The Symmetric and Sub-anti-symmetric Sulutions of Matrix Equation A~TXB=B and ITS Optimal

Using the general singular value decomposition of matrices, the necessary and sufficient conditions for the existence and the expressions for the symmetric and sub-anti-symmetric solutions of the linear matrix equation $ATXA=B$ are established. Furthermore, in the solution set of corresponding equation, the expression of the optimal approximation solution togiven matrix is derived.