The reflexive and anti-reflexive solutions of the matrix equation AHXB=C

In this paper, the necessary and sufficient conditions for the solvability of matrix equation AHXB=C over the reflexive or anti-reflexive matrices are given, and the general expression of the solution for a solvable case is obtained. Moreover, the relative optimal approximation problem is considered. The explicit expressions of the optimal approximation solution and the minimum norm solution are both provided.