Hermitian, hermitian R-symmetric, and hermitian R-skew symmetric Procrustes problems Original Research Article

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