The solutions to linear matrix equations AX = B, YA = D with k-involutory symmetries

摘要

Let R is an element of C-mxm and S is an element of C-nxn be nontrivial k -involutions if their minimal polynomials are both X-k - 1 for some k >= 2, i.e., Rk-1 = R-1 not equal R-1 and Sk-1 = S-1 not equal +/-1. We say that A is an element of C-mxn is (R, S, mu) -symmetric if RAS(-1) = zeta(mu)A, and A is (R, S, a, 0 -symmetric if RAS(-alpha) = zeta(mu)A with alpha, mu is an element of{0, 1,..., k - 1} and alpha not equal 0. Let i be one of the subsets of all (R, S, 0 -symmetric and (R, S, a, 0 -symmetric matrices. Given X E Cnxr, Y E Csxm, B E Cmxr. and D is an element of Cx ", we characterize the matrices A in i that minimize parallel to AX B parallel to(2) + parallel to YA - D parallel to(2) (Frobenius norm) under the assumption that R and S are unitary. Moreover, among the set 9(X, Y, B, D) subset of of the minimizers of parallel to AX B parallel to(2) + parallel to YA - D parallel to(2) = min, we find the optimal approximate matrix A E 5 (X, Y, B, D) that minimizes IIA GII to a given unstructural matrix G E C-mxn. We also present the necessary and sufficient conditions such that AX = B, YA = D is consistent in <9. If the conditions are satisfied, we characterize the consistent solution set of all such A. Finally, a numerical algorithm and some numerical examples are given to illustrate the proposed results. (C) 2017 Elsevier Ltd. All rights reserved.