A nonlinear matrix equation connected to interpolation theory.
type: Item
instance of: scholarly article
title: A nonlinear matrix equation connected to interpolation theory.
zbMATH Open document ID: 1039.15007
DOI: 10.1016/S0024-3795(03)00541-X
author: André C. M. Ran
author: Martine C. B. Reurings
published in: Linear Algebra and its Applications
publication date: 2004-03-14T00:00:00Z
review text: The authors study the matrix equation \(X=Q+A^*(\widehat{X}-C)^{-1}A\), where \(Q\) is an \(n\times n\) positive definite matrix, \(C\) is \(mn\times mn\) positive semidefinite, \(A\) is \(mn\times n\) (arbitrary) and \(\widehat{X}\) is \(mn\times mn\), block diagonal, with \(m\) diagonal blocks equal to \(X\). The authors impose the condition \(C<\widehat{Q}\) (i.e. the matrix \(\widehat{Q}-C\) is positive definite) and prove the existence and uniqueness of the solution in a certain class of positive definite matrices. These solutions are important in a problem from optimal interpolation theory, see \textit{L. A. Sakhnovich} [Interpolation theory and its Applications. (Mathematics and Its applications. (Dordrecht). 428 Dordrecht: Kluwer Academic Publishers.) (1997; Zbl 0894.41001), Chapter 7], where existence and uniqueness of the solutions is conjectured.
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 41A05
Mathematics Subject Classification ID: 47A57
zbMATH DE Number: 2056691
zbMATH Keywords: nonlinear matrix equations
zbMATH Keywords: interpolation theory
zbMATH Keywords: generalized Riccati equation
zbMATH Keywords: positive definite matrices
reviewed by: Vladimir P. Kostov
MaRDI profile type: Publication
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full work available at URL: s0024-3795(03)00541-x
OpenAlex ID: W2008620911
description: scientific article; zbMATH DE number 2056691