The reflexive and anti-reflexive solutions of the matrix equation AX=B

An n×n complex matrix P is said to be a generalized reflection matrix if PH=P and P2=I. An n×n complex matrix A is said to be a reflexive (or anti-reflexive) matrix with respect to the generalized reflection matrix P if A=PAP (or A=−PAP). This paper establishes the necessary and sufficient conditions for the existence of and the expressions for the reflexive and anti-reflexive with respect to a generalized reflection matrix P solutions of the matrix equation AX=B. In addition, in corresponding solution set of the equation, the explicit expression of the nearest matrix to a given matrix in the Frobenius norm have been provided.