Fuzzy Symmetric Solutions of Fuzzy Matrix Equations

Abstract

The fuzzy symmetric solution of fuzzy matrix equation 𝐴𝑋^{0𝑥002𝑑𝑐}=𝐵^{0𝑥002𝑑𝑐}, in which 𝐴 is a crisp 𝑚×𝑚 nonsingular matrix and ^0𝑥0𝐵^02𝑑𝑐 is an 𝑚×𝑛 fuzzy numbers matrix with nonzero spreads, is investigated. The fuzzy matrix equation is converted to a fuzzy system of linear equations according to the Kronecker product of matrices. From solving the fuzzy linear system, three types of fuzzy symmetric solutions of the fuzzy matrix equation are derived. Finally, two examples are given to illustrate the proposed method.

1. Introduction

Linear systems always have important applications in many branches of science and engineering. In many applications, at least some of the parameters of the system are represented by fuzzy rather than crisp numbers. So, it is immensely important to develop a numerical procedure that would appropriately treat general fuzzy linear systems and solve them. The concept of fuzzy numbers and arithmetic operations with these numbers was first introduced and investigated by Zadeh [1], Dubois et al. [2], and Nahmias [3]. A different approach to fuzzy numbers and the structure of fuzzy number spaces was given by Puri and Ralescu [4], Goetschell et al. [5], and Wu and Ming [6, 7].

Since Friedman et al. [8, 9] proposed a general model for solving an 𝑛×𝑛 fuzzy linear systems whose coefficients matrix is crisp and the right-hand side is an arbitrary fuzzy numbers vector by an embedding approach in 1998, many works have been done about how to deal with some fuzzy linear systems with more advanced forms such as dual fuzzy linear systems (DFLSs), general fuzzy linear systems (GFLSs), fully fuzzy linear systems (FFLSs), dual full fuzzy linear systems (DFFLSs), and general dual fuzzy linear systems (GDFLSs). These works were performed mainly by Allahviranloo et al. [10–13], Abbasbandy et al. [14–17], Wang et al. [18, 19] and Dehghan et al. [20, 21], among others. However, for a fuzzy matrix equation which always has a wide use in control theory and control engineering, few works have been done in the past decades. In 2010, Guo et al. [22–24] investigated a class of fuzzy matrix equations 𝐴𝑋=𝐵 in which 𝐴 is an 𝑚×𝑛 crisp matrix and the right-hand side matrix 𝐵 is an 𝑚×𝑙 fuzzy numbers matrix by means of the block Gaussian elimination method and the undetermined coefficients method, and they studied least squares solutions of the inconsistent fuzzy matrix equation 𝐴𝑥=𝐵 by using the generalized inverses. In 2011, Allahviranloo and Salahshour [25] obtained fuzzy symmetric approximate solutions of fuzzy linear systems by solving a crisp system of linear equations and a fuzzified interval system of linear equations. Meanwhile, they [26] investigated the maximal and minimal symmetric solutions of full fuzzy linear systems 𝐴𝑥=𝑏̃ by the same approach.

In this paper, we propose a general model for solving the fuzzy matrix equation 𝐴𝑋=𝐵 where 𝐴 is crisp 𝑚×𝑚 nonsingular matrix and 𝐵 is an 𝑚×𝑛 fuzzy numbers matrix with nonzero spreads. The model is proposed in this way, that is, we first convert the fuzzy matrix equation to a fuzzy system of linear equations based on the Kronecker product of matrices and then obtain three types of fuzzy symmetric solutions of the fuzzy matrix equation by solving the fuzzy linear systems. Finally, some examples are given to illustrate our method. The structure of this paper is organized as follows.

In Section 2, we recall the fuzzy number and present the concept of the fuzzy matrix equation and its fuzzy symmetric solutions. The method to solve the fuzzy matrix equation is proposed and the fuzzy symmetric solutions of the fuzzy matrix equation are obtained in detail in Section 3. Some examples are given to illustrate our method in Section 4 and the conclusion is drawn in Section 5.

2. Preliminaries

2.1. Fuzzy Numbers

There are several definitions for the concept of fuzzy numbers (see [1, 2, 4]).

Definition 1. A fuzzy number is a fuzzy set like 𝑢∶𝑅→𝐼=[0,1] which satisfies the following:(1)𝑢is upper semicontinuous,(2)𝑢 is fuzzy convex, that is, 𝑢(𝜆𝑥+(1−𝜆)𝑦)≥min{𝑢(𝑥),𝑢(𝑦)} for all 𝑥,𝑦∈𝑅,𝜆∈[0,1],(3)𝑢 is normal, that is, there exists 𝑥₀∈𝑅 such that 𝑢(𝑥₀)=1,(4)supp𝑢={𝑥∈𝑅∣𝑢(𝑥)>0} is the support of the 𝑢, and its closure cl(supp𝑢) is compact.
Let 𝐸¹ be the set of all fuzzy numbers on 𝑅.

Definition 2. A fuzzy number 𝑢 in parametric form is a pair (𝑢,𝑢) of functions 𝑢(𝑟), 𝑢(𝑟), 0≤𝑟≤1, which satisfies the requirements:(1)𝑢(𝑟) is a bounded monotonic increasing left continuous function,(2)𝑢(𝑟) is a bounded monotonic decreasing left continuous function,(3)𝑢(𝑟)≤𝑢(𝑟), 0≤𝑟≤1.
A crisp number 𝑥 is simply represented by (𝑢(𝑟),𝑢(𝑟))=(𝑥,𝑥), 0≤𝑟≤1. By appropriate definitions the fuzzy number space {(𝑢(𝑟),𝑢(𝑟))} becomes a convex cone 𝐸¹ which could be embedded isomorphically and isometrically into a Banach space.

Definition 3. Let 𝑥=(𝑥(𝑟),𝑥(𝑟)), 𝑦=(𝑦(𝑟),𝑦(𝑟))∈𝐸¹, 0≤𝑟≤1, and real number 𝑘∈𝑅. Then,(1)𝑥=𝑦 iff 𝑥(𝑟)=𝑦(𝑟)and𝑥(𝑟)=𝑦(𝑟),(2)𝑥+𝑦=(𝑥(𝑟)+𝑦(𝑟),𝑥(𝑟)+𝑦(𝑟)),(3)𝑥−𝑦=(𝑥(𝑟)−𝑦(𝑟),𝑥(𝑟)−𝑦(𝑟)),(4)𝑘𝑥=𝑘𝑥(𝑟),𝑘𝑥(𝑟)𝑘,𝑘≥0,𝑥(𝑟),𝑘𝑥(𝑟),𝑘<0. (1)

2.2. Kronecker Product of Matrices and Fuzzy Matrix

The following definitions and results about the Kronecker product of matrices are from [27].

Definition 4. Suppose 𝐴=(𝑎_𝑖𝑗)∈𝑅^𝑚×𝑛,𝐵=(𝑏_𝑖𝑗)∈𝑅^𝑝×𝑞, the matrix in block form: 𝑎𝐴⊗𝐵=¹¹𝐵𝑎₁₂𝐵⋯𝑎_1𝑛𝐵𝑎₂₁𝐵𝑎₁₂𝐵⋯𝑎_2𝑛𝐵𝑎⋮⋮⋮⋮_𝑚1𝐵𝑎_𝑚2𝐵⋯𝑎_𝑚𝑛𝐵∈𝑅^{𝑚𝑝×𝑛𝑞} (2) is said the Kronecker product of matrices 𝐴 and 𝐵, denoted simply by 𝐴⊗𝐵=(𝑎_𝑖𝑗𝐵).

Definition 5. Let 𝐴=(𝑎_𝑖𝑗)∈𝑅^𝑚×𝑛, 𝑎_𝑖=(𝑎_1𝑖,𝑎_2𝑖,…,𝑎_𝑚𝑖)^𝑇,𝑖=1,…,𝑛, the 𝑚𝑛 dimensions vector: (𝐴)=𝑎Vec¹𝑎₂⋮𝑎_𝑛 (3) is called the extension on column of the matrix 𝐴.

Lemma 6. Let 𝐴=(𝑎_𝑖𝑗)∈𝑅^𝑚×𝑛,𝐵=(𝑏_𝑖𝑗)∈𝑅^𝑛×𝑠, and 𝐶=(𝑐_𝑖𝑗)∈𝑅^𝑠×𝑡. Then, Vec(𝐴𝐵𝐶)=𝐶^𝑇⊗𝐴Vec(𝐵). (4)

Definition 7. A matrix 𝐴=(𝑎_𝑖𝑗) is called a fuzzy matrix, if each element 𝑎_𝑖𝑗 of 𝐴 is a fuzzy number, that is, 𝑎_𝑖𝑗=(𝑎_𝑖𝑗(𝑟),𝑎_𝑖𝑗(𝑟)),1≤𝑖≤𝑚,1≤𝑗≤𝑛,0≤𝑟≤1.

Definition 8. Let 𝐴=(𝑎_𝑖𝑗=(𝑎_𝑖𝑗(𝑟),𝑎_𝑖𝑗(𝑟))∈𝐸^𝑚×𝑛, 𝑎_𝑖=(𝑎_1𝑗,𝑎_2𝑗,…,𝑎_𝑚𝑗)^𝑇,𝑗=1,…,𝑛. Then, the 𝑚𝑛 dimensions fuzzy numbers vector: Vec𝐴=𝑎₁𝑎₂⋮𝑎_𝑛 (5) is called the extension on column of the fuzzy matrix 𝐴.

2.3. Fuzzy Matrix Equations

Definition 9. The matrix system: 𝑎₁₁𝑎₁₂⋯𝑎_1𝑚𝑎₂₁𝑎₁₂⋯𝑎_2𝑚𝑎⋮⋮⋮⋮_𝑚1𝑎_𝑚2⋯𝑎_𝑚𝑚𝑥₁₁𝑥₁₂⋯𝑥_1𝑛𝑥₂₁𝑥₁₂⋯𝑥_2𝑛⋮⋮⋮⋮𝑥_𝑚1𝑥_𝑚2⋯𝑥_𝑚𝑛=𝑏̃₁₁𝑏̃₁₂⋯𝑏̃_1𝑛𝑏̃₂₁𝑏̃₁₂⋯𝑏̃_2𝑛𝑏̃⋮⋮⋮⋮_𝑚1𝑏̃_𝑚2⋯𝑏̃_𝑚𝑛, (6) where 𝑎_𝑖𝑗, 1≤𝑖, 𝑗≤𝑚 are crisp numbers and 𝑏̃_𝑖𝑗, 1≤𝑖≤𝑚, 1≤𝑗≤𝑛 are fuzzy numbers, is called a fuzzy matrix equations (FMEs).
Using matrix notation, we have 𝐴𝑋=𝐵. (7)
A fuzzy numbers matrix: 𝑋=𝑥_𝑖𝑗=𝑥_𝑖𝑗(𝑟),𝑥_𝑖𝑗(𝑟),1≤𝑖≤𝑚,1≤𝑗≤𝑛,0≤𝑟≤1, (8) is called a solution of the fuzzy linear matrix equation (6) if 𝑋 satisfies 𝐴𝑋=𝐵. (9) Clearly, Definition 9 is just for the fuzzy matrix equation and its exact solution. In this paper we will discuss its approximate fuzzy symmetric solutions.

3. Method for Solving FMEs

In this section, we will investigate the fuzzy matrix equation (7), that is, convert it to a crisp system of linear equations and a fuzzified interval system of linear equations, define three types of fuzzy approximate symmetric solution and give its solution representation to the original fuzzy matrix equation.

At first, we convert the fuzzy matrix equation (7) to a fuzzy system of linear equations based on the Kronecker product of matrices.

Theorem 10. Let 𝐴=(𝑎_𝑖𝑗) belong to 𝑅^𝑚×𝑛, let 𝑋=(𝑥_𝑖𝑗)=(𝑥_𝑖𝑗(𝑟),𝑥_𝑖𝑗(𝑟)) belong to 𝐸^𝑛×𝑙, and let 𝐵=(𝑏_𝑖𝑗) belong to 𝑅^𝑙×𝑠. Then, Vec𝐴𝑋𝐵=𝐵^𝑇⊗𝐴Vec𝑋. (10)

Proof. Let 𝑋=(𝑥₁,𝑥₂,…,𝑥_𝑛), 𝑥_𝑗=(𝑥_𝑖𝑗(𝑟),𝑥_𝑖𝑗(𝑟))∈𝐸^𝑚, 𝑖=1,2,…,𝑚, 𝑗=1,2,…,𝑙. 𝐵=(𝑏₁,𝑏₂,…,𝑏_𝑙), 𝑏_𝑗∈𝑅^𝑛, 𝑗=1,2,…,𝑙. Then, Vec𝐴𝑋𝐵=Vec𝐴𝑋𝑏₁,𝐴𝑋𝑏₂,…,𝐴𝑋𝑏_𝑙=𝐴𝑋𝑏₁𝐴𝑋𝑏₂⋮𝐴𝑋𝑏_𝑙. (11)
Since 𝐴𝑋𝑏_𝑗=𝐴𝑥₁,𝐴𝑥₂,…,𝐴𝑥_𝑛𝑏_𝑗=𝐴𝑥₁,𝐴𝑥₂,…,𝐴𝑥_𝑙𝑏_1𝑗𝑏_2𝑗⋮𝑏_𝑙𝑗=𝑏_1𝑗𝐴𝑥₁+𝑏_2𝑗𝐴𝑥₂+⋯+𝑏_𝑙𝑗𝐴𝑥_𝑙=𝑏_1𝑗𝐴,𝑏_2𝑗𝐴,…,𝑏_𝑛𝑗𝐴Vec𝑋, (12) we have 𝐴=𝑏Vec𝑋𝐵¹¹𝐴𝑏₂₂𝐴⋯𝑏_𝑙1𝐴𝑏₁₂𝐴𝑏₂₂𝐴⋯𝑏_𝑙2𝐴𝑏⋮⋮⋮⋮_1𝑙𝐴𝑏_2𝑙𝐴⋯𝑏_𝑙𝑠𝐴𝑋=𝐵Vec_𝑇⊗𝐴Vec𝑋. (13)

Theorem 11. The matrix 𝑋∈𝐸^𝑚×𝑛 is the solution of the fuzzy matrix equation (7) if and only if 𝑥=Vec(𝑋)∈𝐸^𝑚𝑛 is the solution of the following linear fuzzy system: 𝐺𝑥=𝑦, (14) where 𝐺=𝐼_𝑛⊗𝐴 and 𝑦=Vec(𝐵).

Proof. Setting 𝐵=𝐼_𝑛 in (10), we have Vec𝐴𝑋=𝐼_𝑛⊗𝐴Vec𝑋. (15)
Applying the extension operation the Definition 8 to two sides of (7), we also have 𝐺𝑥=𝑦, (16) where 𝐺=𝐼_𝑛⊗𝐴 is an 𝑚𝑛×𝑚𝑛 matrix and 𝑦=Vec(𝐵) is an 𝑚𝑛 fuzzy numbers vector. Thus, the 𝑋 is the solution of (7) which is equivalent to that 𝑥=Vec(𝑋) which is the solution of (14).
For simplicity, we denote 𝑝=𝑚𝑛 in (7), thus 𝑔𝐺=¹¹𝑔₁₂⋯𝑔_1,𝑝𝑔₂₁𝑔₁₂⋯𝑔_2,𝑝𝑔⋮⋮⋮⋮_𝑝,1𝑔_𝑝,2⋯𝑔_𝑝,𝑝,𝑦=𝑦₁𝑦₂⋮𝑦_𝑝 (17) in (14).

The following definitions show what the fuzzy symmetric solutions of the fuzzy matrix equation are.

Definition 12 (see [28]). The united solution set (USS), the tolerable solution set (TSS), and the controllable solution set (CSS) for the system (14) are, respectively, as follows: 𝑋_∃∃=𝑥∈𝑅^𝑝,𝑋∶𝐺𝑥∩𝑦≠𝜙_∀∃=𝑥∈𝑅^𝑝,𝑋∶𝐺𝑥⊆𝑦_∃∀=𝑥∈𝑅^𝑝∶𝐺𝑥⊇𝑦. (18)

Definition 13. A fuzzy vector 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑝)^⊺ given by 𝑥_𝑖=[𝑥_𝑖(𝑟),𝑥_𝑖(𝑟)], 1≤𝑖≤𝑝, 0≤𝑟≤1 is called the minimal symmetric solution of the fuzzy matrix equation (7) which is placed in CSS if for any arbitrary symmetric solution 𝑧=(𝑧₁,𝑧₂,…,𝑧_𝑝)^⊺, which is placed in CSS, that is, 𝑥(1)=𝑧(1), we have (𝑧⊇𝑥),thatis,𝑧_𝑖⊇𝑥_𝑖,thatis,𝜎𝑧_𝑖≥𝜎𝑥_𝑖,𝑖=1,2,…,𝑝, (19) where 𝜎𝑧_𝑖 and 𝜎𝑥_𝑖 are symmetric spreads of 𝑧_𝑖 and 𝑥_𝑖, respectively.

Definition 14. A fuzzy vector 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑝)^⊺ given by 𝑥_𝑖=[𝑥_𝑖(𝑟),𝑥_𝑖(𝑟)],1≤𝑖≤𝑝,0≤𝑟≤1 is called the maximal symmetric solution of the fuzzy matrix equation (7) which is placed in TSS if for any arbitrary symmetric solution 𝑧=(𝑧₁,𝑧₂,…,𝑧_𝑝)^⊺, which is, placed in TSS, that is 𝑥(1)=𝑧(1), we have (𝑥⊇𝑧),thatis,𝑥_𝑖⊇𝑧_𝑖,thatis,𝜎𝑥_𝑖≥𝜎𝑧_𝑖,𝑖=1,2,…,𝑝, (20) where 𝜎𝑧_𝑖 and 𝜎𝑥_𝑖 are symmetric spreads of 𝑧_𝑖 and 𝑥_𝑖, respectively.
Secondly, in order to solve the fuzzy matrix equation (7), we need to consider the fuzzy system of linear equation (14). For the fuzzy linear system (14), we can extend it into to a crisp system of linear equations and a fuzzified interval system of linear equations to obtain its fuzzy symmetric solutions.

Theorem 15 (see [25]). The fuzzy linear system (14) can be extended into a 𝑝×𝑝 crisp function system of linear equations: 𝑔₁₁𝑔₁₂⋯𝑔_1,𝑝𝑔₂₁𝑔₁₂⋯𝑔_2,𝑝𝑔⋮⋮⋮⋮_𝑝,1𝑔_𝑝,2⋯𝑔_𝑝,𝑝𝑥₁𝑥₁⋮𝑥_𝑝=𝑦₁(1)𝑦₂(1)⋮𝑦_𝑝(1)𝑔₁₁,𝑥₁−𝛼₁(𝑟),𝑥₁+𝛼₁(𝑟)+⋯+𝑔_1𝑝𝑥_𝑝−𝛼₁(𝑟),𝑥_𝑝+𝛼₁(𝑟)=𝑦₁(𝑟),𝑦₁(𝑟),𝑔₂₁𝑥₁−𝛼₁(𝑟),𝑥₁+𝛼₁(𝑟)+⋯+𝑔_2𝑝𝑥_𝑝−𝛼₁(𝑟),𝑥_𝑝+𝛼₁(𝑟)=𝑦₂(𝑟),𝑦₂(𝑟),⋮𝑔_𝑝1𝑥₁−𝛼_𝑝(𝑟),𝑥₁+𝛼_𝑝(𝑟)+⋯+𝑎_𝑝𝑝𝑥_𝑝−𝛼_𝑝(𝑟),𝑥_𝑝+𝛼_𝑝(𝑟)=𝑦_𝑝(𝑟),𝑦_𝑝(𝑟), (21)(22) where 𝑦_𝑖(1)∈𝑅, 𝑖=1,2,…,𝑝 and 𝛼_𝑖(𝑟),𝑖=1,2,…,𝑝 are unknown spreads.
Now, one solves the crisp linear system (21) to obtain 𝑥_𝑖,𝑖=1,2,…,𝑝, that is, existed uniquely since det(𝐺)=det(𝐼_𝑛⊗𝐴)=det(𝐴)^𝑛≠0 and solve the interval equations (22) to obtain 𝛼_𝑖(𝑟),𝑖=1,2,…,𝑝.
So, without loss of generality and for simplicity to express the theory, it is assumed that the coefficients matrix 𝐺 is positive. Then, 𝑖th equation of interval system (22) is 𝑔_𝑖1𝑥₁−𝛼_𝑖(𝑟),𝑥₁+𝛼_𝑖(𝑟)+⋯+𝑔_𝑖𝑝𝑥_𝑝−𝛼_𝑖(𝑟),𝑥_𝑝+𝛼_𝑖(𝑟)=𝑦_𝑖(𝑟),𝑦_𝑖(𝑟), (23) it can be rewritten in parametric form: ^𝑝_𝑗=1𝑔_𝑖𝑗𝑥_𝑗−𝛼_𝑖(𝑟)=𝑦_𝑖(𝑟),𝑖=1,2,…,𝑝,_𝑝𝑗=1𝑔_𝑖𝑗𝑥_𝑗+𝛼_𝑖(𝑟)=𝑦_𝑖(𝑟),𝑖=1,2,…,𝑝. (24)(25)
So, after some computations and replacing 𝛼_𝑖(𝑟) with 𝛼_𝑖1(𝑟) in (24) and replacing 𝛼_𝑖(𝑟) with 𝛼_𝑖2(𝑟) in (25), (24), and (25), they are transformed, respectively, to 𝛼_𝑖1(𝑟)=𝑓₁𝑥₁,…,𝑥_𝑝,𝑔_𝑖1,…,𝑔_𝑖𝑝,𝑦_𝑖(𝑟)𝛼,𝑖=1,2,…,𝑝,_𝑖2(𝑟)=𝑓₂𝑥₁,…,𝑥_𝑝,𝑔_𝑖1,…,𝑔_𝑖𝑝,𝑦_𝑖(𝑟),𝑖=1,2,…,𝑝. (26) However, 𝛼_𝑖1(𝑟) is function of 𝑥₁,…,𝑥_𝑝,𝑔_𝑖1,…,𝑔_𝑖𝑝,𝑦_𝑖(𝑟), 𝛼_𝑖2(𝑟) is function of 𝑥₁,…,𝑥_𝑝,𝑔_𝑖1,…,𝑔_𝑖𝑝,𝑦_𝑖(𝑟) such that 𝛼_𝑖1(𝑟) and 𝛼_𝑖2(𝑟) are obtained spreads of 𝑖th equation in system (22). Perhaps, 𝛼_𝑖1(𝑟) and 𝛼_𝑖2(𝑟) do not satisfy the rest of interval equations (22). Therefore, one should determine the reasonable spreads according to decision makers. To this end, three type of spreads are proposed as follows: 𝛼_𝐿(𝑟)=min𝛼_𝑖1(𝑟),𝛼_𝑖2(𝑟𝛼),𝑖=1,2,…,𝑝,0≤𝑟≤1,_𝑈(𝑟)=max𝛼_𝑖1(𝑟),𝛼_𝑖2(𝑟𝛼),𝑖=1,2,…,𝑝,0≤𝑟≤1,_𝜆(𝑟)=𝜆𝛼_𝑈(𝑟)+(1−𝜆)𝛼_𝐿(𝑟)[].,𝑖=1,2,…,𝑝,0≤𝑟≤1,𝜆∈0,1 (27) Hence, by such computations, the fuzzy vector solution of system (7) under proposed spreads (27) will be as follows. For 𝑖=1,2,…,𝑝,0≤𝑟,𝜆≤1: 𝑋_𝐿=𝑥₁(𝑟),…,𝑥_𝑝(𝑟)^𝑡,𝑥_𝑖(𝑟)=𝑥_𝑖−𝛼_𝐿(𝑟),𝑥_𝑖+𝛼_𝐿(𝑟),𝑋_𝑈=𝑥₁(𝑟),…,𝑥_𝑝(𝑟)^𝑡,𝑥_𝑖(𝑟)=𝑥_𝑖−𝛼_𝑈(𝑟),𝑥_𝑖+𝛼_𝑈(𝑟),𝑋_𝜆=𝑥₁(𝑟),…,𝑥_𝑝(𝑟)^𝑡,𝑥_𝑖(𝑟)=𝑥_𝑖−𝛼_𝜆(𝑟),𝑥_𝑖+𝛼_𝜆(𝑟). (28) Now, it is shown that this method always gives us a fuzzy vector solution provided that the right-hand side of system (7) be a triangular fuzzy vector with nonzero left and right spreads.

Theorem 16. Let the right-hand side of the system (14), be 𝑦(𝑟)=(𝑦₁(𝑟),…,𝑦_𝑝(𝑟))^𝑡, where 𝑦_𝑖(𝑟)=[𝑦_𝑖(1)−𝜎_𝑖(1−𝑟),𝑦_𝑖(1)+𝛽_𝑖(1−𝑟)],𝑖=1,2,…,𝑝 and let 𝛼_𝐿(𝑟),𝛼_𝑈(𝑟) and 𝛼_𝜆(𝑟) be defined by (27), then 𝛼_𝐿(𝑟),𝛼_𝑈(𝑟), and 𝛼_𝜆(𝑟) are positive for all 0≤𝑟,𝜆≤1, such that 𝛼_𝐿(𝑟)𝜎=min^𝑖(1−𝑟)∑𝑔_𝑖𝑗,𝛽_𝑖(1−𝑟)∑𝑔_𝑖𝑗,𝛼_𝑈(𝑟)𝜎=max^𝑖(1−𝑟)∑𝑔_𝑖𝑗,𝛽_𝑖(1−𝑟)∑𝑔_𝑖𝑗,𝛼_𝜆(𝑟)𝜎=𝜆max^𝑖(1−𝑟)∑𝑔_𝑖𝑗,𝛽_𝑖(1−𝑟)∑𝑔_𝑖𝑗+()𝜎1−𝜆min^𝑖(1−𝑟)∑𝑔_𝑖𝑗,𝛽_𝑖(1−𝑟)∑𝑔_𝑖𝑗. (29)(30)(31)

Proof. Let us consider the 𝑖th row of interval equations (22), then by applying (24)-(25), we have _𝑗∈𝐻𝑔_𝑖𝑗𝑥_𝑗−𝛼_𝑖1(𝑟)+_𝑗∈𝑄𝑔_𝑖𝑗𝑥_𝑗+𝛼_𝑖1(𝑟=_𝑝𝑗=1)𝑔_𝑖𝑗𝑥_𝑗−^𝑝_𝑗=1𝑔_𝑖𝑗𝛼_𝑖1(𝑟)=𝑦_𝑖(1)−𝜎_𝑖(1−𝑟_𝑗∈𝐻),𝑔_𝑖𝑗𝑥_𝑗+𝛼_𝑖2(𝑟)+_𝑗∈𝑄𝑔_𝑖𝑗𝑥_𝑗−𝛼_𝑖2(𝑟)=^𝑝_𝑗=1𝑔_𝑖𝑗𝑥_𝑗−^𝑝_𝑗=1𝑔_𝑖𝑗𝛼_𝑖2(𝑟)=𝑦_𝑖(1)+𝛽_𝑖(1−𝑟), (32) where 𝐻 and 𝑄 include positive and negative components of coefficient matrix 𝐺, respectively. Also, it is obvious that 𝜎_𝑖,𝛽_𝑖, and the denominator are positive numbers. Therefore, 𝛼_𝐿(𝑟)>0.
Since that 𝑦_𝑖(1)=𝑦_𝑖(1)=∑^𝑝_𝑗=1𝑔_𝑖𝑗𝑥_𝑗, it is sufficient to show (29), that is, 𝛼_𝑖1(𝑟)=𝜎_𝑖(1−𝑟)∑𝑔_𝑖𝑗,𝛼_𝑖2(𝑟)=𝛽_𝑖(1−𝑟)∑𝑔_𝑖𝑗,𝑖=1,2,…,𝑝,0≤𝑟≤1. (33)
For results (30)-(31), the proofs are similar.

Theorem 17. Consider spreads (29)–(31) and corresponding solutions 𝑋_𝐿,𝑋_𝑈, then we one gets(1)𝑋_𝐿∈𝑇𝑆𝑆,(2)𝑋_𝑈∈𝐶𝑆𝑆.
In addition, one can find the maximal and minimal solutions of fuzzy linear system (7) which are placed in 𝑇𝑆𝑆 and 𝐶𝑆𝑆 when the cores of compared solutions in each cases are equal.

Theorem 18. 𝑋_𝐿 is maximal symmetric solution in 𝑇𝑆𝑆, 𝑋_𝑈 is minimal symmetric solution in 𝐶𝑆𝑆.

Proof. Using definitions of TSS and CSS, the proofs are obvious.

Moreover, we could express our proposed method by algorithm as follows.

Algorithm 19.  (1) We convert the fuzzy linear matrix equation (7) to a fuzzy system of linear equations (14) based on the Kronecker product of matrices.(2) We solve system (21) and obtain its crisp solution, that is, 𝑥=(𝑥₁,…,𝑥_𝑞), 𝑥_𝑖∈𝑅.(3) By applying crisp solution (solution of 1-cut), system (14) is transformed to the system of interval equations (22).(4) The spread of all elements of fuzzy vector solution will be obtained by solving system (22), whereas, spreads are named as 𝛼_𝑖1(𝑟), 𝛼_𝑖2(𝑟), respectively, 𝑖=1,2,…,𝑞, 0≤𝑟≤1.(5) The symmetric spreads can be assessed using (27).(6) The fuzzy vector solutions are derived by (28).

4. Numerical Examples

Example 20. Consider the following fuzzy matrix system: 1−113𝑥₁₁𝑥₁₂𝑥₂₁𝑥₂₂=()()𝑟,2−𝑟)(1+𝑟,3−𝑟4+𝑟,7−2𝑟)(4+2𝑟,6−2𝑟. (34) By calculations, we know that the exact solution of above fuzzy matrix system is 118+58𝑟,238−78𝑟168+48𝑟,248−48𝑟78+18𝑟,118−38𝑟488𝑟,8−48𝑟, (35) it admits a strong fuzzy solution.
By Theorems 10 and 11, the original fuzzy matrix equation is equivalent to the following fuzzy linear system 𝐺𝑥=𝑦, that is, 1−1001300001−10013𝑥₁₁𝑥₂₁𝑥₁₂𝑥₂₂=()()()()𝑟,2−𝑟4+𝑟,7−2𝑟1+𝑟,3−𝑟4+2𝑟,6−2𝑟. (36) Then, 1-cut of system is 1−100𝑥1300001−10013¹¹𝑥₂₁𝑥₁₂𝑥₂₂=1524. (37) Therefore, the crisp solution is 𝑥=(2,1,2.5,0.5)^⊺. Now, the system of interval equations (22) is as follows: 2−𝛼₁(𝑟),2+𝛼₁(𝑟)−1−𝛼₁(𝑟),1+𝛼₁(𝑟)=(𝑟,2−𝑟),2−𝛼₂(𝑟),2+𝛼₂(𝑟)+31−𝛼₂(𝑟),1+𝛼₂(𝑟)=(4+𝑟,7−2𝑟),2.5−𝛼₃(𝑟),2.5+𝛼₃(𝑟)−0.5−𝛼₃(𝑟),0.5+𝛼₃(𝑟=(),)1+𝑟,3−𝑟2.5−𝛼₄(𝑟),2.5+𝛼₄(𝑟)+30.5−𝛼₄(𝑟),0.5+𝛼₄(𝑟)=(2+2𝑟,6−2𝑟). (38) Hence, the following results are obtained for all 𝑟∈[0,1] as 𝛼₁₁(𝑟)=𝛼₁₃(𝑟)=𝛼₁₄(𝑟)=1−𝑟2,𝛼₁₂(𝑟)=1−𝑟4,𝛼₂₁(𝑟)=𝛼₂₂(𝑟)=𝛼₂₃(𝑟)=𝛼₂₄(𝑟)=1−𝑟2, (39) and applying (27) we get for all 0≤𝑟,𝜆≤1, 𝛼_𝐿(𝑟)=1−𝑟4,𝛼_𝑈(𝑟)=1−𝑟2,𝛼_𝜆(𝑟)=(1−𝑟)(1+𝜆)4. (40) Thus, the fuzzy symmetric solutions of the (14) are obtained as follows:𝑥_𝐿(𝑟)=2−1−𝑟4,2+1−𝑟41−1−𝑟4,1+1−𝑟42.5−1−𝑟4,2.5+1−𝑟40.5−1−𝑟4,0.5+1−𝑟4,𝑥_𝑈(𝑟)=2−1−𝑟2,2+1−𝑟21−1−𝑟2,1+1−𝑟22.5−1−𝑟2,2.5+1−𝑟20.5−1−𝑟2,0.5+1−𝑟2,𝑥_𝜆(𝑟)=()2−1−𝑟)(1+𝜆4(),2+1−𝑟)(1+𝜆4()1−1−𝑟)(1+𝜆4(),1+1−𝑟)(1+𝜆4()2.5−1−𝑟)(1+𝜆4(),2.5+1−𝑟)(1+𝜆4()0.5−1−𝑟)(1+𝜆4(),0.5+1−𝑟)(1+𝜆4. (41)
According to Theorem 15, we know that the fuzzy approximate symmetric solutions of the original fuzzy matrix equation 𝐴𝑋=𝐵 are𝑋_𝐿(𝑟)=𝑥₁₁𝑥₁₂𝑥₂₁𝑥₂₂=2−1−𝑟4,2+1−𝑟42.5−1−𝑟4,2.5+1−𝑟41−1−𝑟4,1+1−𝑟40.5−1−𝑟4,0.5+1−𝑟4,𝑋_𝑈(𝑟)=𝑥₁₁𝑥₁₂𝑥₂₁𝑥₂₂=2−1−𝑟2,2+1−𝑟22.5−1−𝑟2,2.5+1−𝑟21−1−𝑟2,1+1−𝑟20.5−1−𝑟2,0.5+1−𝑟2,𝑋_𝜆(𝑟)=𝑥₁₁𝑥₁₂𝑥₂₁𝑥₂₂=()2−1−𝑟)(1+𝜆2(),2+1−𝑟)(1+𝜆2()2.5−1−𝑟)(1+𝜆2(),2.5+1−𝑟)(1+𝜆2()1−1−𝑟)(1+𝜆2(),1+1−𝑟)(1+𝜆2()0.5−1−𝑟)(1+𝜆2(),0.5+1−𝑟)(1+𝜆2, (42)respectively.

Example 21. Consider the fuzzy matrix system: 10−11𝑥₁₁𝑥₁₂𝑥₁₃𝑥₂₁𝑥₂₂𝑥₂₃=()().1+𝑟,3−𝑟)(5+𝑟,7−𝑟)(2+𝑟,4−𝑟6+𝑟,8−𝑟)(3+𝑟,5−𝑟)(𝑟,2−𝑟 (43) The exact solution of above fuzzy matrix system is ()()1+𝑟,3−𝑟)(5+𝑟,7−𝑟)(2+𝑟,4−𝑟9,9)(6,6)(4,4, (44) and it is a weak fuzzy solution.
By Theorems 10 and 11, the original fuzzy matrix equation is equivalent to the following fuzzy linear system 𝐺𝑥=𝑦, that is, 100000−11000000100000−11000000100000−11𝑥₁₁𝑥₂₁𝑥₁₂𝑥₂₂𝑥₁₃𝑥₂₃=()()()()()()1+𝑟,3−𝑟6+𝑟,8−𝑟5+𝑟,7−𝑟3+𝑟,5−𝑟2+𝑟,4−𝑟𝑟,2−𝑟. (45) Then, 1-cut of system is 100000𝑥−11000000100000−11000000100000−11¹¹𝑥₂₁𝑥₁₂𝑥₂₂𝑥₁₃𝑥₂₃=276431. (46) Therefore, the crisp solution is 𝑥=(2,9,6,10,3,4)^⊺. Now, the system of interval equations (22) is as follows: 2−𝛼₁(𝑟),2+𝛼₁(𝑟)=(),−1+𝑟,3−𝑟2−𝛼₂(𝑟),2+𝛼₂(𝑟)+9−𝛼₂(𝑟),9+𝛼₂(𝑟)=(6+𝑟,8−𝑟),6−𝛼₃(𝑟),6+𝛼₃(𝑟=(),−)5+𝑟,7−𝑟6−𝛼₄(𝑟),6+𝛼₄(𝑟)+10−𝛼₄(𝑟),10+𝛼₄(𝑟)=(3+𝑟,5−𝑟),3−𝛼₅(𝑟),3+𝛼₅(𝑟)=(),−2+𝑟,4−𝑟3−𝛼₆(𝑟),3+𝛼₆(𝑟)+4−𝛼₆(𝑟),4+𝛼₆(𝑟)=(𝑟,2−𝑟). (47) Hence, the following results are obtained for all 𝑟∈[0,1] as 𝛼₁₁(𝑟)=𝛼₁₃(𝑟)=𝛼₁₅(𝑟)𝛼=1−𝑟,₁₂(𝑟)=𝛼₁₄(𝑟)=𝛼₁₆(𝑟)=1−𝑟2,𝛼₂₁(𝑟)=𝛼₂₃(𝑟)=𝛼₂₅(𝑟)𝛼=1−𝑟,₂₂(𝑟)=𝛼₂₄(𝑟)=𝛼₂₆(𝑟)=1−𝑟2, (48) and applying (27), we get for all 0≤𝑟,𝜆≤1, 𝛼_𝐿(𝑟)=1−𝑟2,𝛼_𝑈(𝑟)𝛼=1−𝑟,_𝜆(𝑟)=(1−𝑟)(1+𝜆)2. (49) Thus, the fuzzy symmetric solutions of the (14) are obtained as follows: 𝑥_𝐿(𝑟)=2−1−𝑟2,2+1−𝑟29−1−𝑟2,9+1−𝑟26−1−𝑟2,6+1−𝑟210−1−𝑟2,10+1−𝑟23−1−𝑟2,3+1−𝑟24−1−𝑟2,4+1−𝑟2,𝑥_𝑈(𝑟)=(())(()((()((()((()((()(,2−1−𝑟,2+1−𝑟9−1−𝑟,9+1−𝑟))6−1−𝑟,6+1−𝑟))10−1−𝑟,10+1−𝑟))3−1−𝑟,3+1−𝑟))4−1−𝑟,4+1−𝑟))𝑥_𝜆(𝑟)=()2−1−𝑟)(1+𝜆2(),2+1−𝑟)(1+𝜆2()9−1−𝑟)(1+𝜆2(),9+1−𝑟)(1+𝜆2()6−1−𝑟)(1+𝜆2(),6+1−𝑟)(1+𝜆2()10−1−𝑟)(1+𝜆2(),10+1−𝑟)(1+𝜆2()3−1−𝑟)(1+𝜆2(),3+1−𝑟)(1+𝜆2()4−1−𝑟)(1+𝜆2(),4+1−𝑟)(1+𝜆2, (50) respectively.
According to Theorem 11, we know that the fuzzy approximate symmetric solutions of the original fuzzy matrix equation 𝐴𝑋=𝐵 are 𝑋_𝐿(𝑟)=𝑥₁₁𝑥₁₂𝑥₁₃𝑥₂₁𝑥₂₂𝑥₂₃=2−1−𝑟2,2+1−𝑟26−1−𝑟2,6+1−𝑟23−1−𝑟2,3+1−𝑟29−1−𝑟2,9+1−𝑟210−1−𝑟2,10+1−𝑟24−1−𝑟2,4+1−𝑟2,𝑋_𝑈(𝑟)=𝑥₁₁𝑥₁₂𝑥₁₃𝑥₂₁𝑥₂₂𝑥₂₃=()(),𝑋1+𝑟,3−𝑟)(5+𝑟,7−𝑟)(2+𝑟,4−𝑟8+𝑟,10−𝑟)(9+𝑟,11−𝑟)(3+𝑟,5−𝑟_𝜆(𝑟)=𝑥₁₁𝑥₁₂𝑥₁₃𝑥₂₁𝑥₂₂𝑥₂₃=𝒜2−2𝒜,2+2𝒜6−2𝒜,6+2𝒜3−2𝒜,3+2𝒜9−2𝒜,9+2𝒜10−2𝒜,10+2𝒜4−2𝒜,4+2, (51)respectively, where 𝒜 denotes (1−𝑟)(1+𝜆).

5. Conclusion

In this work, we presented a model for solving fuzzy matrix equations 𝐴𝑋=𝐵 in which 𝐴 is crisp 𝑚×𝑚 nonsingular matrix and 𝐵 is an 𝑚×𝑛 arbitrary fuzzy numbers matrix with nonzero spreads. The model was proposed in this way, that is, we converted the fuzzy linear matrix equation to a fuzzy system of linear equations based on the Kronecker product of matrices, and then we extended the fuzzy linear system into a crisp system of linear equations and a fuzzified interval system of linear equations. The fuzzy symmetric solutions of the fuzzy linear matrix equation were derived from solving the crisp systems of linear equations. Numerical examples showed that our method is feasible to solve this type of fuzzy matrix equations.

Acknowledgments

The work is supported by the Natural Scientific Funds of PR China (71061013) and the Youth Scientific Research Promotion Project of Northwest Normal University (NWNU-LKQN-1120).