Iterative algorithms for the minimum-norm solution and the least-squares solution of the linear matrix equations A1XB1+C1XTD1=M1,A2XB2+C2XTD2=M2

In this paper, two iterative algorithms are proposed to solve the linear matrix equations A1XB1+C1XTD1=M1,A2XB2+C2XTD2=M2. When the matrix equations are consistent, by the first algorithm, a solution X∗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 X0X0 can be derived by finding the minimum-norm solution of a new matrix equations A1X∼B1+C1X∼TD1=M1,A2X∼B2+C2X∼TD2=M2. 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.