Hermitian, Hermitian -Symmetric, and Hermitian -Skew Symmetric Procrustes Problems

Let be a nontrivial unitary involution; i.e., . We say that is -symmetric (-skew symmetric) if = = −A). Let be one of the following subsets of : (i) hermitian matrices; (ii) hermitian -symmetric matrices; (iii) hermitian -skew symmetric matrices. Given , we characterize the matrices in that minimize (Frobenius norm), and, given an arbitrary , we find the unique matrix among the minimizers of in that minimizes . We also obtain necessary and sufficient conditions for existence of such that = , and, assuming that the conditions are satisfied, characterize the set of all such