On iterative solutions of a class of matrix equations in systems and control

In this paper, we present a general family of iterative methods to solve linear equations, which includes the well-known Jacobi and Gauss-Seidel iterations as its special cases. We give the necessary and sufficient conditions for convergence of the iterative solutions. Furthermore, the methods are extended to solve coupled Sylvester matrix equations. In our approach, we regard the unknown matrices to be solved as the system parameters to be identified, and propose a least squares iterative algorithm by applying a hierarchical identification principle. We prove that the iterative solution consistently converges to the exact solution for any initial value. The algorithms proposed require less storage capacity than the existing numerical ones. Finally, the algorithms are tested on computer and the results verify the theoretical findings.