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

Abstract

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 , ζ 2 , … , ζ k - 1 , where ζ = e 2 π i / 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 μ A , and A is ( R , S , α , μ ) -symmetric if RAS - α = ζ μ A . 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.