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

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)