Jordan chains of $h$-cyclic matrices, II

TL;DR

This paper establishes a spectral characterization of $h$-cyclic matrices by proving that a previously known necessary condition on Jordan chains is also sufficient, and extends results to singular matrices and circulant matrices.

Contribution

It proves the sufficiency of a necessary condition for Jordan chains of $h$-cyclic matrices and provides new characterizations of circulant matrices.

Findings

Necessary condition on Jordan chains is sufficient for $h$-cyclic matrices.

Spectral characterization of nonsingular $h$-cyclic matrices.

New characterization of circulant matrices.

Abstract

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.