Minimization Problems for Generalized Reflexive and Generalized Anti-Reflexive Matrices

Let R ∈ Cm×m and S ∈ Cn×n be nontrivial unitary involutions, i.e., R = R = R−1 = ±Im and S = S = S−1 = ±In. A ∈ Cm×n is said to be a generalized reflexive (anti-reflexive) matrix if RAS = A (RAS = −A). Let ρ be the set of m × n generalized reflexive (anti-reflexive) matrices. Given X ∈ Cn×p, Z ∈ Cm×p, Y ∈ Cm×q and W ∈ Cn×q, we characterize the matrices A in ρ that minimize ‖AX−Z‖2+‖Y HA−WH‖2, and, given an arbitrary à ∈ Cm×n, we find a unique matrix among the minimizers of ‖AX − Z‖2 + ‖Y A − WH‖2 in ρ that minimizes ‖A − Ã‖. We also obtain sufficient and necessary conditions for existence of A ∈ ρ such that AX = Z, Y A = W, and characterize the set of all such matrices A if the conditions are satisfied. These results are applied to solve a class of left and right inverse eigenproblems for generalized reflexive (anti-reflexive) matrices. Keywords—approximation, generalized reflexive matrix, generalized anti-reflexive matrix, inverse eigenvalue problem.