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

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