(Anti-)Hermitian generalized (anti-)Hamiltonian solution to a system of matrix equations (Q1718620)

Summary: We mainly solve three problems. Firstly, by the decomposition of the (anti-)Hermitian generalized (anti-)Hamiltonian matrices, the necessary and sufficient conditions for the existence of and the expression for the (anti-)Hermitian generalized (anti-)Hamiltonian solutions to the system of matrix equations \(A X = B\), \(X C = D\) are derived, respectively. Secondly, the optimal approximation solution \(\min_{X \in K} \parallel \widehat{X} - X \parallel\) is obtained, where \(K\) is the (anti-)Hermitian generalized (anti-)Hamiltonian solution set of the above system and \(\widehat{X}\) is the given matrix. Thirdly, the least squares (anti-)Hermitian generalized (anti-)Hamiltonian solutions are considered. In addition, algorithms about computing the least squares (anti-)Hermitian generalized (anti-)Hamiltonian solution and the corresponding numerical examples are presented.