Riccati equations for continuous- and discrete-time systems
cites work: Families of linear-quadratic problems: Continuity properties
description: scientific article; zbMATH DE number 540059