On the q-numerical range of matrices and matrix polynomials

Abstract

The q-numerical range (0≤ q ≤ 1) of an n × n matrix polynomial is defined by In this article, we investigate the boundary and the shape of Wq (P), using the notion of local dimension. We also obtain that the q-numerical range of first-order matrix polynomials is always simply connected. Moreover, the special cases of 2×2 matrices and matrix polynomials are considered. In particular, the boundary of the q-numerical range of a 2×2 matrix polynomial of degree m lies on an algebraic curve of degree at most 8m.