Analytical best approximate Hermitian and generalized skew-Hamiltonian solution of matrix equation AXAH+CYCH=F

In this paper, we consider the matrix equation AXAH+CYCH=F, where A, C and F are given matrices with appropriate sizes and [X,Y] is an unknown Hermitian and generalized skew-Hamiltonian matrix pair. Based on matrix differential calculus and projection theorem in inner product spaces, we exploit the best approximate solution [X̂,Ŷ] in the set S to a given matrix pair [X∗,Y∗], where S signifies the least-squares Hermitian and generalized skew-Hamiltonian solution set of the matrix equation AXAH+CYCH=F. The analytical expression of the best approximate solution is presented by applying the canonical correlation decomposition and the generalized singular value decomposition. Finally, a numerical algorithm and an illustrated example are given.