Least-squares symmetric and skew-symmetric solutions of the generalized Sylvester matrix equation ∑i=1sAiXBi+∑j=1tCjYDj=E

The generalized Sylvester matrix equation ∑i=1sAiXBi+∑j=1tCjYDj=E with unknown matrices X and Y   is encountered in many system and control applications. In this paper, a direct method is established to solve the least-squares symmetric and skew-symmetric solutions of the equation by using the Kronecker product and the generalized inverses and, the expression of the solution set SS are provided. Moreover, an optimal approximation between a given matrix pair and the affine subspace SS is discussed, and an explicit formula for the unique optimal approximation solution is presented. Finally, two numerical examples are given which demonstrate that the introduced algorithm is quite efficient.