Explicit solution of the matrix equation \(AXB-CXD=E\)
type: Item
instance of: scholarly article
title: Explicit solution of the matrix equation \(AXB-CXD=E\)
zbMATH Open document ID: 0682.15013
DOI: 10.1016/0024-3795(89)90708-8
author: Maite Gassó
author: V. Hernández
published in: Linear Algebra and its Applications
publication date: 1989-01-01T00:00:00Z
review text: The authors consider the unique solution of the matrix equation (1) \(AXB- CXD=E\) where \(X\in {\mathbb{K}}^{n\times p}\) is an unknown matrix and \(A,C\in {\mathbb{K}}^{m\times n}\), \(B,D\in {\mathbb{K}}^{p\times q}\), \(E\in {\mathbb{K}}^{n\times p}\) \(({\mathbb{K}}={\mathbb{R}}\) or \({\mathbb{C}})\). Note that equation (1) is equivalent to \((2)\quad (\lambda C-A)XB-CX(\lambda B-D)=- E.\) In Section 2, when \(m=n\), \(p=q\) and B or C is nonsingular, a theoretical approach to the solution of (1) is presented on the basis of the relative Cayley-Hamilton theorem. Under hypothesis (i) the pencils \(\lambda\) C-A and \(\lambda\) B-D are regular, and (ii) the spectra of these pencils have an empty intersection, the unique solution is obtained by means of the inversion of an \(m\times m\) and \(p\times p\) matrix. This paper is strongly influenced by \textit{A. Jameson}'s work [SIAM J. Appl. Math. 16, 1020-1023 (1968; Zbl 0169.352)]. Section 3 deals with the case of singular pencils. It is shown that if (iii) the pencils \(\lambda\) C-A and \(\lambda\) B-D are right and left invertible, respectively, and (iv) they have disjoint spectra, then the equation (1) has a solution. The proof is based on the reduction to the case of regular pencils by using the same technique applied \textit{P. Van Dooren} [Lecture Notes Math. 973, 58-73 (1983; Zbl 0517.65022)] to the matrix equation in Y and Z \((\lambda C-A)Y-Z(\lambda B-D)=\lambda F-E.\)
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 15A22
zbMATH DE Number: 4116725
zbMATH Keywords: Sylvester equation
zbMATH Keywords: Lyapunov equation
zbMATH Keywords: matrix pencil
zbMATH Keywords: matrix equation
zbMATH Keywords: Cayley-Hamilton theorem
zbMATH Keywords: singular pencils
MaRDI profile type: Publication
full work available at URL: 0024-3795(89)90708-8
OpenAlex ID: W2095033200
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description: scientific article; zbMATH DE number 4116725