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## Perron Frobenius Theorems for the Numerical Range of Semi-Monic Matrix Polynomials

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Author(s) : Karl-Heinz Förster , Paul Kallus

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 03-2015

MSC 2000

15A48 Positive matrices and their generalizations; cones of matrices
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory

Abstract :
We present an extension of the Perron-Frobenius theory to the numerical ranges of semi-monic Perron-Frobenius polynomials, namely matrix polynomials of the form $Q(\lambda) = \lambda^m - (\lambda^lA_l + \cdots + A_0) = \lambda^m - A(\lambda),$ where the coefficients are entrywise nonnegative matrices. Our approach relies on the function $\beta \mapsto \text{numerical radius } A(\beta)$ and the infinite graph $G_m(A_0,\ldots, A_l)$. Our main result describes the cyclic distribution of the elements of the numerical range of $Q(\cdot)$ on the circles with radius $\beta$ satisfying $\beta^m =\text{numerical radius } A(\beta)$

Keywords : Perron-Frobenius theory, numerical range, matrix polynomials