Trapezoidal Fully Fuzzy Sylvester Matrix Equation with Arbitrary Coefficients

Almost every existing method for solving trapezoidal fully fuzzy Sylvester matrix equation restricts the coefficient matrix and the solution to be positive fuzzy numbers only. In this paper, we develop new analytical methods to solve a trapezoidal fully fuzzy Sylvester matrix equation with restricted and unrestricted coefficients. The trapezoidal fully fuzzy Sylvester matrix equation is transferred to a system of crisp equations based on the sign of the coefficients by using Ahmd arithmetic multiplication operations between trapezoidal fuzzy numbers. The constructed method not only obtain a simple crisp system of linear equation that can be solved by any classical methods but also provide a widen the scope of the trapezoidal fully fuzzy Sylvester matrix equation in scientific applications. Furthermore, these methods have less steps and conceptually easy to understand when compared with existing methods. To illustrate the proposed methods numerical examples are solved.


Introduction
Sylvester matrix equations played a prominent role in various areas including control theory (Datta, 2004), medical imaging acquisition systems, model reduction of linear time invariant systems (Antoulas, 2005;Sorensen & Antoulas, 2002) and stochastic control, signal processing, image processing and filtering (Bouhamidi & Jbilou, 2007). When information is imprecise and only some vague knowledge about the actual values of the parameters is available, it is convenient to use fuzzy numbers (D. J. Dubois, 1980;Zadeh, 1965). Considering to any uncertainty problems that can be occurred at any time, the crisp Sylvester matrix equation has to be adapted to the fuzzy environment. Triangular fully fuzzy Sylvester matrix equation (TFFSME) has been studied in (Dubois & Prade, 1978;Malkawi et al., 2015;Shang et al., 2015) where, the TFFSME is converted into a system of crisp linear equations by applying Dubois and Prade's arithmetic operator for multiplication. Then Kronecker product and the matrix inversion method has been applied to obtain the positive fuzzy solution. Nevertheless, these methods are restricted to non-singular positive TFFSME only. To overcome the shortcoming in these methods, Daud, Ahmad and Malkawi (2018c) obtained positive solution for singular TFFSME where the solution is obtained by using the method of pseudoinverse. Moreover, Daud, et al., (2018b) proposed another algorithm for solving TFFSME with arbitrary coefficients which utilized the Kaufmann and Gupta's arithmetic multiplication operator (Kauffman & Gupta, 1991). However, they were able to find positive fuzzy solution only. Recently, in (Elsayed et al., 2020), the authors considered solution of trapezoidal fully fuzzy Sylvester matrix equation (TrFFSME) by transforming TrFFSME to a system of four equations where the positive and negative fuzzy solutions are obtained by applying Kronecker product and Vec-operator method. A study was conducted by Dookhitram et al., (2015) on the TFFSME in the form # % − % % = # , which used the α-cuts expansion approach in the parameters. The method proposed has an advantage in the sense that it provides maximal and minimal symmetric solutions of the TFFSME, however, the method required long fuzzy operations process in obtaining the solution. Similarly, Daud et al., (2018a) proposed an algorithm for solving TFFSME with arbitrary coefficients. However, the method was restricted only for positive fuzzy solutions. Most of the analytical methods proposed for solving TFFSME in the literature are based on Dubois and Prade's arithmetic operator for multiplication which is restricted only for positive fuzzy numbers with very small fuzziness (Fortin et al., 2008) and therefore, most of the analytical methods are restricted to positive coefficients and positive fuzzy solution only. In addition, many researchers have applied Kaufmann and Gupta's arithmetic multiplication operator for solving TFFSME with arbitrary coefficients however, their methods are limited to positive fuzzy solution only for systems with triangular fuzzy numbers (TFNs) only. Sign and size of fuzzy systems are restricted in most of the existing studies due to the nature of analytical methods used and arithmetic operations applied. Where, many methods are limited only for small sized systems with positive fuzzy numbers. Therefore, in order to deal with this shortcoming, in this paper we develop new methods for solving TrFFSME with arbitrary coefficients by using Ahmd arithmetic multiplication operations between trapezoidal fuzzy numbers (TrFNs). The main intent of the proposed methods is to avoid the complexity procedure in the previous methods by introducing new systems of linear equations equivalent to the fuzzy systems which can be solved in one step by MATLAB or Mathematica. The proposed method is able to solve large size systems. In addition, it can also be applied to TFFSME and fully fuzzy matrix equation (FFME) with both (TFNs) and (TrFNs). This paper is organized as follows, In section 2, the preliminary concepts and arithmetic operations of trapezoidal fuzzy numbers are discussed. In section 3, new methods for solving TrFFSME are developed. In section 4, numerical examples are solved. In section 5, conclusion about the proposed methods and achieved results will be drawn.

III)
Arbitrary, if at least one element of # is near zero TrFNs.
Definition 2.10 The Kronecker sum of two matrices ⨁ can be considered as a matrix sum defined by where is a square matrix of order and is a square matrices of order , and 6 , 5 are identity matrices of order and respectively and ⨂ represents the Kronecker product.

Definition 2.11
The Vec-operator generates a column vector from a matrix by stacking the column

Proposed method
In this section the TrFFSME in Eq. (2.8) is converted to an equivalent systems of crisp Sylvester matrix equations based on the sign of fuzzy numbers used by applying Ahmd arithmetic multiplication operators in Eq.(2.2) to Eq. (2.6). Where, the complexity procedure in the literature of TrFFSME can be avoided and the fuzzy solution can be obtained in one step by MATLAB R2020a or Mathematica12. The proposed methods are discussed in the following section.

TrFFSME with arbitrary coefficients.
In the following theorem we obtain a system of nonlinear equations equivalent to the TrFFSME in Eq. (2.8) where the coefficients are arbitrary, and the fuzzy solution is positive.

Remark 3.1
The left-hand side of the non-linear equations obtained in Eq. (3.1) can be reduced to linear system by applying the following:

Remark 3.2
Any × TrFFSME with arbitrary coefficient can converted to a linear system of 2 × 2 equations. It is worth mention that the systems of equation in Eq. (3.1) after reducing it by Remark 3.1 to linear system, can be solved by many classical methods. However, in this paper the matrix inversion method will be used, where the solution is presented in the form:
Then the TrFFSME in Eq.(2.8) can be written as follows: Proof: Straightforward similar to Theorem 3.2.
Then the TrFFSME in Eq.(2.8) can be written as follows:

4)
Proof: Straightforward similar to Theorem 3.2. In the following theorem a system of crisp linear Sylvester matrix equation equivalent to the TrFFSME in Eq. (2.8) when the coefficients are negative, and the fuzzy solution is positive.
Then the TrFFSME in Eq.(2.8) can be written as follows: Proof: Straightforward similar to Theorem 3.2.
In the following subsection we apply Bartle's Stewart method (BSM) to the system of equations in Eq.(3.2) to obtain the positive fuzzy solution. It worth mentioning that, the same method can be also applied to the other systems of equations in Eq. (3.3) to Eq. (3.5).

Generalized BSM for TrFFSME
In this section the generalized BSM is amended to solve the system of equations in Eq. (3.2) as follows: Step 1 ' . Consequently, they can be written as: ' . Applying Kronecker product and Vec-operator gives: Many analytical method can be used to solve for # , $ , & and ' . However, in this paper we will apply Gaussian elimination and back substitution method.

Numerical Examples
In this section, the proposed methods are illustrated by solving two examples.
Step 4 The positive fuzzy solution % = r ( 4, 5, 2, 1 ) ( 4, 5, 3, 1 ) ( 3, 4, 2, 2 ) ( 3, 4, 2, 1 ) s, is feasible. Figure 2 shows the positive fuzzy solution % . To the best of our knowledge, the proposed methods are the first approaches that can be applied to different fuzzy systems with arbitrary coefficients without any amendments. For example, it can be applied to TrFFSME in the form # % + % % = # with TFNs whenever, the mean values in the TrFNs used are equal. In addition, it can also be applied to the FFME in the form # % = # with arbitrary coefficients if we allow % = 0, in # % + % % = # . Therefore, the proposed method is able to solve the following fuzzy systems: FFSME with TFNs and TrFNs and FFME with arbitrary coefficients.

Conclusion
In this paper, a new methods are developed to solve the TrFFSME # % + % % = # with restricted and unrestricted coefficient matrices, based on new fuzzy arithmetic operation between TrFNs. Different linear systems are developed corresponding to the TrFFSME based on the sign of the coefficients, where the solution can be obtained directly by MATLAB or Mathematica. In addition, the proposed methods are applicable for arbitrary FFSME with TFNs and TrFNs and arbitrary FFME with TFNs and TrFNs. As future work the proposed method will be applied to TrFFSME with trapezoidal bipolar fuzzy numbers.