Open Access
 Issue Wuhan Univ. J. Nat. Sci. Volume 26, Number 6, December 2021 459 - 463 https://doi.org/10.1051/wujns/2021266459 17 December 2021

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## 0 Introduction

Throughout this paper, denotes the set of all m×n complex matrices. stand for the rank, conjugate transpose and the range (column space) of a matrix , respectively. The Moore-Penrose inverse, denoted by A +, is the unique solution X of the following Penrose equations

In addition, and are two oblique projectors included by .

The rank and generalized inverse of matrices are powerful tools to characterize relations among matrices, matrix equations, or matrix expressions. Two seminal references are Ref. [1] by Marsaglia and Styan, and Ref. [2] by Meyer. Some fundamental rank equalities and inequalities related to generalized inverses of block matrices were established in the two papers. Since then, the main results have widely been applied to dealing with various problems in the theory of generalized inverses of matrices and their applications, over complex field, even quaternion field, see, e.g., Refs.[3-9] and therein.

The Yang-Baxter equation has close relations to several mathematics subjects such as braid groups and knot theory, which has been widely studied, see, e.g., Refs.[10-15]. It happens that there is a similar case, the classic parameter-free Yang-Baxter equation has the same format as the following matrix equation(1)where A is any given matrix, and X is unknown to be determined. Hence, the matrix equation (1) can be referred to as Yang-Baxter matrix equation. Generally, it is difficult to obtain all the solutions of the equation because of its nonlinearity. Recently, Ding and his partners[15,16] have made several outstanding results. For instance, in Ref.[16], they have completed the existence proof based on the Brouwer fixed point via the direct iteration when the given A is a nonsingular quasi-stochastic matrix such that the inverse A -1 is a stochastic one. In addition, Cibotarica et al [17] studied equation (1) under the assumption that the given matrix A was idempotent. By means of the Jordan decomposition of A , they deduced general solution of the matrix equation mentioned above.

However, as is pointed out above, there may be many other solutions of this matrix equation which are not found. In this paper, letting A be an n×n matrix, we shall make analysis on the relation between the matrices A and B via considering an associated rank optimization, where , and some rank formulas will be established, which may provide ideas for finding all the solutions and analyzing the structures of the solutions of the Yang-Baxter matrix equation (1).

Actually, for some , suppose that matrix equation is consistent, then we have the following equality-constrained rank optimization problem(2)

Obviously, if the minimal rank arrives at zero, then it is equivalent to (1).

The outline of this paper is as follows. In Section 1, some basic formulas on the rank of matrices or matrix expressions will be given. In Section 2, the rank optimization (2) will be studied, after that some rank formulas with respect to the Yang-Baxter matrix equation (1) will be shown. As applications, the general solution of (1) will be expressed under some constrained conditions. Finally, we conclude this paper with some remarks.

## 1 Preliminary Knowledge

In this section, we introduce some necessary results with respect to the ranks of block matrix and matrix expressions.

Lemma 1   [1,2] Let , and . Then

(a) .

(b) .

(c)

where , .

(d) In particular, if and , then .

Lemma 2   [3] Given matrices , and (i=1,2), and letwith variables . Then

The following lemma is well-known (see, e.g., Ref. [18]).

Lemma 3   Let A , B and C be given matrices with appropriate sizes. Then matrix equation is consistent, if and only if one of the following conditions holds:

(a) .

(b) and .

(c) and .

In this case, the general solution of the matrix equation is , where U and V with appropriate sizes are arbitrary.

## 2 Main Results

The optimization problem (2) is indeed the extremal ranks problem upon the nonlinear matrix expression under equality constraint. Tian[5,19] developed an algebraic linearization method to solve the rank and inertia problems of some nonlinear (Hermitian) matrix expressions. The key technique of this method derives from the following block matrix operation where and are given. This equality implies that(3)

On the other hand, for the given matrices A and B as in (1) and (2), applying Lemma 3 to matrix equation , we know that this equation is consistent if and only if(4)In that case, the general solution of which is(5)with U , V .

On the basis of the above analysis, we obtain the following theorem.

Theorem 1   Let matrices A and B be given as in (1) and (2), then(6)

Proof   Substituting (5) into , then it follows from (3) and (4) that

This identity implies that the rank optimization problem (2) is equivalent to find the minimal rank of the linear matrix-valued function , namely,(7)Applying Lemma 2 to yields(8)where

Now, we simplify (8) by the elementary block matrix operations. Noting that the consistency conditions in (4), it is clear thati.e.,

These equalities as well as (4) will be frequently used in the sequel. Thus, by the elementary transformations of block matrices, Lemma 1 (a)-(c) follow that(9) (10) (11)Furthermore, sinceit then follows from (4) and Lemma 1(d) that (12)Putting (12) into (11) deduces that(13)

Similarly, we have(14) (15) (16) (17) (18)Hence, substituting equalities (9), (10), (13)-(18) into (8), from (7) we know that (6) holds.

From Theorem 1, we obtain another main result of this paper on the Yang-Baxter matrix equation (1) and rank optimization problem (2), which reveals the relation between the matrices A and B included in Yang-Baxter matrix equation (1).

Theorem 2   Let A and B be given matrices satisfying (4). Then the Yang-Baxter matrix equation is consistent, if and only if

(a) .

(b) .

Proof   According to (6), Yang-Baxter matrix equation (1) is consistent, if and only ifand

Furthermore, the first equality can be rewritten as(19)and notice that the left items of (19) are nonnegative, which implies that (a) and (b) hold true.

From this theorem, one can obtain the following conclusion.

Corollary 1   Suppose that the matrix A in Theorem 1 is idempotent, i.e., A 2= A , then(20)

Moreover, matrix equations is consistent, if and only if . In this case, the general solution of (1) can be expressed as(21)where idempotent W commuting with A is arbitrary, U 3 is an arbitrary matrix with order n, and U 1, U 2, satisfyIn particular, let , then the general solution commuting with A of the Yang-Baxter matrix equation (1) is(22)where any .

Proof   Combining (4) and the above assumptions, we get and , which implies from (6) that (20) is true. It is easy to verify that (21) and (22) are the required general solutions.

Remark 1   Let be the Jordan form of A with rank r , then we can verify that (21) is indeed the same as (6) in Ref. [17], while (22) is the special case of (21) when the matrices C and D are null matrices.

Corollary 2   Let B = A in Theorem 1, then

That is to say, the Yang-Baxter matrix equation (1) satisfies if and only if .

Proof   From (6) we know that the result holds true.

## 3 Conclusion

In this paper, we have studied the Yang-Baxter matrix equation (1). By using the rank of matrices, this equation is converted into solving an associated minimal rank optimization problem, i.e., (2). After that, some analytical formulas with respect to the known matrices were given. As applications, the general solutions of (1) were derived under some assumptions. These rank formulas may be useful for finding all the solutions and analyzing the structures of the solutions of the Yang-Baxter matrix equation. How to express the general solution of (1) is an interesting but challenging problem.

## References

1. Marsaglia G, Styan G. Equalities and inequalities for ranks of matrices [J]. Linear Multilinear Algebra , 1974, 2 : 269-292. [Google Scholar]
2. Meyer Jr C. Generalized inverses and ranks of block matrices [J]. SIAM J Appl Math , 1973, 25 :597-602. [Google Scholar]
3. Tian Y G. Upper and lower bounds for ranks of matrix expressions using generalized inverses [J]. Linear Algebra Appl , 2002, 355 (1-3): 187-214. [Google Scholar]
4. Tian Y G. On properties of BLUEs under general linear regression models [J]. J Stat Plan Inf , 2013, 143 (4): 771-782. [CrossRef] [Google Scholar]
5. Tian Y G. Extremal ranks of a quadratic matrix expression with applications [J]. Linear Multilinear Algebra , 2011, 59 (6): 627-644. [Google Scholar]
6. Xiong Z P, Qin Y Y. On the inverse of a special Schur complement [J]. Appl Math Comput , 2012, 218 (14): 7679-7684. [Google Scholar]
7. Wang Q W, Yu S W, Zhang Q. The real solutions to a system of quaternion matrix equations with applications [J]. Comm Algebra , 2009, 37 (1): 2060-2079. [CrossRef] [MathSciNet] [Google Scholar]
8. Wang Q W, Zhang X, He Z H. On the Hermitian structures of the solution to a pair of matrix equations [J]. Linear Multilinear Algebra , 2012, 61 (1): 73-90. [Google Scholar]
9. Rashedi S, Ebadi G, Biswas A. The maximal and minimal ranks of a quaternion matrix expression with applications [J]. J Egypt Math Soc , 2013, 21 (3): 175-183. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
10. Drinfeld V. On some unsolved problems in quantum group theory [J]. Lecture Notes in Mathematics , 1992, 1510 : 1-8. [Google Scholar]
11. Etingof P, Schedler T, Soloviev A. Set-theoretical solutions of the quantum Yang-Baxter equation [J]. Duke Math J , 1999, 100 : 169-178. [CrossRef] [MathSciNet] [Google Scholar]
12. Gateva-Ivanova T, Bergh M. Semi-groups of I-type [J]. J Algebra , 1998, 206 : 97-112. [CrossRef] [MathSciNet] [Google Scholar]
13. Felix F. Nonlinear Equations, Quantum Groups And Duality Theorems: A Primer on the Yang-Baxter Equation [D]. Buffalo: The State University of New York at Buffalo, 2001. [Google Scholar]
14. Yang C N, Ge M L. Braid Group, Knot Theory, and Statistical Mechanics [M]. Singapore: World Scientific, 1989. [Google Scholar]
15. Ding J, Rhee N. Spectral solutions of the Yang-Baxter matrix equation [J]. J Math Anal Appl , 2013, 402 : 567-573. [CrossRef] [MathSciNet] [Google Scholar]
16. Ding J, Rhee N. A nontrivial solution to a stochastic matrix equation [J]. East Asian J Appl Math , 2012, 2 : 277-284. [CrossRef] [MathSciNet] [Google Scholar]
17. Cibotarica A, Ding J, Kolibal J, et al . Solutions of the Yang-Baxter matrix equation for an idempotent [J]. Numer Algebra Control Opt , 2013, 3 (2): 347-352. [Google Scholar]
18. Ben-Israel A, Greville T. Generalized Inverses: Theory and Applications [M]. New York: R.E. Krieger Publishing Company, 1980. [Google Scholar]
19. Tian Y G. Solving optimization problems on ranks and inertias of some constrained nonlinear matrix functions via an algebraic linearization method [J]. Nonlinear Anal , 2012, 75 (2): 717-734. [Google Scholar]

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