Abstract: This paper discusses the least-squares solutions of matrix equation AX=Bfor anti-symmetric ortho-symmetric matrices on the linear manifold and its optimal approximation problems. The general expression of the least-squares solution set is given.In addition,the expression of the optimal approximation solution to the given matrix is obtained.Numerical methods of the optimal approximation to a given matrix and numerical experiments are described.

Key words: Matrix equation, Linear manifold, Anti-symmetric ortho-symmetric matrix, Optimal approximation