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

Abstract

Let A D PQ T , where P and Q are two n 2 complex matrices of full column rank such that Q T P is singular. We solve the quadratic matrix equation AXA D XAX. Together with a previous paper devoted to the case that Q T P is nonsingular, we have completely solved the matrix equation with any given matrix A of rank-two.

Unification Theories: New Results and Examples

“…Note that if A is "small" then D − AXA could be regarded as a deformation of D. 12can be interpreted as a "generalized eigenvalue problem" (see, for example, [12]). 12is a type of Yang-Baxter matrix equation (see, for example, [13,14]) if D = O n and X = −Y.…”

All Solutions of the Yang–Baxter-Like Matrix Equation for Nilpotent Matrices of Index Two

“…In [15], when A is a diagonalisable matrix, the authors proposed numerical methods to calculate solutions of (1) by applying the mean ergodic theorem. When A is a low rank matrix, all solutions of (1) have been found in [16][17][18] for the noncommuting case. In [19], the authors have obtained explicit solutions when A is an idempotent matrix.…”

Some non-commuting solutions of the Yang-Baxter-like matrix equation

“…One possible reason is that solving a polynomial system of n 2 quadratic equations with n 2 unknowns is a challenging topic [7]. In the past several years, some special cases of (1) have been obtained for various classes of matrices A with different approaches in [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]. Because finding general solutions of the Yang-Baxter-like matrix Eq.…”

“…Up to now, there are only isolated results toward this goal for special classes of the given matrix A, e.g., [18][19][20][21][22][23][24][25][26]. All solutions have been constructed for rank-1 matrices A in [23], rank-2 matrices A in [24,25], non-diagonalizable elementary matrices A in [26], idempotent matrices A ( = )…”