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