LAUSR

Abstract Let A be an arbitrary square matrix and its Jordan canonical form is P − 1 A P = J = diag ( J 1 ( λ 1 ) , ⋯ , J q ( λ q ) ) with P an invertible matrix. λ 1 , ⋯ , λ q are different eigenvalues of matrix A. The commuting solution problem of the matrix equation A X A = X A X is equivalent to the problem J i ( λ i ) Y ( i ) = Y ( i ) J i ( λ i ) , J i ( λ i ) 2 Y ( i ) = J i ( λ i ) ( Y ( i ) ) 2 with Y = P − 1 X P = 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 H η ( δ 1 , δ 2 ) 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 λ i = 0 or λ i ≠ 0 with respect to the i-th Jordan block J i ( λ i ) .