Iterative algorithms for the minimum-norm solution and the least-squares solution of the linear matrix equations A_1XB_1 + C _1X~TD_1 = M_1, A_2XB _2 + C_2X~TD_2 = M_2

摘要

In this paper, two iterative algorithms are proposed to solve the linear matrix equations A_1XB_1 + C_1X~TD _1 = M_1, A_2XB_2 + C_2X ~TD_2 = M_2. When the matrix equations are consistent, by the first algorithm, a solution ~X can be obtained within finite iterative steps in the absence of roundoff-error for any initial value, furthermore, the minimum-norm solution can be got by choosing a special kind of initial matrix. Additionally, the unique optimal approximation solution to a given matrix A_1X?B_1 + C_1X? ~TD_1 = M_1, A_2X?B_2 + C_2X?~TD_2 = M_2. When the matrix equations are inconsistent, we present the second algorithm to find the least-squares solution with the minimum-norm. Finally, two numerical examples are tested by MATLAB, the results show that these iterative algorithms are efficient.