Iterative algorithms for the minimum-norm solution and the least-squares solution of the linear matrix equations [formula omitted

Abstract

In this paper, two iterative algorithms are proposed to solve the linear matrix equations A 1 XB 1 + C 1 X T D 1 = M 1 , A 2 XB 2 + C 2 X T D 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 X 0 can be derived by finding the minimum-norm solution of a new matrix equations A 1 X ∼ B 1 + C 1 X ∼ T D 1 = M 1 , A 2 X ∼ B 2 + C 2 X ∼ T D 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.