The higher rank numerical range of nonnegative matrices

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