An iterative method for the skew-symmetric solution and the optimal approximate solution of the matrix equation AXB=CAXB=C

In this paper, an iterative method is constructed to solve the linear matrix equation AXB=CAXB=C over skew-symmetric matrix X  . By the iterative method, the solvability of the equation AXB=CAXB=C over skew-symmetric matrix can be determined automatically. When the equation AXB=CAXB=C is consistent over skew-symmetric matrix X  , for any skew-symmetric initial iterative matrix X1X1, the skew-symmetric solution can be obtained within finite iterative steps in the absence of roundoff errors. The unique least-norm skew-symmetric iterative solution of AXB=CAXB=C can be derived when an appropriate initial iterative matrix is chosen. A sufficient and necessary condition for whether the equation AXB=CAXB=C is inconsistent is given. Furthermore, the optimal approximate solution of AXB=CAXB=C for a given matrix X0X0 can be derived by finding the least-norm skew-symmetric solution of a new corresponding matrix equation AX˜B=C˜. Finally, several numerical examples are given to illustrate that our iterative method is effective.