A finite iterative method for solving a pair of linear matrix equations (AXB,CXD)=(E,F)(AXB,CXD)=(E,F)

In this paper, an efficient iterative method is presented to solve a pair of linear matrix equations (AXB,CXD)=(E,F)(AXB,CXD)=(E,F) with real matrix X. By this iterative method, the solvability of the matrix equations pair can be determined automatically. When the pair of matrix equations are consistent, then, for any initial matrix X0, a solution can be obtained within finite iteration steps in the absence of roundoff errors, and the least norm solution can be obtained by choosing a special kind of initial matrix. In addition, the unique optimal approximation solution X^ to a given matrix X¯ in Frobenius norm can be obtained by finding the least norm solution of a new pair of matrix equations (AX∼B,CX∼D)=(E∼,F∼), where E∼=E-AX¯B,F∼=F-CX¯D. The given numerical examples demonstrate that the iterative method is quite efficient.