Research Article Solving Complex Fuzzy Linear Matrix Equations

In this paper, a kind of complex fuzzy linear matrix equation A 􏽥 XB � 􏽥 C , in which 􏽥 C is a complex fuzzy matrix and A and B are crisp matrices, is investigated by using a matrix method. The complex fuzzy matrix equation is extended into a crisp system of matrix equations by means of arithmetic operations of fuzzy numbers. Two brand new and simpliﬁed procedures for solving the original fuzzy equation are proposed and the correspondingly suﬃcient condition for strong fuzzy solution are analysed. Some examples are calculated in detail to


Introduction
e uncertainty of the parameters is involved in the process of actual mathematical modeling, which is often represented and computed by the fuzzy numbers. e theory and computation of linear systems related with fuzzy numbers always play an important role in the fuzzy mathematics. In the past decades, there has a great enormous investigation in the study of fuzzy mathematics and its applications. e definition of fuzzy numbers and their arithmetic operations were first introduced by Zadeh [1], Dubois and Prade [2], and Nahmias [3]. A different approach to fuzzy numbers and the fuzzy number spaces was studied by Puri and Ralescu [4], Goetschell and Voxman [5], and Wu and Ma [6,7].
It is well known that some matrix systems such as Lyapunov, Sylvester, and Stein matrix equations always have wide use in science and technology field. So, the investigation on fuzzy matrix systems has been paid attention by some scholars in past decades. In 2009, Allahviranloo et al. [21] investigated the fuzzy matrix equation, AXB � C. In 2018, AmirfakhrianIn et al. [22] presented a new algorithm for calculating the fuzzy linear matrix equation with the form AXB � C by another way. In 2011, Gong and Guo [23] discussed inconsistent fuzzy linear systems and studied its least squares fuzzy solution. In 2014, Gong et al. [24] studied the general dual fuzzy matrix systems AX + B � CX + D based on the LR fuzzy numbers. In 2017, Guo et al. [25,26] studied the fuzzy matrix system of the form XA � B by a matrix method and made a further investigation to dual fuzzy matrix equation AX + B � CX + D. In 2018, Guo et al. introduced the complex fuzzy matrix equation ZC � W and proposed a general model to deal with it. At the same year, they considered the approximate fuzzy inverse and its simple application in fully fuzzy linear systems [27]. In 2019, Guo and Shang [28] put up a new method for solving linear fuzzy matrix equations, AXB � C.
For complex fuzzy linear systems, few researchers have developed methods to investigate them in the past decades. e fuzzy complex numbers were introduced firstly by Buckley [29] in 1989. In 2000, Qiu et al. [30] restudied the sequence and series of fuzzy complex numbers and their convergence by considering the n × n fuzzy complex linear systems. In 2009, Rahgooy et al. [31] applied the fuzzy complex linear system of linear equations to described circuit analysis problem. In 2014, Behera and Chakraverty discussed the fuzzy complex system of linear equations by the embedding method and modified arithmetic operations of the complex fuzzy numbers later [8,18].
In this paper, we propose a matrix method to deal with complex fuzzy linear matrix equation, AXB � C. At first, we introduce the complex fuzzy matrix and its operation with the crisp number. en, we convert the complex fuzzy matrix equation to a model which is a crisp linear system of function matrix equations. e fuzzy solution of the original fuzzy equation is gained by solving the model, that is, a crisp system of matrix equations. en, conditions of the strong fuzzy solution are also discussed. Finally, two illustrating examples are given. Our results enrich fuzzy linear system theory.

Preliminaries
Definition 1 (see [1]). A fuzzy number is a fuzzy set such as u: } is the support of the u, and its closure cl(suppu) is compact Let E 1 be the set of all fuzzy numbers on R.

Solving Complex Fuzzy Linear Matrix Systems
Definition 6 (see [32]). A matrix A � (a ij ) is called a complex fuzzy matrix if each element a ij of A is a complex fuzzy number. Let A � (a ij ) � (m ij (r), m ij (r))+ i(n ij (r), n ij (r)), i, j � 1, 2, . . . , n, and the complex fuzzy Definition 7. For any two arbitrary complex fuzzy matrix X � P + iQ and Y � U + iV, where P, Q, U, and V are fuzzy numbers matrices; their arithmetic is as follows: e fuzzy matrix equation (3) can be extended to a crisp function linear matrix system as follows: where and in which the elements a + ij of matrix A + and a − ij of matrix A − are determined by the following way: if Since and So, equation (9) can be rewritten as In comparison with the coefficients of i, we obtain and Mathematical Problems in Engineering i.e., Denoting in the matrix form, the above matrix equations can be written as and us, we obtain equations (5)-(7) as follows: where and By the matrix operation, the above linear matrix equations are equivalent with It is completed the proof.
In a similar way, we could obtain another model for solving equation (3).

Theorem 2.
e fuzzy linear matrix equation (3) can be extended to a crisp function linear matrix system as follows: where and By the matrix operation, the above linear matrix equations are equivalent with Proof. e proof is similar to eorem 1.
Theorem 3 (see [28]). Let S belong to R m×n , T belong to R p×q , and C belong to R m×q . en, the minimal solution X * of the matrix equation SXT � C is expressed by In order to solve the fuzzy matrix equation (3), we need to consider the systems of linear equations (21) or (26). It seems that we have obtained the minimal solution of the function linear system (21) as Meanwhile, we obtain the minimal solution of the function linear system (26) as where G † is the Moore-Penrose generalized inverse of matrix G. However, the solution matrix from equations (28) or (29) Definition 8. Let X(r) � (m ij (r), m ij (r)) + i(n ij (r), n ij (r)), 1 ≤ i, j ≤ n be the minimal solution of model (21) or (26). e complex fuzzy number matrix W � (p ij (r), p ij (r)) + i(q ij (r), q ij (r)), 1 ≤ i, j ≤ n defined by and q ij (r) � min n ij (r), n ij (r), n ij (1), n ij (1) , q ij (r) � max n ij (r), n ij (r), n ij (1), n ij (1) , If (m ij (r), m ij (r)), 1 ≤ i and j ≤ n, and (n ij (r), n ij (r)), 1 ≤ i and j ≤ n, are all fuzzy numbers, then p ij (r) � m ij (r), p ij (r) � m ij (r), q ij (r) � n ij (r), and q ij (r) � n ij (r),

≤ i, j ≤ n, and W � [P(r), P(r)] + i[Q(r), Q(r)] is called a strong complex fuzzy minimal solution of fuzzy linear matrix systems (3). Otherwise, W � [P(r), P(r)]+ i[Q(r), Q(r)] is called a weak complex fuzzy minimal solution of fuzzy linear matrix systems (3).
To illustrate expression (28) or (29) to be a fuzzy solution matrix, we now discuss the generalized inverses of the nonnegative symmetric matrix, in a special structure [33].

Theorem 4. If
where where Proof. Let We obtain By adding and then subtracting the two parts of the above equations, we obtain and consequently, e proof is completed. e key points to make the solution matrix being a strong fuzzy solution is that are appropriate fuzzy numbers matrices, i.e., each element in which is an appropriate triangular fuzzy number. By the following analysis, a sufficient condition is that It is completed the proof. For the model equation (26), we have the following result.

Theorem 5. If
and where Mathematical Problems in Engineering We know the condition that T + ≥ 0 is equivalent to conditions E ≥ O and F ≥ O by eorem 4. Since It admits a bounded monotonic increasing continuous function matrix.
On the contrary, we have i.e., Now that A † , E − F ≥ 0, and We know that (53) us, the above complex fuzzy matrix equation has a strong complex fuzzy minimal solution as 6 Mathematical Problems in Engineering

X � M(r), M(r) + i N(r), N(r) .
(54) It is completed the proof. For the model equation (21), we have the following result by the similar analysis.

Theorem 6. If
and (56) e fuzzy matrix equation (3) has a strong fuzzy minimal solution as follows: where Proof. e proof is similar to eorem 5.

Numerical Examples
Example 1. Consider the following complex fuzzy linear matrix system: By eorem 2, the original fuzzy matrix equation is extended into the following system of linear matrix equation (17): where and U(r) � 1 + r 4 + 3r From (31), the solution of the computing model is

Mathematical Problems in Engineering
and By eorem 2, the original fuzzy matrix equation is extended into the following system of linear matrix equation (26): From (32), the solution of the computing model is It means which admits a weak fuzzy minimal solution by Definition 5.

Conclusion
In this paper, we put up a scheme for calculating complex fuzzy matrix equations AXB � C, where A and B are m × m and n × n crisp matrices, respectively, and C is an m × n complex fuzzy matrix. e fuzzy approximate solution of the original fuzzy matrix equation was derived from solving the model which is a function linear matrix system. In addition, the conditions of strong fuzzy approximate solution were analysed. Numerical examples showed the effectiveness of the proposed method. We should point out that a fact, i.e., it is rather difficult to obtain the strong complex fuzzy solution of the system in general. Our method can be applied to solve all kinds of semicomplex fuzzy matrix equations. We will study the fully complex fuzzy matrix equation and its applications on traffic control and decision making on this basis in next step.

Data Availability
No data were used related with practical problems.

Conflicts of Interest
e authors declare that they have no conflicts of interest, financial, or otherwise.