  McDonald and Paparella [Linear Algebra Appl. 498 (2016), 145--159] gave a
necessary condition on the structure of Jordan chains of $h$-cyclic matrices.
In this work, that necessary condition is shown to be sufficient. As a
consequence, we provide a spectral characterization of nonsingular, $h$-cyclic
matrices. In addition, we provide results for the Jordan chains corresponding
to the eigenvalue zero of singular matrices. Along the way, a new
characterization of circulant matrices is given.
