Best approximate solution of matrix equation AXB+CY D=E

We derive an analytical expression of the best approximate solution in the least-squares solution set SE of the matrix equation AXB + CYD = E to a given matrix pair (X-f, Y-f), where A, B, C, D, and E are given matrices of suitable sizes. Our work is based on the projection theorem in the finite-dimensional inner product space, and we use the generalized singular value decomposition and the canonical correlation decomposition. Moreover, we establish a direct method for computing this best approximate solution. An algorithm for finding the best approximate solution is described in detail, and an example is used to show the feasibility and effectiveness of our algorithm.