Hostname: page-component-7dd5485656-dk7s8 Total loading time: 0 Render date: 2025-10-31T01:29:05.262Z Has data issue: false hasContentIssue false
  • English
  • Français

The Generalized Rayleigh Quotient

Published online by Cambridge University Press:  20 November 2018

M. V. Pattabhiraman
M. V. Pattabhiraman*
Affiliation:
University of Calgary, Alberta, Canada
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we generalize the concept of the Rayleigh quotient to a complex Banach space. Lord Rayleigh investigated the quotient

(1)

considered as a function of the components of q, in the case of a symmetric matrix pencil Aλ+C with A positive definite. It is known that R(q) has a stationary value when q is a characteristic vector of Aλ+C and that

(2)

where q i is a characteristic vector corresponding to the characteristic value λ i

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 17 , Issue 2 , 01 June 1974 , pp. 251 - 256
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Hille, E. and Philips, R. S., Functional analysis and semigroups, American Mathematical Society Colloquium Publications, 1965.Google Scholar
2. Kummer, H., Zur Praktischen Behandlung nichtlinearer Eigenwertaufgaben abgeschlossener linear er operator en, Giessen, 1964, Mitteilunger aus dem Mathem Seminar Giessen.Google Scholar
3. Lancaster, P., Lambda matrices and vibrating systems, Pergamon Press, 1966.Google Scholar
4. Ostrowski, A. M., On the convergence of the Rayleigh quotient iteration for the computation of the characteristic vectors, Archive for Rational Mechanics and Analysis, Vol. 1, No. 3, 1958, pp. 233-241.Google Scholar
5. Pattabhiraman, M. V. and Lancaster, P., Spectral properties of a polynomial operator, Research Paper #45, Dept. of Math., Univ. of Calgary, 1968, Numenshe Mathematik, Vol. 13, 1969, pp. 247-259.Google Scholar
6. Wilansky, A., Functional Analysis, Blaisdell Publishing Company, New York, 1964.Google Scholar