An optimal approximation problem for a matrix equation

Chang and Wang discussed the consistent conditions for the symmetric solutions of the linear matrix equation (A^TXA, B^TXB)=(C, D) and obtained the general expressions for its symmetric solution and symmetric least-squares solution, but they could not give the minimum Frobenius norm symmetric solution for this equation or the related least-squares problem. In this paper, the explicit analytical expression of the symmetric optimal approximation solution (or the minimum Frobenius norm symmetric solution as a special case) for the least-squares problem of this linear matrix equation is obtained by using the projection theorem in the Hilbert space, the quotient singular value decomposition and the canonical correlation decomposition in the matrix theory for efficient tools; therefore, this result gives an answer to the open problem in Chang and Wang's paper in 1993.