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

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.


Introduction
We are interested in finding all solutions of the quadratic matrix equation: where the given A and the unknown X are n × n square complex matrices with n ≥ 2. e above equation (1) is called the Yang-Baxter-like matrix equation because it is similar to the classical parameter-free Yang-Baxter equation [1,2]. e Yang-Baxter equation was first introduced by Yang in 1967 and then by Baxter in 1972 in the study of statistical mechanics. e Yang-Baxter equation has been extensively researched by mathematicians and physicists in knot theory, braid group theory, and so on [3][4][5][6].
Obviously, the Yang-Baxter matrix equation has two trivial solutions X � 0 and X � A. However, we are interested in finding nontrivial solutions. Finding all solutions of equation (1) is a hard work for a general matrix A. Indeed, we can reformulate (1) into a system of polynomial equations, so it is equivalent to solving a system of n 2 quadratic polynomial equations in n 2 variables. To find all solutions is not an easy task even if for a 3 × 3 matrix [7]. 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. A family of commuting solutions of (1) were constructed for those eigenvalues of A that are non-semisimple in [12,13]. Some researchers have also proposed some numerical methods for finding commuting solutions. For instance, in [14], when A is a nonsingular matrix such that its inverse is a stochastic matrix, Ding and Rhee found nontrivial solutions of (1) via Brouwer's fixed point theorem. 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. However, for a general matrix A, it is difficult to characterize and determine all the solutions of (1), even if for nilpotent matrix. e purpose of this paper is to find all the solutions of equation (1) under the assumption that A is a nilpotent matrix of index 2. We first give a system of matrix equations of smaller sizes to find all solutions of the original matrix equation in Section 2. In the next two sections, we study all solutions for AXA � AXA when A is rank one and rank two, respectively. Finally, we present two examples of our solution result in Section 5 and conclude with Section 6. 0 0 (3) appears k times with k the rank of A, and 0 denotes the (n − 2k) × (n − 2k) zero matrix. So, there exists a nonsingular matrix W such that Lemma 1. Let two matrices A and X satisfy AXA � XAX. en, for any nonsingular matrix U, the matrices Conversely, if Y satisfies the above equations for a given B � U −1 AU, then X � UYU −1 satisfies AXA � XAX.
According to Lemma 1, we know that solving (1) can be reduced to solving the simplified matrix equation: So, solution X to (1) can be expressed as X � WYW −1 , where W satisfies A � WJW −1 . us, in our analysis below, we find all solutions of (6). Let Y be partitioned the same way as J into the (k + 1) × (k + 1) block matrix: where which is equivalent to the system Note that the above system does not contain M. is means that M is arbitrary. Because finding all solutions of (10) is very difficult, we focus on finding all solutions of (10) when k � 1 and k � 2.

All Solutions for the Nilpotent Matrix of Index 2 with Rank 2
When k � 2, this means where 0 is the (n − 4) × (n − 4) zero matrix. Let Let Y be partitioned the same way as J into the 2 × 2 block matrix: en, JYJ � YJY is equivalent to the system: All solutions of this system were obtained in [18]. We have the following results.

Examples
We give two examples to illustrate our results.

Example 1. Let
en, there exists such that A � WJW −1 . By eorem 1, we obtain all solutions of (1) as e second example is a matrix with rank 2.

Conclusions
In this paper, we obtain all solutions for the Yang-Baxterlike (1) when A is a nilpotent matrix of index 2. We first obtain a system of matrix equations of smaller sizes to find all the solutions of the original matrix equation. For a special case, that is, k � 1 and k � 2, we derive all solutions of the Yang-Baxter-like matrix equation in detail. However, for an arbitrary nilpotent matrix, finding all solutions of (1) is a hard work. We hope to solve this problem in the future.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.