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

TL;DR

This paper characterizes when the numerical range of a linear pencil equals the entire complex plane, linking it to the convex hull of the joint numerical range and exploring implications for Hermitian pencils.

Contribution

It provides a new characterization of linear pencils with the entire complex plane as their numerical range and extends classical results for Hermitian pencils.

Findings

Numerical range equals  iff 0 is in the convex hull of the joint numerical range.

If the numerical range is  and certain positivity conditions hold, then the matrices share a common isotropic vector.

Improves classical results on Hermitian linear pencils.

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 $A + A^*, B + B^* \geq 0$, then $A$ and $B$ have a common isotropic vector. Moreover, we improve the classical result which describes Hermitian linear pencils.