A note on the nonzero spectrum of irreducible matrices

TL;DR

This paper extends existing conditions for the nonzero eigenvalues of primitive matrices to the broader class of irreducible matrices over positive reals and integers, providing a more comprehensive spectral characterization.

Contribution

It generalizes prior spectral conditions from primitive to irreducible matrices over _+ and _+ matrices, broadening the theoretical framework.

Findings

Extended spectral conditions to irreducible matrices

Unified understanding of eigenvalue multisets for broader matrix classes

Provides necessary and sufficient conditions for nonzero eigenvalues

Abstract

In this note we extend the necessary and sufficient conditions of Boyle-Handleman 1991 and Kim-Ormes-Roush 2000 for a nonzero eigenvalue multiset of primitive matrices over $\R_+$ and $\Z_+$, respectively, to irreducible matrices.