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arXiv · 2205.05051

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

Vadym Koval·Patryk Pagacz
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Abstract

The main purpose of this article is to show that the numerical range of a linear pencil $λA + B$ 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 $λA + B$ 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.

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Vadym Koval, Patryk Pagacz. 2022-05-20. Matrix pencils with the numerical range equal to the whole complex plane. https://doi.org/10.1016/j.laa.2022.10.021

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