A Note on the Level Sets of a Matrix Polynomial and Its Numerical Range

Abstract

Let P(λ) be an n x n matrix polynomial with bounded numerical range W(P) and let n > 2. If Ω is a connected subset of W(P), then the set

$$\mathop \cup \limits_{\omega \in \Omega } \left\{ {x \in {{\Bbb C}^n}:{x^*}P\left( \omega \right)x = 0,{x^*}x = 1} \right\}$$

is also connected. As a consequence, if P(λ) is selfadjoint, then every $\omega \in \overline {\left( {W\left( P \right)\backslash \mathbb{R}} \right)} \cap \mathbb{R}$ is a multiple root of the equation ${\text{ }}x_\omega ^*P\left( \lambda \right){x_\omega } = 0$ for some unit ${x_\omega } \in {{\Bbb C}^n}$.