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

Let R@?C^m^x^m and S@?C^n^x^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^x^n is a generalized reflexive matrix if RGS=G. The set of all mxn generalized reflexive matrices is denoted by GRC^m^x^n. In this paper, a sufficient and necessary condition for the matrix equation AXB=D, where A@?C^p^x^m,B@?C^n^x^q and D@?C^p^x^q, to have a solution X@?GRC^m^x^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^x^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.

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