^{1}

^{2}

^{1}

^{2}

The fuzzy matrix equations

Since many real-world engineering systems are too complex to be defined in precise terms, imprecision is often involved in any engineering design process. Fuzzy systems have an essential role in this fuzzy modeling, which can formulate uncertainty in actual environment. In many matrix equations, some or all of the system parameters are vague or imprecise, and fuzzy mathematics is a better tool than crisp mathematics for modeling these problems, and hence solving a fuzzy matrix equation is becoming more important. The concept of fuzzy numbers and arithmetic operations with these numbers were first introduced and investigated by Zadeh [

In the past decades, many researchers have studied the fuzzy linear equations such as fuzzy linear systems (FLS), dual fuzzy linear systems (DFLS), general fuzzy linear systems (GFLS), fully fuzzy linear systems (FFLS), dual fully fuzzy linear systems (DFFLS), and general dual fuzzy linear systems (GDFLS). These works were performed mainly by Friedman et al. [

To make the multiplication of fuzzy numbers easy and handle the fully fuzzy systems, Dubois and Prade [

In Section

A fuzzy number is a fuzzy set like

supp

Let

A fuzzy number

The definition of a right shape function

Clearly,

Also, two LR fuzzy numbers

For arbitrary LR fuzzy numbers

Addition:

Multiplication:

If

if

if

Scalar multiplication:

A matrix

Let

The matrix system:

Using matrix notation, we have

A fuzzy numbers matrix

Up to the rest of this paper, we will discuss the nonnegative solution

First, we extend the fuzzy linear matrix system (

The fuzzy linear matrix system (

We denote

Secondly, in order to solve the fuzzy linear matrix equation (

Let

Let

Our hypotheses on

On the other hand, because

When

For linear matrix equations

By the pseudoinverse of matrices, we solve model (

Let

At last, we give a sufficient condition for nonnegative fuzzy minimal norm least squares solution of FFLME (

Let

Since

Now that

The following Theorems give some results for such

The inverse of a nonnegative matrix

Let

There exists a permutation matrix

Consider the fully fuzzy linear matrix system

By Theorem

Now that the matrix

By Definition

since

Consider the following fuzzy matrix system:

By Theorem

By the same way, we obtain that the solutions of the above three systems of linear matrix equations are as follows:

Since

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

The work is supported by the Natural Scientific Funds of China (no. 71061013) and the Youth Research Ability Project of Northwest Normal University (NWNU-LKQN-1120).