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

Abstract

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

1. Introduction

We are interested in finding all solutions of the quadratic matrix equation:where the given and the unknown are square complex matrices with . The 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]. The Yang–Baxter equation was first introduced by Yang in 1967 and then by Baxter in 1972 in the study of statistical mechanics. The Yang–Baxter equation has been extensively researched by mathematicians and physicists in knot theory, braid group theory, and so on [3–6].

Obviously, the Yang–Baxter matrix equation has two trivial solutions and . However, we are interested in finding nontrivial solutions. Finding all solutions of equation (1) is a hard work for a general matrix . Indeed, we can reformulate (1) into a system of polynomial equations, so it is equivalent to solving a system of quadratic polynomial equations in variables. To find all solutions is not an easy task even if for a matrix [7]. Most solutions obtained so far are commuting ones for particular choices of matrices . 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 that are non-semisimple in [12, 13]. Some researchers have also proposed some numerical methods for finding commuting solutions. For instance, in [14], when 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 is a diagonalisable matrix, the authors proposed numerical methods to calculate solutions of (1) by applying the mean ergodic theorem. When is a low rank matrix, all solutions of (1) have been found in [16–18] for the noncommuting case. In [19], the authors have obtained explicit solutions when is an idempotent matrix. However, for a general matrix , it is difficult to characterize and determine all the solutions of (1), even if for nilpotent matrix.

The purpose of this paper is to find all the solutions of equation (1) under the assumption that 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 when is rank one and rank two, respectively. Finally, we present two examples of our solution result in Section 5 and conclude with Section 6.

2. All Solutions for the Nilpotent Matrix of Index 2

Let be a nilpotent matrix with index 2. Clearly, matrix has 0 as its only eigenvalue and the minimal polynomial of is . So, we can write the Jordan canonical form of asin which the Jordan blockappears times with the rank of , and 0 denotes the zero matrix. So, there exists a nonsingular matrix such that

Lemma 1. Let two matrices and satisfy . Then, for any nonsingular matrix , the matrices and satisfy

Conversely, if satisfies the above equations for a given , then satisfies .

According to Lemma 1, we know that solving (1) can be reduced to solving the simplified matrix equation:

So, solution to (1) can be expressed as , where satisfies . Thus, in our analysis below, we find all solutions of (6). Let be partitioned the same way as into the block matrix:where is , and for , and

Then, the equation becomeswhich is equivalent to the system

Note that the above system does not contain . This means that is arbitrary. Because finding all solutions of (10) is very difficult, we focus on finding all solutions of (10) when and .

3. All Solutions for the Nilpotent Matrix of Index 2 with Rank 1

When , this means that the rank of is 1. Then,where 0 is the zero matrix. Let be partitioned the same way as into the block matrix:where , , , and . According to (4), we obtain

Obviously, the last equation of (13) is also a Yang–Baxter matrix equation. Next, we solve the last equation in (13).

Lemma 2. The solutions of the equation are

Proof. LetThen, the equation becomesThat is,This leads to and , and is arbitrary.
By Lemma 2, all solutions of the last equation of (13) are , , and . Substituting such matrices into the first three equations of (13), we obtain the following result.

Theorem 1. Suppose that is a nilpotent matrix with rank 1. Then, all solutions of (1) are , where is partitioned as (12) in which is an arbitrary matrix such that

Proof. When , then . So, we just need to solve the first two equations in (13). The second equation (13) is . That is,Since , we have . Thus, . This gives (18).
Let . Then, and . So, the second and the third equations of (13) are satisfied. Next, we only have to solve the first equation of (13), i.e.,from which , is arbitrary or , is arbitrary. This gives (19).
If , then . So, the second equation of (13) is satisfied. From the first equation , we obtainwhich leads to . Since , we obtain . Thus, . This gives (20).

4. All Solutions for the Nilpotent Matrix of Index 2 with Rank 2

When , this meanswhere 0 is the zero matrix. Let . Then,

Let be partitioned the same way as into the block matrix:where is , , , and is . Then, is equivalent to the system:

All solutions of this system were obtained in [18]. We have the following results.

Lemma 3 (see Lemma 4.1 in [18]). The solutions of the equation are

Theorem 2 (see Theorem 4.2 in [18]). Suppose is a nilpotent matrix with rank 2, and is partitioned as (4). Then, all solutions of (1) are , where is partitioned as (10) in which is an arbitrary matrix such that with , in the left matrix, and , in the second one,

5. Examples

We give two examples to illustrate our results.

Example 1. LetThen, there existssuch that . By Theorem 1, we obtain all solutions of (1) asThe second example is a matrix with rank 2.

Example 2. LetAll solutions of (11) are as given by Lemma 3. Multiplying out with , we find all solutions of (1) arewith , in the left matrix, and in the second one,

6. Conclusions

In this paper, we obtain all solutions for the Yang–Baxter-like (1) when 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, and , 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

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 11861008), China Postdoctoral Science Foundation (no. 2018M641974), Natural Science Foundation of Jiangxi Province (no. 20192BAB201008), China Scholarship Council (no. 201909865004), Research fund of Gannan Normal University (Nos. YJG-2018-11 and 18zb04), and Key Disciplines Coordinate Innovation Projects of Gannan Normal University.