The (P,Q) generalized 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 (P,Q) generalized reflexive (or anti-reflexive) matrix with respect to the generalized reflection matrix dual (P,Q) if A=PAQ (or A=-PAQ). This paper establishes the necessary and sufficient conditions for the existence of and the expressions for the (P,Q) generalized reflexive and anti-reflexive 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 has been provided.