On the intrinsic structure of the solution set to the Yang-Baxter-like matrix equation
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.
scientific article; zbMATH DE number 7481813
scientific article