Procrustes problems and associated approximation problems for matrices with k-involutory symmetries

We say that a matrix R∈C^n×n is k-involutary if its minimal polynomial is x^k-1 for some k?2, so R^k-1=R^-1 and the eigenvalues of R are 1,ζ,ζ²,…,ζ^k-1, where ζ=e^2πi/k. Let α,μ∈{0,1,…,k-1}. If R∈C^m×m, A∈C^m×n, S∈C^n×n and R and S are k-involutory, we say that A is (R,S,μ)-symmetric if RAS^-1=ζ^μA, and A is (R,S,α,μ)-symmetric if RAS^-α=ζ^μA.Let L be the class of m×n(R,S,μ)-symmetric matrices or the class of m×n(R,S,α,μ)-symmetric matrices. Given X∈C^n×t and B∈C^m×t, we characterize the matrices A in L that minimize ‖AX-B‖ (Frobenius norm), and, given an arbitrary W∈C^m×n, we find the unique matrix A∈L that minimizes both ‖AX-B‖ and ‖A-W‖. We also obtain necessary and sufficient conditions for existence of A∈L such that AX=B, and, assuming that the conditions are satisfied, characterize the set of all such A.