Solving the system of generalized Sylvester matrix equations over the generalized centro-symmetric matrices

The system of generalized Sylvester matrix equations [Formula: see text] (including Sylvester and Lyapunov matrix equations as special cases) has nice applications in various branches of control and system theory. In the present paper, we consider this system over the generalized centro-symmetric matrix X. By extending the Jacobi and the Gauss–Seidel iterations and by applying the hierarchical identification principle, we propose a gradient-based iterative algorithm for finding the generalized centro-symmetric solution of the system. It is shown that the iterative algorithm consistently converges to the generalized centro-symmetric solution for any initial generalized centro-symmetric matrix. A numerical experiment is also given to show the effectiveness of the proposed algorithm.