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

Let R∈Cm×m and S∈Cn×n be nontrivial k-involutions if their minimal polynomials are both xk−1 for some k≥2, i.e., Rk−1=R−1≠±I and Sk−1=S−1≠±I. We say that A∈Cm×n is (R,S,μ)-symmetric if RAS−1=ζμA, and A is (R,S,α,μ)-symmetric if RAS−α=ζμA with α,μ∈{0,1,…,k−1} and α≠0. Let S be one of the subsets of all (R,S,μ)-symmetric and (R,S,α,μ)-symmetric matrices. Given X∈Cn×r, Y∈Cs×m, B∈Cm×r and D∈Cs×n, we characterize the matrices A in S that minimize ‖AX−B‖2+‖YA−D‖2 (Frobenius norm) under the assumption that R and S are unitary. Moreover, among the set S(X,Y,B,D)⊂S of the minimizers of ‖AX−B‖2+‖YA−D‖2=min, we find the optimal approximate matrix A∈S(X,Y,B,D) that minimizes ‖A−G‖ to a given unstructural matrix G∈Cm×n. We also present the necessary and sufficient conditions such that AX=B,YA=D is consistent in S. 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.