Rational solutions of the matrix equation \(p(X) = A\)
type: Item
instance of: scholarly article
title: Rational solutions of the matrix equation \(p(X) = A\)
author: Gilbert J. Groenewald
author: Gerrit Goosen
author: Dawie Janse van Rensburg
author: André C. M. Ran
author: Madelein Thiersen
published in: ELA. The Electronic Journal of Linear Algebra
publication date: 2024-10-29T00:00:00Z
review text: Given a polynomial \(p(\lambda)\) with rational coefficients, and given an \(n \times n\) rational matrix \(A\), this work seeks to identify conditions, based on the characteristic polynomial \(f(\lambda)\) of \(A\) and the polynomial \(p(\lambda)\) under which a rational solution a rational solution \(X\) exists for the matrix equation \(p(X)=A\). One such a condition is mentioned in a paper by \textit{R. Reams} [Linear Algebra Appl. 258, 187--194 (1997; Zbl 0884.15008)]. This work extends Theorem 1 from R. Reams' paper, generalizing the analysis beyond the case \(p(\lambda) = \lambda^m\) with odd \(m\) is considered. The paper concludes by proposing a broader problem: given a rational polynomial \(R(x; y)\) in two variables, can we find a pair of rational matrices \((X; Y )\) such that \(R(X; Y ) = 0\)?
reviewed by: F. Panjeh Ali Beik
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 15B33
Mathematics Subject Classification ID: 12D05
Mathematics Subject Classification ID: 12F10
zbMATH DE Number: 7935111
zbMATH Keywords: matrix polynomial equation
zbMATH Keywords: rational solution
zbMATH Keywords: nonderogatory matrix
MaRDI profile type: Publication
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DOI: 10.13001/ELA.2024.8677
description: scientific article; zbMATH DE number 7935111