The Hermitian reflexive solutions to the matrix inverse problem AX=BAX=B

In this paper, the necessary and sufficient conditions for the solvability of the matrix inverse problem AX=BAX=B in Hermitian reflexive matrix set HJn×nHJn×n are given, and the solution expression of this matrix inverse problem is also presented. Moreover, an optimal approximation to any matrix in the above solution set in Frobenius norm is provided.