The Hermitian {P,k+1}-(anti-)reflexive solutions of a linear matrix equation

In this paper, we first establish the necessary and sufficient conditions for the existence and the explicit expressions of the Hermitian {P,k+1}-(anti-)reflexive solutions of the matrix equation AX=BAX=B, and meanwhile the best approximation solution is considered. Then, if the solvability conditions are not satisfied, the least squares Hermitian {P,k+1}-(anti-)reflexive solutions and the least squares Hermitian {P,k+1}-(anti-)reflexive solutions with the minimum norm of the above matrix equation are respectively derived. In addition, two algorithms are shown to compute the least squares Hermitian {P,k+1}-(anti-)reflexive solutions, and the corresponding numerical examples are also given to illustrate the feasibility of the algorithms.