In this note, we apply a hierarchical identification principle to study solving the Sylvester and Lyapunov matrix equations. In our approach, we regard the unknown matrix to be solved as system parameters to be identified, and present a gradient iterative algorithm for solving the equations by minimizing certain criterion functions. We prove that the iterative solution consistently converges to the true solution for any initial value, and illustrate that the rate of convergence of the iterative solution can be enhanced by choosing the convergence factor (or step-size) appropriately. Furthermore, the iterative method is extended to solve general linear matrix equations. 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.

In this paper we study coupled matrix equations, which are encountered in many systems and control applications. First, we extend the well-known Jacobi and Gauss--Seidel iterations and present a large family of iterative methods, which are then applied to develop iterative solutions to coupled Sylvester matrix equations. The basic idea is to regard the unknown matrices to be solved as parameters of a system to be identified and to obtain the iterative solutions by applying a hierarchical identification principle. Next, we generalize the Sylvester equations to general coupled matrix equations, and propose a gradient-based iterative algorithm for the solutions, using a block-matrix inner product---the star $(\star)$ product; we prove that the iterative algorithm always converges to the (unique) solutions for any initial values. One advantage of the algorithms proposed is that they require less storage space in implementation than existing numerical methods. Finally, we test the algorithms and show their eff...