Issue 
Wuhan Univ. J. Nat. Sci.
Volume 26, Number 6, December 2021



Page(s)  459  463  
DOI  https://doi.org/10.1051/wujns/2021266459  
Published online  17 December 2021 
Mathematics
CLC number: O124.6
Some Rank Formulas for the YangBaxter Matrix Equation AXA = XAX
School of Mathematics and Statistics, Tianshui Normal University, Tianshui
741001, Gansu, China
† To whom correspondence should be addressed. Email: liangml2005@163.com; liangmaolin@tsnu.edu.cn
Received:
20
July
2021
Let A be an arbitrary square matrix, then equation AXA =XAX with unknown X is called YangBaxter matrix equation. It is difficult to find all the solutions of this equation for any given A. In this paper, the relations between the matrices A and B are established via solving the associated rank optimization problem, where B =AXA = XAX, and some analytical formulas are derived, which may be useful for finding all the solutions and analyzing the structures of the solutions of the above YangBaxter matrix equation.
Key words: YangBaxter matrix equation / rank / generalized inverse
Biography: DAI Lifang, female, Master, research direction: numerical linear algebra with applications. Email: dailf2005@163.com
Foundation item: Supported by the National Natural Science Foundation of China (11961057) , the Science and Technology Project of Gansu Province ( 21JR1RE287 and 2021B221) , the Fuxi Scientific Research Innovation Team of Tianshui Normal University (FXD202003) , and the Science Foundation ( CXT201936 and CXJ202011) as well as the Education and Teaching Reform Project of Tianshui Normal University ( JY202004 and JY203008)
© Wuhan University 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 MoorePenrose 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.[39] and therein.
The YangBaxter equation has close relations to several mathematics subjects such as braid groups and knot theory, which has been widely studied, see, e.g., Refs.[1015]. It happens that there is a similar case, the classic parameterfree YangBaxter 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 YangBaxter 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 quasistochastic 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 YangBaxter matrix equation (1).
Actually, for some , suppose that matrix equation is consistent, then we have the following equalityconstrained 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 YangBaxter 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 wellknown (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 matrixvalued 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 YangBaxter matrix equation (1) and rank optimization problem (2), which reveals the relation between the matrices A and B included in YangBaxter matrix equation (1).
Theorem 2 Let A and B be given matrices satisfying (4). Then the YangBaxter matrix equation is consistent, if and only if
(a) .
(b) .
Proof According to (6), YangBaxter 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 YangBaxter 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 YangBaxter 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 YangBaxter 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 YangBaxter matrix equation. How to express the general solution of (1) is an interesting but challenging problem.
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