Generalized reflexive solutions of the matrix equation AXB=D and an associated optimal approximation problem

Let R∈Cm×m and S∈Cn×n be nontrivial unitary involutions, i.e., RH=R=R−1≠ImRH=R=R−1≠Im and SH=S=S−1≠InSH=S=S−1≠In. We say that G∈Cm×n is a generalized reflexive matrix if RGS=GRGS=G. The set of all m×nm×n generalized reflexive matrices is denoted by GRCm×n. In this paper, a sufficient and necessary condition for the matrix equation AXB=D, where A∈Cp×m,B∈Cn×q and D∈Cp×q, to have a solution X∈GRCm×n is established, and if it exists, a representation of the solution set SXSX is given. An optimal approximation between a given matrix X̃∈Cm×n and the affine subspace SXSX is discussed, an explicit formula for the unique optimal approximation solution is presented, and a numerical example is provided.