Trapezoidal fully fuzzy Sylvester matrix equation with arbitrary coefficients

In this paper, a new analytical method for solving trapezoidal fully fuzzy Sylvester matrix equations (TrFFSME) with arbitrary coefficients is proposed. Sylvester matrix equations (SME) have a significant impact in various fields such as control theory, medical imaging systems, and model reduction. In the presence of uncertainty, classical SMEs are not sufficient to handle such problems. Therefore, SMEs with fuzzy numbers are an effective way to model matrix equations in such situations. Previous literature has only provided solutions to fully fuzzy Sylvester matrix equations (FFSME) with positive coefficients. However, the proposed method in this paper can solve TrFFSME without any restriction on the sign of the coefficients. The TrFFSME is converted to an equivalent system of four crisp SMEs using a newly developed arithmetic fuzzy multiplication operation. The resulting system of SME is solved by fuzzy Bartels -Stewart method. The proposed method extends the scope of the TrFFSME in scientific applications and provides a direct method to the fuzzy solution. Numerical examples are provided to illustrate the effectiveness of the proposed method.