An iterative method for the skew-symmetric solution and the optimal approximate solution of the matrix equation A X B=C

In this paper, an iterative method is constructed to solve the linear matrix equation A X B = C over skew-symmetric matrix X. By the iterative method, the solvability of the equation A X B = C over skew-symmetric matrix can be determined automatically. When the equation A X B = C is consistent over skew-symmetric matrix X, for any skew-symmetric initial iterative matrix X(1), 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 A X B=C can be derived when an appropriate initial iterative matrix is chosen. A sufficient and necessary condition for whether the equation AXB = C is inconsistent is given. Furthermore, the optimal approximate solution of A X B = C for a given matrix X(0) can be derived by finding the least-norm skew-symmetric solution of a new corresponding matrix equation AXB = C. Finally, several numerical examples are given to illustrate that our iterative method is effective. (C) 2006 Elsevier B.V. All rights reserved.