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The fuzzy symmetric solution of fuzzy matrix equation

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 [

Since Friedman et al. [

In this paper, we propose a general model for solving the fuzzy matrix equation

In Section

There are several definitions for the concept of fuzzy numbers (see [

A fuzzy number is a fuzzy set like

Let

A fuzzy number

A crisp number

Let

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

Suppose

Let

Let

A matrix

Let

The matrix system:

Using matrix notation, we have

A fuzzy numbers matrix:

In this section, we will investigate the fuzzy matrix equation (

At first, we convert the fuzzy matrix equation (

Let

Let

Since

The matrix

Setting

Applying the extension operation the Definition

For simplicity, we denote

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

The united solution set (USS), the tolerable solution set (TSS), and the controllable solution set (CSS) for the system (

A fuzzy vector

A fuzzy vector

Secondly, in order to solve the fuzzy matrix equation (

The fuzzy linear system (

Now, one solves the crisp linear system (

So, without loss of generality and for simplicity to express the theory, it is assumed that the coefficients matrix

So, after some computations and replacing

Let the right-hand side of the system (

Let us consider the

Since that

For results (

Consider spreads (

In addition, one can find the maximal and minimal solutions of fuzzy linear system (

Using definitions of

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

We convert the fuzzy linear matrix equation (

We solve system (

By applying crisp solution (solution of 1-cut), system (

The spread of all elements of fuzzy vector solution will be obtained by solving system (

The symmetric spreads can be assessed using (

The fuzzy vector solutions are derived by (

Consider the following fuzzy matrix system:

By Theorems

According to Theorem

Consider the fuzzy matrix system:

By Theorems

According to Theorem

In this work, we presented a model for solving fuzzy matrix equations

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).