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

On the commuting solutions to the Yang-Baxter-like matrix equation for identity matrix minus special rank-two matrices

“…Suppose A = I − PQ T is such that Λ = Λ 3 in its Jordan form (3) with det(Q T P) = 0. Then all the commuting solutions of (1) are X = WYW −1 with Y partition as (9), such that M is an arbitrary (n − 4) × (n − 4) projection matrix, the vectors S, S T belong to the null space of I − M and the vectors T, T T belong to the null space of M, when t 11 + t 33 = 1, (i) t 13 = 0, t 31 0, then t 11 = 0 or 1, t 33 = 1 − t 11 and t 31 is an arbitrary non-zero complex number, t 12 and t 32 are two arbitrary complex numbers.…”

“…We limit the task to only finding the solutions that commute with A. Some solutions can be obtained in [6] when the matrix A is a special class of Jordan forms, and a more general result was proved in [9] when the matrix A is a class of diagonalizable matrices, but the general solution has still never been obtained for arbitrary matrices A. In a recent paper [20], the author have found all the solutions of (1), where the given n × n complex matrix A = PQ T , with two n × 2 matrices P and Q, with the assumption that Q T P is singular.…”

All solutions of the Yang–Baxter-like matrix equation for rank-one matrices

“…Such spectral solutions are also commuting ones, that is, they commute with the given matrix. In the special case that A is diagonalizable, all the commuting solutions of (1) are found in [9]. In particular, all the commuting solutions with A satisfying A k = A for a positive integer k 3 have been obtained, and further more when A is a Householder matrix so that A 3 = A, all the non-commuting solutions of (1) have also been constructed.…”

“…In a recent paper, the spectral perturbation technique of [9] for Householder matrices were extended in [10] to more general elementary matrices. But still there has not been a successful method that leads to all the solutions of the Yang-Baxter-like matrix equation when A is arbitrary.…”

On the Yang-Baxter-like matrix equation for rank-two matrices

“…By restricting the task to only finding the solutions that commute with A, several solution results have been obtained in [10] for matrices A of special Jordan forms, and a more general result was proved in [15] for the class of diagonalizable matrices. However, no general result has been found so far for non-all associated with eigenvalue 0.…”