Approximate Solution of LR Fuzzy Sylvester Matrix Equations

Abstract

The fuzzy Sylvester matrix equation 𝐴𝑋+𝑋𝐵=𝐶 in which 𝐴,𝐵 are 𝑚×𝑚 and 𝑛×𝑛 crisp matrices, respectively, and 𝐶 is an 𝑚×𝑛 LR fuzzy numbers matrix is investigated. Based on the Kronecker product of matrices, we convert the fuzzy Sylvester matrix equation into an LR fuzzy linear system. Then we extend the fuzzy linear system into two systems of linear equations according to the arithmetic operations of LR fuzzy numbers. The fuzzy approximate solution of the original fuzzy matrix equation is obtained by solving the crisp linear systems. The existence condition of the LR fuzzy solution is also discussed. Some examples are given to illustrate the proposed method.

1. Introduction

System of simultaneous matrix equations plays a major role in various areas such as mathematics, physics, statistics, engineering, and social sciences. In many problems in various areas of science, which can be solved by solving a linear matrix equation, some of the system parameters are vague or imprecise, and fuzzy mathematics is better than crisp mathematics for mathematical modeling of these problems, and hence solving a linear matrix equation where some or all elements of the system are fuzzy is important. The concept of fuzzy numbers and arithmetic operations with these numbers were first introduced and investigated by Zadeh [1], Dubois and Prade [2], and Nahmias [3]. A different approach to fuzzy numbers and the structure of fuzzy number spaces was given by Puri and Ralescu [4], Goetschel and Voxman [5], and Wu and Ma [6, 7].

Since Friedman et al. [8] proposed a general model for solving an 𝑛×𝑛 fuzzy linear systems whose coefficients matrix are crisp and the right-hand side is an arbitrary fuzzy number vector by a embedding approach, some works [9–20] have been done about how to deal with some advanced fuzzy linear systems such as dual fuzzy linear systems (DFLS), general fuzzy linear systems (GFLS), full fuzzy linear systems (FFLS), dual full fuzzy linear systems (DFFLS), and general dual fuzzy linear systems (GDFLS). However, for a fuzzy linear matrix equation which always has a wide use in control theory and control engineering, few works have been done in the past decades. In 2009, Allahviranloo et al. [21] studied the fuzzy linear matrix equation, (FLME) of the form 𝐴𝑋𝐵=𝐶. By using the parametric form of the fuzzy number, they derived necessary and sufficient conditions for the existence of the set of fuzzy solutions and designed a numerical procedure for calculating the solutions of the fuzzy matrix equations. In 2011, Guo and Gong [22, 23] investigated a class of simple fuzzy matrix equations 𝐴𝑋=𝐵 by the undetermined coefficients method and studied least squares solutions of the inconsistent fuzzy matrix equation 𝐴𝑋=𝐵 by using generalized inverses of matrices. In 2011, Guo [24] studied the approximate solution of fuzzy Sylvester matrix equations with triangular fuzzy numbers. Lately, Guo and Shang [25] considered the fuzzy symmetric solutions of fuzzy matrix equation 𝐴𝑋=𝐵.

The LR fuzzy number and its arithmetic operations were first introduced by Dubois and Prade. We know that triangular fuzzy numbers are just specious cases of LR fuzzy numbers. In particular, Allahviranloo et al. [26] have showed us that the weak fuzzy solution of fuzzy linear systems 𝐴𝑥=𝑏̃ does not exist sometimes when 𝑥,𝑏̃ are denoted by triangular fuzzy numbers. Recently, he also considered the LR fuzzy linear systems [27] by the linear programming with equality constraints. In this paper we consider the fuzzy approximate solution of LR fuzzy Sylvester matrix equation 𝐴𝑋+𝑋𝐵=𝐶. In fact, the fuzzy Sylvester matrix equation 𝐴𝑋+𝑋𝐵=𝐶 has numerous applications in control theory, signal processing, filtering, model reduction, decoupling techniques for ordinary and partial differential equations, and block-diagonalization of matrices and so on. But there was little research work on it. The contributions of this paper is to generalize Dubois’ definition and arithmetic operation of LR fuzzy numbers and then use this result to solve fuzzy Sylvester matrix systems numerically. The structure of this paper is organized as follows.

In Section 2, we recall the LR fuzzy number, generalize the definition of LR fuzzy numbers, and present the concept of the LR fuzzy Sylvester matrix equation. The model to the fuzzy Sylvester matrix equation is proposed in detail and the fuzzy approximate solution of the original fuzzy matrix systems is derived from solving the crisp systems of linear equations in Section 3. Some examples are given in Section 4 and the conclusion is drawn in Section 5.

2. Preliminaries

2.1. The LR Fuzzy Numbers

Definition 1 (see [1]). A fuzzy number is a fuzzy set like 𝑢∶𝑅→𝐼=[0,1] which satisfies(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) Supp 𝑢={𝑥∈𝑅∣𝑢(𝑥)>0} is the support of the 𝑢, and its closure cl(supp 𝑢) is compact.
Let 𝐸¹ be the set of all fuzzy numbers on 𝑅.

Definition 2 (see [2]). 𝐴 fuzzy number 𝑀 is said to be an LR fuzzy number if𝜇_𝑀(𝑥)=𝐿𝑚−𝑥𝛼𝑅,𝑥≤𝑚,𝛼>0,𝑥−𝑚𝛽,𝑥≥𝑚,𝛽>0, (1) where 𝑚 is the mean value of 𝑀 and 𝛼 and 𝛽 are left and right spreads, respectively. The function 𝐿(⋅) is called the left shape function of an LR fuzzy number and it satisfies:(1)𝐿(𝑥)=𝐿(−𝑥);(2)𝐿(0)=1 and 𝐿(1)=0;(3)𝐿(𝑥) is nonincreasing on [0,∞).
The definition of a right shape function 𝑅(⋅) is usually similar to that of 𝐿(⋅).
An LR fuzzy number 𝑀 is symbolically shown as 𝑀=𝑚,𝛼,𝛽_LR. (2)
Clearly, 𝑀=(𝑚,𝛼,𝛽)_LR is positive (negative) if and only if 𝑚−𝛼>0(𝑚+𝛽<0).
Also, two LR fuzzy numbers 𝑀=(𝑚,𝛼,𝛽)_LR and 𝑁=(𝑛,𝛾,𝛿)_LR are said to be equal, if and only if 𝑚=𝑛, 𝛼=𝛾, and 𝛽=𝛿.
Noticing that 𝛼,𝛽>0 in Definition 2, which limits its applications, we extend the definition of LR fuzzy numbers as follows.

Definition 3 (generalized LR fuzzy numbers). (1) If 𝛼<0 and 𝛽>0, we define 𝑀=(𝑚,0,max{−𝛼,𝛽})_LR and 𝜇_𝑀(𝑥)=𝑅0,𝑥<𝑚,𝑥−𝑚max−𝛼,𝛽,𝑥≥𝑚. (3)
(2) If 𝛼>0 and 𝛽<0, we define 𝑀=(𝑚,max{𝛼,−𝛽},0)_LR and 𝜇_𝑀(𝑥)=𝐿𝑥−𝑚max𝛼,−𝛽,𝑥≥𝑚,0,𝑥<𝑚. (4)
(3) If 𝛼<0 and 𝛽<0, we define 𝑀=(𝑚,−𝛽,−𝛼)_LR and 𝜇_𝑀(𝑥)=𝐿𝑚−𝑥𝑅−𝛽,𝑥<𝑚,𝑥−𝑚−𝛼,𝑥≥𝑚. (5)

Based on the extension principle, the arithmetic operations for LR fuzzy numbers were defined. For arbitrary LR fuzzy numbers 𝑀=(𝑚,𝛼,𝛽)_LR and 𝑁=(𝑛,𝛾,𝛿)_LR, we have(1) addition 𝑀+𝑁=𝑚,𝛼,𝛽_LR+𝑛,𝛾,𝛿_LR=𝑚+𝑛,𝛼+𝛾,𝛽+𝛿_LR, (6)(1) scalar multiplication 𝜆×𝑀=𝜆×𝑚,𝛼,𝛽_LR=𝜆𝑚,𝜆𝛼,𝜆𝛽_LR,𝜆>0,𝜆𝑚,−𝜆𝛽,−𝜆𝛼_RL,𝜆<0. (7)

2.2. Some Results on Matrix Theory

Let 𝐴 be an 𝑚×𝑛 real matrix and (⋅)^⊤ denote the transpose of a matrix (⋅). We recall that a generalized inverse 𝐺 of 𝐴 is an 𝑛×𝑚 matrix which satisfies one or more of Penrose equations(1)(2)(3)𝐴𝐺𝐴=𝐴,𝐺𝐴𝐺=𝐺,)(𝐴𝐺^⊤=𝐴𝐺,(4)(𝐺𝐴)^⊤=𝐺𝐴. (8)

For a subset {𝑖,𝑗,𝑘} of set {1,2,3,4}, the set of 𝑛×𝑚 matrices satisfying the equations contained in {𝑖,𝑗,𝑘} is denoted by 𝐴{𝑖,𝑗,𝑘}. A matrix in 𝐴{𝑖,𝑗,𝑘} is called an {𝑖,𝑗,𝑘}-inverse of 𝐴 and is denoted by 𝐴{𝑖,𝑗,𝑘}. In particular, the matrix 𝐺 is called a {1}-inverse or a 𝑔-inverse of 𝐴 if it satisfies (1). As usual, the 𝑔-inverse of 𝐴 is denoted by 𝐴⁻. If 𝐺 satisfies (2) then it is called a {2}-inverse and If 𝐺 satisfies (1) and (2) then it is called a reflexive inverse or a {1,2}-inverse of 𝐴. The Moore-Penrose inverse of 𝐴 is the matrix 𝐺 which satisfies (1)–(4). Any matrix 𝐴 admits a unique Moore-Penrose inverse, denoted by 𝐴^†.

Lemma 4 (see [28]). For a system of linear equations 𝐴𝑥=𝑏. (9) When it is consistent, its solution can be expressed by 𝑥=𝐺𝑏 in which 𝐺∈𝐴{1}; when it has an infinite number of solutions, its minimal norm solution can be expressed by 𝑥=𝐺𝑏 in which 𝐺∈𝐴{1,4}. When it is inconsistent, its least squares solutions can be expressed by 𝑥=𝐺𝑏 in which 𝐺∈𝐴{1,3}. In particular, 𝑥=𝐴^†𝑏 is the minimum norm least square solution to the above linear system.

Definition 5 (see [28]). 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} (10) is said to be the Kronecker product of matrices 𝐴 and 𝐵, which denoted simply 𝐴⊗𝐵=(𝑎_𝑖𝑗𝐵).

2.3. The Fuzzy Sylvester Matrix Equation

Definition 6. A matrix 𝐴=(𝑎_𝑖𝑗) is called an LR fuzzy matrix, if each element 𝑎_𝑖𝑗 of 𝐴 is an LR fuzzy number.
For example, we represent 𝑚×𝑛 LR fuzzy matrix 𝐴=(𝑎_𝑖𝑗), that 𝑎_𝑖𝑗=(𝑎_𝑖𝑗,𝛼_𝑖𝑗,𝛽_𝑖𝑗)_LR with new notation 𝐴=(𝐴,𝑀,𝑁), where 𝐴=(𝑎_𝑖𝑗), 𝑀=(𝛼_𝑖𝑗) and 𝑁=(𝛽_𝑖𝑗) are three 𝑚×𝑛 crisp matrices. Particularly, an 𝑛 dimensions LR fuzzy numbers vector 𝑥 can be denoted by (𝑥,𝑥^𝑙,𝑥^𝑟), where 𝑥=(𝑥_𝑖), 𝑥^𝑙=(𝑥^𝑙_𝑖), and 𝑥^𝑟=(𝑥^𝑟_𝑖) are three 𝑛 dimensions crisp vectors.

Definition 7. The matrix system 𝑎₁₁𝑎₁₂⋯𝑎_1𝑚𝑎₂₁𝑎₁₂⋯𝑎_2𝑚𝑎⋯⋯⋯⋯_𝑚𝑛1𝑎_𝑚2⋯𝑎_𝑚𝑚𝑥₁₁𝑥₁₂⋯𝑥_1𝑛𝑥₂₁𝑥₁₂⋯𝑥_2𝑛𝑥⋯⋯⋯⋯_𝑚1𝑥_𝑚2⋯𝑥_𝑚𝑛+𝑥₁₁𝑥₁₂⋯𝑥_1𝑛𝑥₂₁𝑥₁₂⋯𝑥_2𝑛𝑥⋯⋯⋯⋯_𝑚1𝑥_𝑚2⋯𝑥_𝑚𝑛𝑏₁₁𝑏₁₂⋯𝑏_1𝑛𝑏₂₁𝑏₁₂⋯𝑏_2𝑛𝑏⋯⋯⋯⋯_𝑛1𝑏_𝑛2⋯𝑏_𝑛𝑛=𝑐̃₁₁𝑐̃₁₂⋯𝑐̃_1𝑛𝑐̃₂₁𝑐̃₁₂⋯𝑐̃_2𝑛𝑐̃⋯⋯⋯⋯_𝑚1𝑐̃_𝑚2⋯𝑐̃_𝑚𝑛, (11) where 𝐴 and 𝐵 are real matrices and 𝐶 is an LR fuzzy matrix, that is, 𝑎_𝑖𝑗,𝑏_𝑖𝑗∈𝑅, 𝑐̃_𝑖𝑗∈𝐸, is called an LR fuzzy Sylvester matrix equations (LRFSMEs).
Using matrix notation, we have 𝐴𝑋+𝑋𝐵=𝐶. (12) Up to the rest of this paper, we suppose that 𝐶 is a positive LR fuzzy numbers matrix and use the formulas given in Definition 3.
An LR fuzzy numbers matrix 𝑋=𝑥_𝑖𝑗,𝑥^𝑙_𝑖𝑗,𝑥^𝑟_𝑖𝑗LR,1≤𝑖≤𝑚,1≤𝑗≤𝑛 (13) is called an LR solution of the fuzzy Sylvester matrix equation (11) if 𝑋 satisfies (12).

3. Method for Solving LRFSMEs

In this section we investigate the fuzzy Sylvester matrix equations (12). Firstly, we set up a computing model for solving LRFSME. Then we define the LR fuzzy solution of LRFSME and obtain its solution representation by means of generalized inverses of matrices. Finally, we give a sufficient condition for strong fuzzy approximate solution to the fuzzy Sylvester matrix equation.

3.1. The Model

At first, we convert the fuzzy Sylvester matrix equation (12) into an LR fuzzy system of linear equations based on the Kronecker product of matrices.

Definition 8. Let 𝐴=(𝑎_𝑖𝑗)=(𝑎_𝑖𝑗,𝑎^𝑙_𝑖𝑗,𝑎^𝑟_𝑖𝑗)_LR∈𝐸^𝑚×𝑛, 𝑎_𝑖=(𝑎_1𝑖,𝑎_2𝑖,…,𝑎_𝑚𝑖)^𝑇, 𝑖=1,…,𝑛. Then the 𝑚𝑛 dimensions fuzzy numbers vector 𝐴=𝑎Vec¹𝑎₂⋮𝑎_𝑛 (14) is called the extension on column of the fuzzy matrix 𝐴.

Theorem 9 (see [25]). Let 𝐴=(𝑎_𝑖𝑗) belong to 𝑅^𝑚×𝑚, 𝑋=(𝑥_𝑖𝑗)=(𝑥_𝑖𝑗,𝑥^𝑙_𝑖𝑗,𝑥^𝑟_𝑖𝑗) belong to 𝐸^𝑚×𝑛, and 𝐵=(𝑏_𝑖𝑗) belong to 𝑅^𝑛×𝑛. Then Vec𝐴𝑋𝐵=𝐵^𝑇⊗𝐴Vec𝑋. (15)

Theorem 10. Let 𝐴=(𝑎_𝑖𝑗) belong to 𝑅^𝑚×𝑚, 𝑋=(𝑥_𝑖𝑗)=(𝑥_𝑖𝑗,𝑥^𝑙_𝑖𝑗,𝑥^𝑟_𝑖𝑗) belong to 𝐸^𝑚×𝑛, and 𝐵=(𝑏_𝑖𝑗) belong to 𝑅^𝑛×𝑛. Then Vec𝐴𝑋+𝑋𝐵=𝐼_𝑛⊗𝐴+𝐵^𝑇⊗𝐼_𝑚Vec𝑋, (16) where 𝐼_𝑛 and 𝐼_𝑚 denote unit matrices with order 𝑛 and order 𝑚, respectively.

Proof. Setting 𝐵=𝐼_𝑛 in (15), we have Vec𝐴𝑋=𝐼_𝑛⊗𝐴Vec𝑋. (17) Similarly, the result Vec𝑋𝐵=𝐵^𝑇⊗𝐼_𝑚Vec𝑋 (18) is obvious when we replace 𝐴 with 𝐼_𝑚 in (15).
We combine (17) and (18) and obtain the following conclusion: 𝐴𝐴𝑋=𝐼Vec𝑋+𝑋𝐵=Vec+Vec𝑋𝐵_𝑛⊗𝐴+𝐵^𝑇⊗𝐼_𝑚Vec𝑋. (19)

Theorem 11. The matrix 𝑋∈𝐸^𝑚×𝑛 is the solution of the fuzzy linear matrix equation (12) if and only if that 𝑥=Vec(𝑋) is the solution of the following [LR] linear fuzzy system: 𝐺𝑥=𝑦, (20) where 𝐺=(𝐼_𝑛⊗𝐴+𝐵^𝑇⊗𝐼_𝑚) and 𝑦=Vec(𝐶).

Proof. Applying the extension operation to two sides of (12) and according to the Definition 8 and Theorem 10, we have 𝐺𝑥=𝑦, (21) where 𝐺=(𝐼_𝑛⊗𝐴+𝐵^𝑇⊗𝐼_𝑚) is an 𝑚𝑛×𝑚𝑛 matrix and 𝑦=Vec(𝐶) is an 𝑚𝑛 LR fuzzy numbers vector. Thus the 𝑋 is the solution of (16) is equivalent to that 𝑥=Vec(𝑋) is the solution of (20).

For simplicity, we denote 𝑝=𝑚𝑛 in (20); thus 𝑔𝐺=¹¹𝑔₁₂⋯𝑔_1,𝑝𝑔₂₁𝑎₁₂⋯𝑔_2,𝑝𝑔⋯⋯⋯⋯_𝑝,1𝑔_𝑝,2⋯𝑔_𝑝,𝑝,𝑦𝑦=¹𝑦₂⋮𝑦_𝑝. (22)

Secondly, we extend the fuzzy LR linear system (16) into two systems of linear equations according to arithmetic operations of LR fuzzy numbers.

Theorem 12. The LR fuzzy linear system (16) can be extended into the following two crisp systems of linear equations: 𝑆𝑥𝐺𝑥=𝑦,_𝑙𝑥^𝑟=𝑦^𝑙𝑦^𝑟, (23)(24) where 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑝)^𝑇,  𝑥^𝑙=(𝑥^𝑙₁,𝑥^𝑙₂,…,𝑥^𝑙_𝑝)^𝑇, 𝑥^𝑟=(𝑥^𝑟₁,𝑥^𝑟₂,…,𝑥^𝑟_𝑝)^𝑇, 𝑦^𝑙=(𝑦^𝑙₁,𝑦^𝑙₂,…,𝑦^𝑙_𝑝)^𝑇, 𝑦^𝑟=(𝑦^𝑟₁,𝑦^𝑟₂,…,𝑦^𝑟_𝑝)^𝑇, 𝑆 is a 2𝑝×2𝑝 matrix, and (𝑠_𝑖𝑗),1≤𝑖, 𝑗≤2𝑝 are determined as follows: 𝑔_𝑖𝑗≥0⇒𝑠_𝑖𝑗=𝑠_{𝑝+𝑖,𝑝+𝑗}=𝑔_𝑖𝑗,  𝑔_𝑖𝑗<0⇒𝑠_{𝑖,𝑗+𝑝}=𝑠_{𝑝+𝑖,𝑗}=−𝑔_𝑖𝑗,1≤𝑖, 𝑗≤𝑝, and any 𝑠_𝑘𝑡 which is not determined by the above items is zero, 1≤𝑘, 𝑡≤2𝑝.

Proof. Let 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑝)^⊤, 𝑥_𝑗=(𝑥_𝑗,𝑥^𝑙_𝑗,𝑥^𝑟_𝑗)_LR, and let 𝑔_𝑖 be the 𝑖th row of 𝐺,𝐺∈ℝ^𝑝×𝑝,𝑔_𝑖=(𝑔_𝑖1,𝑔_𝑖2,…,𝑔_𝑖,𝑝), 𝑖=1,…,𝑝. We can represent 𝐺𝑥 in the form [𝐺𝑥]_𝑖=𝑔_𝑖𝑥,𝑖=1,…,𝑝. (25)
Let 𝑄⁺_𝑖={𝑗∶𝑔_𝑖𝑗≥0} and 𝑄⁻_𝑖={𝑗∶𝑔_𝑖𝑗<0}. We have [𝐺𝑥]_𝑖=_{𝑗∈𝑄+𝑖}𝑔_𝑖𝑗𝑥_𝑗+_{𝑗∈𝑄−𝑖}𝑔_𝑖𝑗𝑥_𝑗,𝑖=1,…,𝑝, (26) that is, [𝐺𝑥]_𝑖=_{𝑗∈𝑄+𝑖}𝑔_𝑖𝑗𝑥_𝑗+_{𝑗∈𝑄−𝑖}𝑔_𝑖𝑗𝑥_𝑗,_{𝑗∈𝑄+𝑖}𝑔_𝑖𝑗𝑥^𝑙_𝑗−_{𝑗∈𝑄−𝑖}𝑔_𝑖𝑗𝑥^𝑟_𝑗,_{𝑗∈𝑄+𝑖}𝑔_𝑖𝑗𝑥^𝑟_𝑗−_{𝑗∈𝑄−𝑖}𝑔_𝑖𝑗𝑥^𝑙_𝑗LR,𝑖=1,…,𝑝. (27) Considering the given LR fuzzy vector 𝑦=(𝑦₁,𝑦₂,…,𝑦_𝑝)^⊤,𝑦_𝑖=(𝑦_𝑖,𝑦^𝑙_𝑖,𝑦^𝑟_𝑖)_LR for the right-hand side of (16), we can write the system (27) as 𝑔_𝑖𝑥,_{𝑗∈𝑄+𝑖}𝑔_𝑖𝑗𝑥^𝑙_𝑗−_{𝑗∈𝑄−𝑖}𝑔_𝑖𝑗𝑥^𝑟_𝑗,_{𝑗∈𝑄+𝑖}𝑔_𝑖𝑗𝑥^𝑟_𝑗−_{𝑗∈𝑄−𝑖}𝑔_𝑖𝑗𝑥^𝑙_𝑗LR=𝑦_𝑖,𝑦^𝑙_𝑖,𝑦^𝑟_𝑖LR,𝑖=1,…,𝑝, (28) where 𝑥=(𝑥₁,…,𝑥_𝑝)^𝑇.
Suppose 𝐺𝑥=𝑦 has a solution. Then, the corresponding mean value 𝑥=(𝑥₁,…,𝑥_𝑝)^𝑇 of the solution must lie in the following crisp linear systems: 𝐺𝑥=𝑦, (29) that is, 𝑔₁₁𝑔₁₂⋯𝑔_1,𝑝𝑔₂₁𝑎₁₂⋯𝑔_2,𝑝𝑔⋯⋯⋯⋯_𝑝,1𝑔_𝑝,2⋯𝑔_𝑝,𝑝𝑥₁𝑥₂⋮𝑥_𝑝=𝑦₁𝑦₂⋮𝑦_𝑝. (30) Meanwhile, the left spread 𝑥^𝑙=(𝑥^𝑙₁,…,𝑥^𝑙_𝑝)^𝑇 and the right spread 𝑥^𝑟=(𝑥^𝑟₁,…,𝑥^𝑟_𝑝)^𝑇 of the solution are obtained by solving the following crisp linear systems: 𝑠₁₁𝑠₁₂⋯𝑠_1,2𝑝𝑠₂₁𝑠₁₂⋯𝑠_2,2𝑝𝑠⋯⋯⋯⋯_2𝑝,1𝑠_2𝑝,2⋯𝑠_2𝑝,2𝑝𝑥^𝑙₁𝑥^𝑙₂⋮𝑥^𝑙_𝑝𝑥^𝑟₁𝑥^𝑟₂⋮𝑥^𝑟_𝑝=𝑦^𝑙₁𝑦^𝑙₂⋮𝑦^𝑙_𝑝𝑦^𝑟₁𝑦^𝑟₂⋮𝑦^𝑟_𝑝, (31) in which (𝑠_𝑖𝑗), 1≤𝑖,𝑗≤2𝑝 are determined as follows: 𝑔_𝑖𝑗≥0⇒𝑠_𝑖𝑗=𝑠_{𝑝+𝑖,𝑝+𝑗}=𝑔_𝑖𝑗,  𝑔_𝑖𝑗<0⇒𝑠_{𝑖,𝑗+𝑝}=𝑠_{𝑝+𝑖,𝑗}=−𝑔_𝑖𝑗, 1≤𝑖,𝑗≤𝑝,and any 𝑠_𝑘𝑡 which is not determined by the above items is zero, 1≤𝑘,𝑡≤2𝑝. Moreover, 𝑆 has the following structure: 𝑆=𝐸𝐹𝐹𝐸≥0 (32) and satisfies 𝐺=𝐸−𝐹.

3.2. Computing the Model

In order to solve the LR fuzzy Sylvester matrix equation (12), we need to consider the LR fuzzy system of linear equations (20). In order to solve (20), we need to consider the systems of linear equations (23) and (24). For instance, when matrix 𝐺 in (23) and matrix 𝑆 in (24) are both invertible, their solutions are expressed by 𝑥=𝐺⁻¹𝑥𝑦,_𝑙𝑥^𝑟=𝑆⁻¹𝑦^𝑙𝑦^𝑟, (33)(34) respectively.

The following Lemma shows when the matrix 𝑆 is nonsingular and how to calculate 𝑆⁻¹.

Lemma 13 (see [8]). The matrix 𝑆 is nonsingular if and only if the matrices 𝐺=𝐸−𝐹 and 𝐸+𝐹 are both nonsingular. If 𝑆⁻¹ exists it must have the same structure as 𝑆, that is, 𝑆⁻¹=𝐿𝐻𝐻𝐿, (35) where 𝐿=(1/2)((𝐸+𝐹)⁻¹+(𝐸−𝐹)⁻¹),  𝐻=(1/2)((𝐸+𝐹)⁻¹−(𝐸−𝐹)⁻¹).
It seems that we have obtained the solution of the original fuzzy linear matrix system (20) as follows: 𝑥=𝑥,𝑥^𝑙,𝑥^𝑟_LR=𝐺⁻¹𝑦,𝐼_𝑝𝑂𝑆⁻¹𝑦^𝑙𝑦^𝑟,𝑂𝐼_𝑝𝑆⁻¹𝑦^𝑙𝑦^𝑟_LR, (36) where 𝐼_𝑝 is a 𝑝 order unit matrix and 𝑂 is a 𝑝 order null matrix. But the solution vector may still not be an appropriate LR fuzzy numbers vector except for 𝑥^𝑙≥0, 𝑥^𝑟≥0. So one gives the definition of LR fuzzy solutions to (11) by the fuzzy linear systems (20) as follows.

Definition 14. Let 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑝)^⊤, 𝑥_𝑗=(𝑥_𝑗,𝑥^𝑙_𝑗,𝑥^𝑟_𝑗)_LR, 𝑗=1,…,𝑝. If 𝑥=𝑥₁,…,𝑥_𝑝^⊤ (37) is an exact solution of (23), 𝑥^𝑙=(𝑥^𝑙₁,…,𝑥^𝑙_𝑝)^⊤ and 𝑥^𝑟=(𝑥^𝑟₁,…,𝑥^𝑟_𝑝)^⊤ are an exact solution of (24); respectively, such that 𝑥^𝑙≥0,𝑥^𝑟≥0, we call 𝑥=(𝑥,𝑥^𝑙,𝑥^𝑟)_LR an LR fuzzy solution of (11).
When linear equation (23) or (24) is inconsistent, we can consider its approximate solution. An approximation solution which is often used is the least squares solution of (23) or (24), defined by minimizing the Frobenius norm of (𝑦−𝐺𝑥) or (𝑌−𝑆𝑋).
For instance, we seek 𝑋^∗∈𝑅^2𝑝 to (31) such that 𝑌−𝑆𝑋^∗_𝐹=min‖𝑌−𝑆𝑋‖_𝐹, (38) that is, minimizing the sum of squares of moduli of (𝑌−𝑆𝑋)‖𝑌−𝑆𝑋‖²_𝐹=^𝑃_𝑖=1𝑏^𝑙_𝑖−^𝑝_𝑗=1𝑠_𝑖𝑗𝑥^𝑙_𝑗+𝑠_{𝑖,𝑝+𝑗}𝑥^𝑟_𝑗²+𝑏^𝑟_𝑖−^𝑝_𝑗=1𝑠_{𝑝+𝑖,𝑗}𝑥^𝑙_𝑗+𝑠_{𝑝+𝑖,𝑝+𝑗}𝑥^𝑟_𝑗². (39) By the generalized inverse theory [28], we know 𝑋^∗=𝑥^𝑙𝑥^𝑟∗=𝑆^{1,3}𝑦^𝑙𝑦^𝑟, (40) where 𝑆^{1,3} is the least squares inverse of the matrix 𝑆.
Now we define the LR fuzzy approximate solution of the fuzzy matrix equations (11) from the fuzzy linear systems (20).

Definition 15. Let 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑝)^⊤, 𝑥_𝑗=(𝑥_𝑗,𝑥^𝑙_𝑗,𝑥^𝑟_𝑗)_LR, 𝑗=1,…,𝑝. If 𝑥=(𝑥₁,…,𝑥_𝑝)^𝑇, or 𝑥^𝑙=(𝑥^𝑙₁,…,𝑥^𝑙_𝑝)^𝑇 and 𝑥^𝑟=(𝑥^𝑟₁,…,𝑥^𝑟_𝑝)^𝑇 are least squares solutions of (23) and (24); respectively, such that 𝑥^𝑙≥0,𝑥^𝑟≥0, then we say 𝑥=(𝑥,𝑥^𝑙,𝑥^𝑟)_LR is an LR fuzzy approximate solution of (11).
Let 𝐺 belong to 𝑅^𝑝×𝑝 and 𝑦 be a 𝑝 arbitrary LR fuzzy numbers vector. Then the solutions of linear systems (23) and (24) can be expressed uniformly by 𝑥=𝐺^†𝑥𝑦,_𝑙𝑥^𝑟=𝑆^†𝑦^𝑙𝑦^𝑟, (41)(42) respectively, no matter (23) and (24) are consistent or not.
It seems that the solution of the LR fuzzy linear system (20) can be expressed as 𝑥=𝑥,𝑥^𝑙,𝑥^𝑟_LR=𝐺^†𝑦,𝐼_𝑝𝑂𝑆^†𝑦^𝑙𝑦^𝑟,𝑂𝐼_𝑝𝑆^†𝑦^𝑙𝑦^𝑟_LR. (43) But it is not the case except for 𝑥^𝑙≥0 and 𝑥^𝑟≥0. In this case, we have a fuzzy set solution not fuzzy number solution. To find the fuzzy number solution, we can approximate the fuzzy set solution by a fuzzy number by one of the approximation methods finally.
Now we give a sufficient condition for LR fuzzy solution to the fuzzy Sylvester matrix equation.

3.3. A Sufficient Condition for LR Fuzzy Solution

To illustrate the expression (43) of an appropriate LR fuzzy solution vector, we now discuss the generalized inverses of matrix 𝑆 in a special structure.

Lemma 16 (see [18]). Let 𝑆=𝐸𝐹𝐹𝐸. (44) Then the matrix 𝑆^†=(𝐸+𝐹)^†+(𝐸−𝐹)^†(𝐸+𝐹)^†−(𝐸−𝐹)^†(𝐸+𝐹)^†−(𝐸−𝐹)^†(𝐸+𝐹)^†+(𝐸−𝐹)^† (45) is the Moore-Penrose inverse of the matrix 𝑆, where (𝐸+𝐹)^†,(𝐸−𝐹)^† are Moore-Penrose inverses of matrices 𝐸+𝐹 and 𝐸−𝐹, respectively.

The key points to make the solution vector be an LR fuzzy solution are 𝑥^𝑙≥0 and 𝑥^𝑟≥0. Since 𝑦^𝑙≥0,𝑦^𝑟≥0 and 𝑥^𝑙=𝐼_𝑝𝑂𝑆^†𝑦^𝑙𝑦^𝑟,𝑥^𝑟=𝑂𝐼_𝑝𝑆^†𝑦^𝑙𝑦^𝑟, (46) the nonnegativity of 𝑥^𝑙 and 𝑥^𝑟 is equivalent to the condition 𝑆^†≥0.

By the above analysis, one has the following conclusion.

Theorem 17. Let 𝐺 belong to 𝑅^𝑝×𝑝 and 𝑆^† nonnegative. Then the solution of the LR fuzzy linear system (12) is expressed by 𝑥=𝐺^†𝑦,𝐼_𝑝𝑂𝑆^†𝑦^𝑙𝑦^𝑟,𝑂𝐼_𝑝𝑆^†𝑦^𝑙𝑦^𝑟_LR, (47) and it admits an LR fuzzy approximate solution.

By further study, one gives a sufficient condition for obtaining nonnegative LR fuzzy solution of fuzzy Sylvester matrix equation (12) when its right-hand side 𝐶 is a positive LR fuzzy numbers matrix.

Theorem 18. Let 𝐺^† and 𝑆^† be nonnegative matrices, and 𝐺^†𝑏−(𝐼_𝑝𝑂)𝑆^†𝑦𝑙_𝑦𝑟≥0. Then the LR fuzzy linear system (12) has a nonnegative LR fuzzy approximate solution as follows: 𝑥=𝐺^†𝑏,𝐼_𝑝𝑂𝑆^†𝑦^𝑙𝑦^𝑟,𝑂𝐼_𝑝𝑆^†𝑦^𝑙𝑦^𝑟_LR. (48)

Proof. Since 𝐺^† is a nonnegative matrix, we have 𝑥=𝐺^†𝑦≥0.
Now that ^𝑥𝑙_𝑥𝑟=𝑆^†𝑦𝑙_𝑦𝑟; therefore, with 𝑆^†≥0 and 𝑦^𝑙≥0 and 𝑦^𝑟≥0, we have ^𝑥𝑙_𝑥𝑟≥0. Thus 𝑥=(𝑥,𝑥^𝑙,𝑥^𝑟)_LR is an LR fuzzy vector which satisfies 𝐺𝑥=𝑦. Since 𝑥−𝑥^𝑙=𝐺^†𝑏−(𝐼_𝑝𝑂)𝑆^†𝑦𝑙_𝑦𝑟≥0, the LR fuzzy linear system (12) has a nonnegative LR fuzzy approximate solution by Definition 2.

The following theorems give some results for such 𝑆⁻¹, 𝑆^{1,3} and 𝑆^† to be nonnegative. As usual, (⋅)^⊤ denotes the transpose of a matrix (⋅).

Theorem 19 (see [29]). The inverse of a nonnegative matrix 𝑆 is nonnegative if and only if 𝑆 is a generalized permutation matrix.

Theorem 20 (see [30]). The matrix 𝑆, of rank 𝑟 with no zero row or zero column, admits a nonnegative {1,3}-inverse if and only if there exist some permutation matrices 𝑃,𝑄 such that 𝑃𝑆𝑄=[𝑅,∗], (49) where 𝑅 is a direct sum of 𝑟 positive, rank-one matrices.

Theorem 21 (see [29]). Let 𝑆 be a 2𝑚×2𝑛 nonnegative matrix with rank 𝑟. Then the following assertions are equivalent.(a)𝑆^†≥0. (b)There exists a permutation matrix 𝑃, such that 𝑃𝑆 has the form 𝐵𝑃𝑆=¹𝐵₂⋮𝐵_𝑟𝑂, (50) where each 𝐵_𝑖 has rank 1 and the rows of 𝐵_𝑖 are orthogonal to the rows of 𝐵_𝑗, whenever 𝑖≠𝑗, the zero matrix may be absent.(c)𝑆^†=^{𝐺𝐶⊤𝐺𝐷⊤}_{𝐺𝐷⊤𝐺𝐶⊤} for some positive diagonal matrix 𝐺. In this case, (𝐶+𝐷)^†=𝐺(𝐶+𝐷)^⊤,(𝐶−𝐷)^†=𝐺(𝐶−𝐷)^⊤. (51)

4. Numerical Examples

Example 22. Consider the following fuzzy matrix system: 10𝑥−11¹¹𝑥₁₂𝑥₁₃𝑥₂₁𝑥₂₂𝑥₁₃+𝑥₁₁𝑥₁₂𝑥₁₃𝑥₂₁𝑥₂₂𝑥₁₃=()10−10−111203,2,1_LR(4,1,1)_LR(2,1,1)_LR(5,2,2)_LR(3,1,2)_LR(7,2,3)_LR. (52) By Theorems 9, 10, and 11, the original fuzzy matrix equation is equivalent to the following LR fuzzy linear system 𝐺𝑥=𝑦, that is, −200010𝑥−11000100002000−1−102−1010100−101−11¹¹𝑥₂₁𝑥₁₂𝑥₂₂𝑥₁₃𝑥₂₃=(5,2,1)_LR(4,2,2)_LR(3,1,1)_LR(5,1,2)_LR(2,1,1)_LR(7,2,3)_LR. (53)
From Theorem 12, the model to above fuzzy linear system is made of the following two crisp systems of linear equations:

−200010𝑥−11000100002000−1−102−1010100−101−11¹¹𝑥₂₁𝑥₁₂𝑥₂₂𝑥₁₃𝑥₂₃=543527,200010000000𝑥010001100000000020000000000002001100001010100000000101010010000000200010100000010001000000000020001100000002100000001010010010000101^𝑙11𝑥^𝑙₂₁𝑥^𝑙₁₂𝑥^𝑙₂₂𝑥^𝑙₁₃𝑥^𝑙₂₃𝑥^𝑟₁₁𝑥^𝑟₂₁𝑥^𝑟₁₂𝑥^𝑟₂₂𝑥^𝑟₁₃𝑥^𝑟₂₃=221112121213. (54)

The coefficient matrices 𝐺 and 𝑆 are both nonsingular; we can obtain the mean value 𝑥, the left spread 𝑥^𝑙, and the right spread 𝑥^𝑟 of the solution to the above fuzzy linear system as follows: 𝑥=𝐺⁻¹𝑦=×543527=,𝑥0.5000−0.2500000.5000.750−0.500−0.2500.250−0.250−0.50000.7500−1.0000.5000.500−0.250−0.5000.5000.500000.50000000.2500.2500.250−0.2500.2501.75001.5000−2.2505.75001.50004.2500𝑋=^𝑙𝑥^𝑟=𝑆⁻¹𝑌=0.75001.37500.25000.50000.50000.37500.25000.6250−0.2500.50000.50000.6250, (55)(56) where𝑆⁻¹=0.50−0.25000000000−0.130.63−0.18−0.330.130.13−0.34−0.13−0.07−0.130.380.3800−0.501.00−0.500.250000.25−0.25−0.13−0.25−0.250.750.25−0.75−0.630.250.250.25000.50000000000.130.380.180.38−0.13−0.13−0.130.130.310.13−0.38−0.380000000.50−0.25000−0.38−0.13−0.06−0.130.380.38−0.130.6−0.18−0.30.130.13−0.500.2500000−0.50100.25−0.75−0.630.250.250.250.25−0.25−0.13−0.25−0.250.75000000000.5000−0.130.130.310.13−0.38−0.380.130.380.180.38−0.13−0.13. (57)

Since 𝑥^𝑟₁₂ is nonpositive, the solution we obtained is not an LR fuzzy solution of the fuzzy linear system 𝐺𝑥=𝑦 and given by 𝑥𝑥=¹¹=(1.7500,0.7500,0.2500)_LR𝑥₂₁=(1.5000,1.3750,0.6250)_LR𝑥₁₂=(−2.250,0.2500,−0.250)_LR𝑥₂₂=(5.7500,0.5000,0.5000)_LR𝑥₁₃=(1.5000,0.5000,0.5000)_LR𝑥₂₃=(4.2500,0.3750,0.6250)_LR. (58)

According to Theorems 12 and Definitions 3 and 14, we know that the solution of the original fuzzy linear matrix equation 𝐴𝑋+𝑋𝐵=𝐶 is a generalized LR fuzzy solution given by𝑥𝑋=¹¹𝑥₁₂𝑥₁₃𝑥₂₁𝑥₂₂𝑥₁₃=(1.750,0.750,0.250)_LR(−2.250,0.250,0.000)_LR(1.500,0.500,0.500)_LR(1.500,1.375,0.625)_LR(5.750,0.500,0.500)_LR(4.250,0.375,0.625)_LR. (59)

Example 23. Consider the fuzzy Sylvester matrix system −20𝑥11¹¹𝑥₁₂𝑥₂₁𝑥₂₂+𝑥₁₁𝑥₁₂𝑥₂₁𝑥₂₂=()−11−105,1,2_LR(3,2,2)_LR(4,2,1)_LR(2,1,1)_LR. (60)
By Theorems 9, 10, and 11, the original fuzzy matrix equation is equivalent to the following LR fuzzy linear system 𝐺𝑥=𝑦, that is, −1010𝑥1201−10−200−111¹¹𝑥₂₁𝑥₁₂𝑥₂₂=(5,1,2)_LR(4,2,1)_LR(3,2,2)_LR(2,1,1)_LR. (61)
From Theorem 12, the model to the above fuzzy linear system is made of the following two crisp systems of linear equations: −1010𝑥1201−10−200−111¹¹𝑥₂₁𝑥₁₂𝑥₂₂=5432𝑥,0010100012010000000010200011010010000010000012011020000001000011^𝑙11𝑥^𝑙₂₁𝑥^𝑙₁₂𝑥^𝑙₂₂𝑥^𝑟₁₁𝑥^𝑟₂₁𝑥^𝑟₁₂𝑥^𝑟₂₂=12212121. (62)(63)
The coefficient matrices 𝐺 and 𝑆 are both singular; we can obtain the mean value 𝑥, the left spread 𝑥^𝑙, and the right spread 𝑥^𝑟 of the solution to the above fuzzy linear system by the same method 𝑥=𝐺^†,𝑥𝑦=−4.33332.33330.66673.6667^𝑙𝑥^𝑟=𝑆^†𝑌=1.3330.0000.3330.6670.6670.0000.6670.333. (64)
Since 𝑥^𝑙_𝑖𝑗,𝑥^𝑟_𝑖𝑗,𝑖,𝑗=1,2 are nonnegative, the solution we obtained is an LR fuzzy approximate solution of the fuzzy linear system 𝐺𝑥=𝑦 and given by 𝑥𝑥=¹¹=(−4.3333,1.3333,0.6667)_LR𝑥₂₁=(2.3333,0.0000,0.0000)_LR𝑥₁₂=(0.6667,0.3333,0.6667)_LR𝑥₂₂=(3.6667,0.6667,0.3333)_LR. (65)
According to Theorems 12 and Definition 15, we know that the fuzzy approximate solution of the original fuzzy linear matrix equation 𝐴𝑋+𝑋𝐵=𝐶 is 𝑥𝑋=¹¹𝑥₁₂𝑥₂₁𝑥₂₂=(−4.3333,1.3333,0.6667)_LR(0.6667,0.3333,0.6667)_LR(2.3333,0.0000,0.0000)_LR(3.6667,0.6667,0.3333)_LR, (66) and it admits an appropriate LR fuzzy approximate solution.

5. Conclusion

In this work we presented a model for solving fuzzy Sylvester matrix equations 𝐴𝑋+𝑋𝐵=𝐶 where 𝐴 and 𝐵 are 𝑚×𝑚 and 𝑛×𝑛 crisp matrices, respectively, and 𝐶 is an 𝑚×𝑛 arbitrary LR fuzzy numbers matrix. The model was proposed in this way, that is, we converted the fuzzy linear matrix equation into an LR fuzzy linear systems, then we extended the fuzzy linear systems into two systems of linear equations. The LR fuzzy solution of the fuzzy matrix equation was derived from solving the crisp systems of linear equations. In addition, the existence condition of LR fuzzy solution was studied. Numerical examples showed that our method is feasible to solve this type of fuzzy matrix equations.

Acknowledgments

The authors were very thankful for the reviewer’s helpful suggestions to improve the paper. This paper was financially supported by the National Natural Science Foundation of China (71061013) and the Youth Scientific Research Ability Promotion Project of Northwest Normal University (nwnu-lkqn-11-20).