On the intrinsic structure of the solution set to the Yang-Baxter-like matrix equation
type: Item
instance of: scholarly article
title: On the intrinsic structure of the solution set to the Yang-Baxter-like matrix equation
zbMATH Open document ID: 1487.15016
author: Nebojša Č. Dinčić
author: Bogdan D. Djordjević
published in: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas. RACSAM
publication date: 2022-02-28T00:00:00Z
review text: The authors consider a matrix equation that is somewhat reminiscent of the Yang-Baxter equation, namely \(AXA=XAX\), where \(A\) is a given square matrix while \(X\) is the unknown matrix of the same size. They call this equation ``Yang-Baxter-like matrix equation''. First, they study the properties of the trivial solutions \(X=0\) and \(X=A\), namely, whether or not they are isolated in the solution set. It turns out that if \(A\) is a nondegenerate matrix, \(X = 0\) is always isolated, while \(X = A\) is isolated under some conditions. Then, the authors study the case when \(A\) is singular (degenerate). In this case, they show that the two trivial solutions can be path-connected in the solution set. Additionally, they prove that every nontrivial non-commuting solution is always contained in some path-connected subset of the solution set, regardless of whether \(A\) is regular or singular. Finally, the authors develop some new methods for obtaining more nontrivial solutions, and point out connections with some related subjects such as generalized inverses of a matrix and Sylvester matrix equation. There are many explicit examples illustrating the theory.
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 15A10
Mathematics Subject Classification ID: 15A16
Mathematics Subject Classification ID: 16T25
zbMATH DE Number: 7481813
zbMATH Keywords: Yang-Baxter-like matrix equation
zbMATH Keywords: Sylvester equation
zbMATH Keywords: nonlinear matrix equations
zbMATH Keywords: generalized inverses
reviewed by: Igor G. Korepanov
describes a project that uses: mftoolbox
MaRDI profile type: Publication
full work available at URL: s13398-022-01214-8
OpenAlex ID: W4210610652
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DOI: 10.1007/S13398-022-01214-8
description: scientific article; zbMATH DE number 7481813