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

  • Panayiotis J. Psarrakos
Conference paper
  • 289 Downloads
Part of the Operator Theory: Advances and Applications book series (OT, volume 130)

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 \({x_\omega } \in {{\Bbb C}^n}\).

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Copyright information

© Springer Basel AG 2002

Authors and Affiliations

  • Panayiotis J. Psarrakos
    • 1
  1. 1.Department of Mathematics and StatisticsUniversity of ReginaReginaCanada

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