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 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.

翻译：本篇文章的主要目的是显示线性铅笔$\lambda A + B$的数值范围等于$\mathbb{C}美元,如果而且只有0.美元属于美元和B美元这一共同数字范围,我们还要证明,如果线性铅笔$\lambda A + B$的数值范围等于$\mathbb{C}美元和$A + A +, B + B + geq 0美元,那么美元和 $B$有一个共同的异向矢量。此外,我们改进了描述赫米契线性铅笔的经典结果。