The higher rank numerical range of nonnegative matrices

TL;DR

This paper explores the properties of the higher rank numerical range of nonnegative matrices, extending Perron-Frobenius theory, and applies these findings to Perron polynomials through their companion matrices.

Contribution

It extends Perron-Frobenius theory to higher rank numerical ranges of nonnegative matrices and applies this to Perron polynomials via companion matrices.

Findings

Extended Perron-Frobenius theory to higher rank numerical ranges.

Characterized elements of maximum modulus in these ranges.

Applied theory to Perron polynomials using companion matrices.

Abstract

In this article the well known "Perron-Frobenius theory" is investigated involving the higher rank numerical range $\Lambda_{k}(A)$ of an irreducible and entrywise nonnegative matrix $A$ and extending the notion of elements of maximum modulus in $\Lambda_{k}(A)$. Further, an application of this theory to the $\Lambda_{k}(L(\lambda))$ of a Perron polynomial $L(\lambda)$ is elaborated via its companion matrix $C_{L}$.