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

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


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


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Lancaster, P., Psarrakos, P., The numerical range of selfadjoint quadratic matrix polynomials, preprint (2000).Google Scholar
  2. [2]
    Li, C.-K., Rodman, L., Numerical range of matrix polynomials, SIAM J. Matrix Anal. Applic. 15 (1994), 1256–1265.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    Lyubich, Y., Separation of roots of matrix and operator polynomials, Integral Equations and Operator Theory 29 (1998), 52–62.MathSciNetCrossRefGoogle Scholar
  4. [4]
    Lyubich, Y., Markus, A.S., Connectivity of level sets of quadratic forms and Hausdorff-Toeplitz type theorems, Positivity 1 (1997), 239–254.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    Maroulas, J., Psarrakos, P., A connection between numerical ranges of selfadjoint matrix polynomials, Linear and Multilinear Algebra 44 (1998), 327–340.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    Ostrowski, A.M., Solutions of equations in Euclidean and Banach spaces, Academic Press, New York 1973.Google Scholar
  7. [7]
    Willard, S., General Topology, Addison-Wesley Publ. Company 1970.zbMATHGoogle Scholar

Copyright information

© Springer Basel AG 2002

Authors and Affiliations

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

Personalised recommendations