The extension of Roth's theorem for matrix equations over a ring
type: Item
instance of: scholarly article
title: The extension of Roth's theorem for matrix equations over a ring
zbMATH Open document ID: 0880.15016
DOI: 10.1016/S0024-3795(96)00286-8
author: Jianzhou Liu
author: Liping Huang
published in: Linear Algebra and its Applications
publication date: 1998-02-08T00:00:00Z
review text: The authors consider the matrix equation \(\sum_{i=0}^k A^iXB_i=C\), where \(A\) is square, \(X, B_0,\dots, C\) are of such sizes that the equation makes sense, and the coefficients of all matrices are in a ring \(R\) with 1. Let \(B(\lambda)= \sum_{i=0}^k B_i\lambda^i\), where \(\lambda\) is an indeterminate over \(R\). The main result (Theorem 2) is that, under suitable conditions, the above equation has a solution \(X\) if, and only if, the matrices \[ \begin{pmatrix} \lambda I-A &-C\\ 0&B(\lambda)\end{pmatrix} \qquad\text{and}\qquad \begin{pmatrix} \lambda I-A &0\\ 0&B(\lambda) \end{pmatrix} \] are equivalent over \(R[\lambda]\). The conditions under which the theorem is established are that either \(R\) is a finitely generated module over its centre \(Z\) or that \(R\) is a division ring and \(A\) satisfies a polynomial equation over \(Z\). The original theorem of \textit{W. E. Roth} [Proc. Am. Math. Soc. 3, 392-396 (1952; Zbl 0047.01901)] deals with the equation \(AX- XB=C\) over a field.
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 15A54
zbMATH DE Number: 1044236
zbMATH Keywords: equivalence
zbMATH Keywords: Roth's similarity theorem
zbMATH Keywords: matrix equation
reviewed by: G. E. Wall
MaRDI profile type: Publication
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Wikidata QID: Q127846835
description: scientific article; zbMATH DE number 1044236