Nongeneric eigenvalue perturbations of Jordan blocks

We show that if an n X n Jordan block is perturbed by an O(ε) upper k-Hessenberg matrix (k subdiagonals including the main diagonal), then generically the eigenvalues split into p rings of size k and one of size r (if r ≠ 0), where n = pk + r. This generalizes the familiar result (k = n, p = 1, r = 0) that generically the eigenvalues split into a ring of size n. We compute the radii of the rings to first order and generalize the result in a number of directions involving multiple Jordan blocks of the same size.