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

摘要

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 S are provided. Moreover, an optimal approximation between a given matrix pair and the affine subspace S 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.