Complete commuting solutions of the Yang–Baxter-like matrix equation for diagonalizable matrices

Let AA be a square matrix that is diagonalizable. We find all the commuting solutions of the quadratic matrix equation AXA=XAXAXA=XAX, by taking advantage of the Jordan form structure of AA, together with the help of a well-known theorem on the uniqueness of a solution to Sylvester’s equation. Two special classes of the given matrix AA are further investigated, including circular matrices and those that are equal to some of their powers. Moreover, all the non-commuting solutions are constructed when AA is a Householder matrix, based on a spectral perturbation result.