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

Abstract

Let A be a nilpotent matrix of index two, and consider the Yang–Baxter-like matrix equation AXA=XAX. We first obtain a system of matrix equations of smaller sizes to find all the solutions of the original matrix equation. When A is a nilpotent matrix with rank 1 and rank 2, we get all solutions of the Yang–Baxter-like matrix equation.

Commuting Outer Inverse-Based Solutions to the Yang–Baxter-like Matrix Equation

“…Many direct methods have recently been constructed to find several classes of solutions to (1) and most of them are based on the structure of A; see, e.g., [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24] and references therein. To illustrate it further, all solutions were investigated in [9] for the matrix A = I − uv T such that v T u = 0, where u and v are n-dimensional vectors.…”

“…All solutions that commute with A were identified in [23] provided A 3 = 0. In [24], all the solutions were observed for A such that A 2 = 0 and has a rank equal to one or two.…”

Explicit solutions of the Yang-Baxter-like matrix equation for a singular diagonalizable matrix with three distinct eigenvalues

“…Finding all non-commuting solutions of Yang-Baxter-like matrix equation ( 1) is still a challenging task when A is arbitrary. Up to now, there are only isolated results toward this goal for special classes of the given matrix A, e.g., [17][18][19][20][21][22][23][24][25][26][27][28]. 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 (A 2 = A) in [19], A 2 = I in [18,20], A 3 = A in [21], A 4 = A in [27], and diagonalizable matrices A with two different eigenvalues in [22].…”