Perron–Frobenius type results on the numerical range

Abstract

We present results connecting the shape of the numerical range to intrinsic properties of a matrix A. When A is a nonnegative matrix, these results are to a large extent analogous to the Perron–Frobenius theory, especially as it pertains to irreducibility and cyclicity in the combinatorial sense. Special attention is given to polygonal, circular and elliptic numerical ranges. The main vehicles for obtaining these results are the Hermitian and skew-Hermitian parts of A, as well as Levinger's transformation aA+(1−a)A ∗ .