The generalized anti-reflexive solutions for a class of matrix equations (BX = C, XD = E)

In this paper, the generalized anti-reflexive solution for matrix equations (BX = C, XD = E), which arise in left and right inverse eigenpairs problem, is considered. With the special properties of generalized anti-reflexive matrices, the necessary and sufficient conditions for the solvability and a general expression of the solution are obtained. Furthermore, the related optimal approximation problem to a given matrix over the solution set is solved. In addition, the algorithm and the example to obtain the unique optimal approximation solution are given.