Commuting Solutions of a Quadratic Matrix Equation for Nilpotent Matrices

Abstract

We solve the quadratic matrix equation AXA = XAX with a given nilpotent matrix A, to find all commuting solutions. We first provide a key lemma, and consider the special case that A has only one Jordan block to motivate the idea for the general case. Our main result gives the structure of all the commuting solutions of the equation with an arbitrary nilpotent matrix.

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

“…The first equation of (14) implies . Suppose that the rank of matrix A is m and the multiplicity of eigenvalue 1 is k.…”

“…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.…”

“…Because finding general solutions of the Yang-Baxter-like matrix Eq. (1) is difficult, almost all the works so far have been toward constructing commuting solutions of the equation; see, e.g., [7][8][9][10][11][12][13][14][15][16][17] and references therein. Commuting solutions mean that the unknown matrix X satisfies the commutability condition = AX XA.…”

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

“…One possible reason is that Yang-Baxter-like matrix equation ( 1) is equivalent to solving a polynomial system of n 2 quadratic equations with n 2 unknowns, which solving this system is a very challenging topic. Almost all the works so far have been toward constructing commuting solutions (AX = XA) of the equation; see, e.g., [5][6][7][8][9][10][11][12][13][14][15][16] and the references therein. Finding all non-commuting solutions of Yang-Baxter-like matrix equation ( 1) is still a challenging task when A is arbitrary.…”

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

“…Most solutions obtained so far are commuting ones for particular choices of matrices A. See, for example, [8] for diagonalizable matrix and [9,10] for nilpotent matrix. In [11], infinitely many solutions of (1) were obtained for any semisimple eigenvalues of the given matrix.…”