Occurrence conditions for eigenvalue rings

Characterize why and under which conditions rings of eigenvalues arise in truncated CMV matrices.

Background

The paper identifies rings of eigenvalues as a characteristic finite-size feature of truncated CMV matrices in the phase-II regime and relates them to severe eigenvalue conditioning. A perturbation of a finite Jordan block is mentioned as one mechanism capable of producing such a ring, but the authors leave open a more intuitive and general characterization of the mechanisms and conditions responsible for their occurrence.