Abstract:

The symmetric solution and the antisymmetric solution of matrix equations AX=B, XD=E were considered. Based on the special properties of symmetric
 (anti-symmetric) matrices and the canonical correlation decomposition (CCD) of a pair of matrices, the necessary and sufficient conditions for the solvability
and the general expression of the symmetric solution (antisymmetric solution) were obtained. Moreover, the related optimal approximation problem to a given matrix over the solution set was solved.

Key words: matrix equations, symmetric matrix, anti-symmetric matrix, optimal approximation, canonical correlation decomposition (CCD)