Numerical approximation of the boundary of numerical range of matrix polynomials

Abstract

The numerical range of an n × n matrix polynomialand plays an important role in the study of matrix polynomials. In this paper, we describe a methodology for the illustration of its boundary, ∂W (P ), using recent theoretical results on numerical ranges and algebraic curves.

A primer of Perron–Frobenius theory for matrix polynomials

“…The numerical range of the Perron polynomial L A (λ) = I λ 3 − A is illustrated in the right part of the figure (the unshaded star-shaped region). It is obtained using the inclusion-exclusion algorithm described in [25] and confirms the above discussion. The eigenvalues of A and L A (λ) are marked with +'s.…”

Perturbation of the q-numerical radius of a weighted shift operator

“…There have been a sequence of interesting papers with results related to the q-numerical range (cf. [1,[5][6][7]13,15,16,19,20]). In the case q = 1, Stout [18] gave a formula for the numerical radius of a general Hilbert-Schmidt class weighted shift operator, as the reciprocity of a minimal positive root of an entire analytic function.…”

Matrix polynomials with spectral radius equal to the numerical radius

“…For extensions to operator polynomials in Hilbert space we refer to [15]. Geometric properties of W (B) are gathered together in [13]. The numerical range is a tool to determine the stability radius [11] or to obtain factorizations [8] of a matrix polynomial.…”