Solution of Fuzzy Matrix Equation System

The main is to develop a method to solve an arbitrary fuzzy matrix equation system by using the embedding approach. Considering the existing solution to n × n fuzzy matrix equation system is done. To illustrate the proposed model a numerical example is given, and obtained results are discussed.


Introduction
The concept of fuzzy numbers and fuzzy arithmetic operations was first introduced by Zadeh 1 , Dubois, and Prade 2 .We refer the reader to 3 for more information on fuzzy numbers and fuzzy arithmetic.Fuzzy systems are used to study a variety of problems including fuzzy metric spaces 4 , fuzzy differential equations 5 , fuzzy linear systems 6-8 , and particle physics 9, 10 .
One of the major applications of fuzzy number arithmetic is treating fuzzy linear systems 11-20 , several problems in various areas such as economics, engineering, and physics boil down to the solution of a linear system of equations.Friedman et al.21 introduced a general model for solving a fuzzy n×n linear system whose coefficient matrix is crisp, and the right-hand side column is an arbitrary fuzzy number vector.They used the parametric form of fuzzy numbers and replaced the original fuzzy n × n linear system by a crisp 2n × 2n linear system and studied duality in fuzzy linear systems Ax Bx y where A and B are real n × n matrix, the unknown vector x is vector consisting of n fuzzy numbers, and the constant y is vector consisting of n fuzzy numbers, in 22 .In 6-8, 23, 24 the authors presented conjugate gradient, LU decomposition method for solving general fuzzy linear systems, or symmetric fuzzy linear systems.Also, Abbasbandy et al. 25 investigated the existence of a minimal solution of general dual fuzzy linear equation system of the form Ax f Bx c, where A and B are real m × n matrices, the unknown vector x is vector consisting of n fuzzy numbers, and the constants f and c are vectors consisting of m fuzzy numbers.

International Journal of Mathematics and Mathematical Sciences
In this paper, we give a new method for solving a n × n fuzzy matrix equation system whose coefficients matrix is crisp, and the right-hand side matrix is an arbitrary fuzzy number matrix by using the embedding method given in Cong-Xin and Min 26 and replace the original n × n fuzzy linear system by two n × n crisp linear systems.It is clear that, in large systems, solving n × n linear system is better than solving 2n × 2n linear system.Since perturbation analysis is very important in numerical methods.Recently, Ezzati 27 presented the perturbation analysis for n × n fuzzy linear systems.Now, according to the presented method in this paper, we can investigate perturbation analysis in two crisp matrix equation systems instead of 2n × 2n linear system as the authors of Ezzati 27 and Wang et al. 28 .

Preliminaries
Parametric form of an arbitrary fuzzy number is given in 29 as follows.A fuzzy number u in parametric form is a pair u, u of functions u r , u r , 0 ≤ r ≤ 1, which satisfy the following requirements: 1 u r is a bounded left continuous nondecreasing function over 0, 1 , 2 u r is a bounded left continuous nonincreasing function over 0, 1 , and The set of all these fuzzy numbers is denoted by E which is a complete metric space with Hausdorff distance.A crisp number α is simply represented by u r u r α, 0 ≤ r ≤ 1.
For arbitrary fuzzy numbers x x r , x r , y y r , y r , and real number k, we may define the addition and the scalar multiplication of fuzzy numbers by using the extension principle as 29 a x y if and only if x r y r and x r y r , b x y x r y r , x r y r , and Definition 2.1.The n × n linear system is as follows: where the given matrix of coefficients A a ij , 1 ≤ i, j ≤ n is a real n × n matrix, the given y i ∈ E, 1 ≤ i ≤ n, with the unknowns x j ∈ E, 1 ≤ j ≤ n is called a fuzzy linear system FLS .The operations in 2.1 is described in next section.
Here, a numerical method for finding solution 21 of a fuzzy n × n linear system is given.
International Journal of Mathematics and Mathematical Sciences 3 Definition 2.2 see 21 .A fuzzy number vector x 1 , x 2 , . . ., x n t given by is called a solution of the fuzzy linear system 2.1 if a ij x j y i .

2.3
If, for a particular i, a ij > 0, for all j, we simply get Finally, we conclude this section by a reviewing on the proposed method for solving fuzzy linear system 21 .
The authors 21 wrote the linear system of 2.1 as follows: SX Y, 2.5 where s ij are determined as follows: and any s ij which is not determined by 2.1 is zero and The structure of S implies that s ij ≥ 0, 1 ≤ i, j ≤ 2n and that is

S
where

2.11
Corollary 2.4 see 30 .The solution of 2.5 is obtained by

Fuzzy Matrix Equation System
A matrix system such as  x j n T is the solution of the following systems: A x j x j y j y j , j 1, 2, . . ., n, 3.5 where y j y j y j 1 y j 1 , y j 2 y j 2 , . . ., y j n y j n T , j 1, 2, . . ., n.
Proof.It is the same as the proof of Theorem 3 in 27 .
For solving 3.2 , we first solve the following system: . . .

3.6
Using matrix notation, we have Suppose that the solution of 3.7 is as 3.8 Let matrices B and C have defined as Section 2. Now using matrix notation for 3.7 , we get in parametric form B − C X r X r Y r Y r .We can write this system as follows:

3.9
International Journal of Mathematics and Mathematical Sciences By substituting X r D − X r and X r D − X r in the first and second equation of above system, respectively, we have

3.12
Therefore, we can solve fuzzy matrix equation system 3.2 by solving 3.7 -3.10 .
Theorem 3.2.Let in 3.3 j 1, also g and G are the number of multiplication operations that are required to calculate (the proposed method in Friedman et al. [21]) and Proof.According to Section 2, we have Therefore, for determining S −1 , we need to compute B C

Conclusions
In this paper, we propose a general model for solving fuzzy matrix equation system.The original system with matrix coefficient A is replaced by two n × n crisp matrix equation systems.
contains the positive entries of A, and C contains the absolute values of the negative entries of A, that is, A B − C.
−1 and B − C −1 .Now, assume that M is n × n matrix and denote by h M the number of multiplication operations that are required to calculate M −1 .It is clear that Remark 3.3.In 3.3 if j 1, then this paper is similar to 27 .Example 3.4.Consider the 2 × 2 fuzzy matrix equation system as follows: Obviously, x 11 , x 12 , x 21 and x 22 , are fuzzy numbers.