Numerical solution to linear matrix equation by finite steps iteration

The matrix equation Σ^l_i=1A_iXB_i = C, which contains the well-known Sylvester matrix equation and Lyapunov matrix equation as special cases, has many important applications in control system theory. This study presents an iterative algorithm to solve such linear matrix equation. It is shown that the proposed algorithm converges to the unique solution to the linear matrix equation at finite steps for arbitrary initial condition. Moreover, if the matrix equation is not consistent, the least squares solution can be obtained by alternatively solving a linear matrix equation in the same form, which can also be solved by the proposed iterative algorithm. Numerical example shows the effectiveness of the proposed approach.