Complex spectrally arbitrary zero–nonzero patterns

Abstract

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 × 3 and 4 × 4 complex spectrally arbitrary zero–nonzero patterns.

Keywords:

• spectrally arbitrary
• zero–nonzero patterns
• complex matrices

AMS Subject Classification:

• 15A18

Acknowledegments

The authors would like to thank the anonymous referee for very carefully reading through this article and making several helpful suggestions.