Abstract
The numerical range of a selfadjoint matrix polynomial \( L\left( \lambda \right) = \sum\nolimits_j^\ell { = 0{\lambda ^j}{A_j}} \) is the set of points μ∈ℂ for which x* L(μ)x = 0 for some nonzero vector x. As for the classical eigenvalue problem (when L(λ) = λ I − A), the spectrum of L(λ) is contained in its numerical range. Properties of the numerical range are investigated with special emphasis on the cases when L(λ) has only real spectrum (and, possibly, the point at infinity) and when the coefficients of the matrix polynomial are real symmetric matrices.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Research supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Barkwell, L. and Lancaster, P. Overdamped and gyroscopic vibrating systems. J. Appl. Mech. 59 (1992), 176–181.
Barkwell, L., Lancaster, P. and Markus, A.S. Gyroscopically stabilized systems: A class of quadratic eigenvalue problems with real spectrum. Canad. J. Math. 44 (1992), 42–53.
Brickman, L. On The Field Of Values Of A Matrix. Proceedings of the American Mathematical Society 12, 1961.
Crawford, C.R. A stable generalized eigenvalue problem. SIAM Journal on Numerical Analysis 13 (1976), 854–860.
Gohberg, I., Lancaster, P. and Rodman, L. Spectral analysis of selfadjoint matrix polynomials. Annals of Math. (1980), 33–71.
Gohberg, I., Lancaster, P. and Rodman, L. Matrix Polynomials. Academic Press, New York, 1982.
Lancaster, P., Markus, A. and Matsaev, V. Perturbations of G-selfadjoint operators and operator polynomials with real spectrum, Operator Theory and its Applications, Birkhäuser Verlag), vol. 87 (1996), 207–221.
Lancaster, P. and Ye, Q. Variational properties and Rayleigh quotient algorithms for symmetric matrix pencils. In Operator Theory: Advances and Applications, vol. 40, pp. 247–278, Birkhäuser Verlag, 1989.
Li, C.K. and Rodman, L. Numerical range of matrix polynomials. SIAM Journal on Matrix Analysis and Applications, 15 (1994), 1256–1265.
Markus, A. Introduction to the Spectral Theory of Polynomial Operator Pencils, Vol. 71, Translations of Math Monographs, American Math. Soc, Providence, 1988.
Maroulas, J. and Psarrakos, P. Geometrical Properties of Numerical Range of Matrix Polynomials. Computers Math. Applic. 31, No. 4/5, pp. 41–47, 1996.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Additional information
Dedicated to Heinz Langer on the occasion of his 60th birthday
Rights and permissions
Copyright information
© 1998 Springer Basel AG
About this chapter
Cite this chapter
Lancaster, P., Maroulas, J., Zizler, P. (1998). The numerical range of selfadjoint matrix polynomials. In: Dijksma, A., Gohberg, I., Kaashoek, M.A., Mennicken, R. (eds) Contributions to Operator Theory in Spaces with an Indefinite Metric. Operator Theory Advances and Applications, vol 106. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8812-7_15
Download citation
DOI: https://doi.org/10.1007/978-3-0348-8812-7_15
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9782-2
Online ISBN: 978-3-0348-8812-7
eBook Packages: Springer Book Archive