Employing an iterative gradient-based approach to solve generalized coupled Sylvester matrix equations over generalized centro-symmetric matrices

In this research, we introduce a novel approach to solving coupled generalized Sylvester matrix (GSM) equations MV + NW = GVH + D, AV + BW = EVF + C by leveraging the structural properties of generalized centro-symmetric matrices (GCSMs). Specifically, we propose a gradient-based iterative technique that exploits the symmetry and pattern inherent in GCSMs to facilitate the solution process. The proposed method offers an efficient computational framework, particularly well-suited for problems where the coefficient matrices and initial conditions are generalized centro-symmetric. Our approach focuses on iteratively refining an initial guess?chosen to be a GCSM?toward a generalized centro-symmetric solution (GCSS) of the coupled GSM equations. A key advantage of this method is its rapid convergence in the idealized setting free of round-off errors. When the matrix equations under consideration are compatible with the structure of the initial GCSM, the method can achieve accurate results in a relatively small number of iterations.