Complex spectrally arbitrary zero-nonzero patterns

In this article, we highlight interesting properties of complex spectrally arbitrary zero-nonzero patterns. In particular, we investigate irreducible complex spectrally arbitrary zero-nonzero patterns for which all Jacobians are zero at every nilpotent realization. We also study complex spectrally arbitrary patterns whose corresponding directed graph does not contain a two-cycle. Lastly, we provide a complete list of all 3 x 3 and 4 x 4 complex spectrally arbitrary zero-nonzero patterns.