The matrix nearness problem for symmetric matrices associated with the matrix equation [ATXA, BTXB] = [C, D]

A direct method, based on the projection theorem in inner products spaces, the generalized singular value decomposition and the canonical correlation decomposition, is presented for finding the optimal approximate solution in the set SE to a given matrix , where SE denotes the least-squares symmetric solution set of the matrix equation [ATXA, BTXB] = [C, D]. The analytical expression of the optimal approximate solution is obtained, and an algorithm for finding this solution is also suggested.