Hermitian, hermitian R-symmetric, and hermitian R-skew symmetric Procrustes problems

Abstract

Let R∈ C n×n be a nontrivial unitary involution; i.e., R=R ∗=R −1≠±I . We say that A∈ C n×n is R-symmetric ( R-skew symmetric) if RAR= A ( RAR=− A). Let S be one of the following subsets of C n×n : (i) hermitian matrices; (ii) hermitian R-symmetric matrices; (iii) hermitian R-skew symmetric matrices. Given Z, W∈ C n×m , we characterize the matrices A in S that minimize ∥ AZ− W∥ (Frobenius norm), and, given an arbitrary E∈ C n×n , we find the unique matrix among the minimizers of ∥ AZ− W∥ in S that minimizes ∥ A− E∥. We also obtain necessary and sufficient conditions for existence of A∈ S such that AZ= W, and, assuming that the conditions are satisfied, characterize the set of all such A.