Arbitrary Generalized Trapezoidal Fully Fuzzy Sylvester Matrix Equation

In the fuzzy literature, researchers have applied the concept of Vec-operator and Kronecker product for solving arbitrary Fuzzy Matrix Equations (FME). However, this approach is limited to positive or negative FMEs and cannot be applied to FMEs with near-zero fuzzy numbers. Therefore, this paper proposes a new analytical method for solving a family of arbitrary FMEs. The proposed method is able to solve Arbitrary Generalized Trapezoidal Fully Fuzzy Sylvester Matrix Equations (AGTrFFSME), in addition to many unrestricted FMEs such as Sylvester, Lyapunov and Stein fully fuzzy matrix equations with arbitrary triangular or trapezoidal fuzzy numbers. The proposed method thus fruitfully removes the sign restriction imposed by researchers and is, therefore, better to use in several engineering and scientific applications. The AGTrFFSME is converted to a system of non-linear equations, which is reduced using new multiplication operations between trapezoidal fuzzy numbers. The feasibility conditions are introduced to distinguish between fuzzy and non-fuzzy solutions to the AGTrFFSME.