Matrix pencils with the numerical range equal to the whole complex plane

Abstract

The main purpose of this article is to show that the numerical range of a linear pencil $\lambda A+B$ is equal to $\mathbb{C}$ if and only if $0$ belongs to the convex hull of the joint numerical range of $A$ and $B$. We also prove that if the numerical range of a linear pencil $\lambda A+B$ is equal to $\mathbb{C}$ and $A+A^{\ast },B+B^{\ast }\ge 0$, then $A$ and $B$ have a common isotropic vector. Moreover, we improve the classical result which describes Hermitian linear pencils.

arXiv comment9