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Main Title: Perron Frobenius Theorems for the Numerical Range of Semi-Monic Matrix Polynomials
Author(s): Förster, Karl-Heinz
Kallus, Paul
Type: Research Paper
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 $\beta^m =\text{numerical radius } A(\beta)$
Subject(s): Perron-Frobenius theory
numerical range
matrix polynomials
Issue Date: 12-Mar-2015
Date Available: 17-Dec-2021
Language Code: en
DDC Class: 510 Mathematik
MSC 2000: 15A48 Positive matrices and their generalizations; cones of matrices
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
15A22 Matrix pencils
15B48 Positive matrices and their generalizations; cones of matrices
Series: Preprint-Reihe des Instituts für Mathematik, Technische Universität Berlin
Series Number: 2015, 03
ISSN: 2197-8085
TU Affiliation(s): Fak. 2 Mathematik und Naturwissenschaften » Inst. Mathematik
Appears in Collections:Technische Universität Berlin » Publications

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