On commuting solutions of the Yang–Baxter-like matrix equation

Let A be an arbitrary square matrix and its Jordan canonical form is P−1AP=J=diag(J1(λ1),⋯,Jq(λq)) with P an invertible matrix. λ1,⋯,λq are different eigenvalues of matrix A. The commuting solution problem of the matrix equation AXA=XAX is equivalent to the problem Ji(λi)Y(i)=Y(i)Ji(λi), Ji(λi)2Y(i)=Ji(λi)(Y(i))2 with Y=P−1XP=diag(Y(1),⋯,Y(q)). We give the structures of the commuting solutions Y(i) in special Toeplitz forms. Based on them, we construct new matrices Hη(δ1,δ2) related to the commuting solutions. Then we propose a method of solving all the commuting Yang–Baxter-like solutions, by which all solutions can be obtained step by step by recursively solving matrix equations in two cases λi=0 or λi≠0 with respect to the i-th Jordan block Ji(λi).