The solutions to linear matrix equations [formula omitted] with [formula omitted]-involutory symmetries.

Let R ∈ C m × m and S ∈ C n × n be nontrivial k -involutions if their minimal polynomials are both x k − 1 for some k ≥ 2 , i.e., R k − 1 = R − 1 ≠ ± I and S k − 1 = S − 1 ≠ ± I . We say that A ∈ C m × n is ( R , S , μ ) -symmetric if R A S − 1 = ζ μ A , and A is ( R , S , α , μ ) -symmetric if R A S − α = ζ μ A 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 ∈ C n × r , Y ∈ C s × m , B ∈ C m × r and D ∈ C s × n , we characterize the matrices A in S that minimize ‖ A X − B ‖ 2 + ‖ Y A − 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 ‖ A X − B ‖ 2 + ‖ Y A − 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 ∈ C m × n . We also present the necessary and sufficient conditions such that A X = B , Y A = 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. [ABSTRACT FROM AUTHOR]