Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle

In this paper, by extending the well-known Jacobi and Gauss–Seidel iterations for Ax = b, we study iterative solutions of matrix equations AXB = F and generalized Sylvester matrix equations AXB + CXD = F (including the Sylvester equation AX + XB = F as a special case), and present a gradient based and a least-squares based iterative algorithms for the solution. It is proved that the iterative solution always converges to the exact solution for any initial values. The basic idea is to regard the unknown matrix X to be solved as the parameters of a system to be identified, and to obtain the iterative solutions by applying the hierarchical identification principle. Finally, we test the algorithms and show their effectiveness using a numerical example.