The (M, N)-symmetric Procrustes problem

An p×q matrix A is said to be (M,N)-symmetric if MAN=(MAN)T for given M∈Rn×p,N∈Rq×n. In this paper, the following (M,N)-symmetric Procrustes problem is studied. Find the (M,N)-symmetric matrix A which minimizes the Frobenius norm of AX-B, where X and B are given rectangular matrices. We use Project Theorem, the singular-value decomposition and the generalized singular-value decomposition of matrices to analysis the problem and to derive a stable method for its solution. The related optimal approximation problem to a given matrix on the solution set is solved. Furthermore, the algorithm to compute the optimal approximate solution and the numerical experiment are given.