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

Abstract

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