The anti-symmetric ortho-symmetric solution of a linear matrix equation and its optimal approximation

This paper discusses the anti-symmetric ortho-symmetric solution of a linear matrix equation and its optimal approximation. By the generalized singular value decomposition of the matrices, the necessary and sufficient conditions for the solvability of the matrix equation and the general form of the anti-symmetric ortho-symmetric solution are given. In addition, the existence and uniqueness of the optimal approximation are proved. Numerical methods of the optimal approximation to a given matrix and numerical experiments are described.