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In this article the well known "Perron-Frobenius theory" is investigatedinvolving the higher rank numerical range $\Lambda_{k}(A)$ of an irreducibleand entrywise nonnegative matrix $A$ and extending the notion of elements ofmaximum modulus in $\Lambda_{k}(A)$. Further, an application of this theory tothe $\Lambda_{k}(L(\lambda))$ of a Perron polynomial $L(\lambda)$ is elaboratedvia its companion matrix $C_{L}$.

The higher rank numerical range of nonnegative matrices