An iterative method for symmetric solutions and optimal approximation solution of the system of matrix equations A1XB1 = C1, A2XB2 = C2

The symmetric solutions of the system of matrix equations A1XB1 = C1, A2XB2 = C2 are too difficult to be obtained by applying matrices decomposition. In this paper, an iterative method is applied to solve this problem. With it, the solvability of this system of matrix equations can be determined automatically, when this system of matrix equations is consistent, its solution can be obtained within finite iterative steps, and its least-norm solution can be obtained by choosing a special kind of initial iterative matrix, furthermore, its optimal approximation solution to a given matrix can be derived by finding the least-norm symmetric solution of a new system of matrix equations A1X˜B1=C˜1,A2X˜B2=C˜2. Finally, numerical examples are given.