Solutions to the Matrix Yang-Baxter Equation

Abstract

This article explores various classes of solutions to the Yang-Baxter type matrix equation, $AXA=XAX$, over the field $K = \mathbb{C}$ or $\mathbb{R}$, by means of the spectral properties of the solution space. We first prove several global constraints on any solution \(X\), including a spectral inclusion \(\sigma(X)\subseteq\sigma(A)\cup\{0\}\) and \(A\)-invariance of \(\ker X\). Specializing to the key case where \(A\) is a single Jordan block, we completely characterised solutions for the case where $A$ is equivalent to a Jordan block. We extend our findings to encompass the general scenario of multiple Jordan blocks under specific conditions. Additionally, novel tools from commutative algebra, such as the Gr\"obner basis, were also applied to arrive at solutions. We construct pencils of solutions that generate new families from known ones under clear algebraic conditions. Together, these results supply a toolbox for analyzing \(AXA=XAX\) beyond diagonalizable \(A\), clarifying how spectral data and Jordan structure govern the solution set.