An inequality for selfadjoint operators on a Hilbert space
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- by Herbert J. Bernstein PDF
- Proc. Amer. Math. Soc. 100 (1987), 319-321 Request permission
Corrigendum: Proc. Amer. Math. Soc. 101 (1987), 394.
Abstract:
An elementary inequality of use in testing convergence of eigenvector calculations is proven. If ${e_\lambda }$ is a unit eigenvector corresponding to an eigenvalue $\lambda$ of a selfadjoint operator $A$ on a Hilbert space $H$, then \[ {\left | {(g,{e_\lambda })} \right |^2} \leq \frac {{{{\left \| g \right \|}^2}{{\left \| {Ag} \right \|}^2} - {{(g,Ag)}^2}}}{{{{\left \| {(A - \lambda I)g} \right \|}^2}}}\] for all $g$ in $H$ for which $Ag \ne \lambda g$. Equality holds only when the component of $g$ orthogonal to ${e_\lambda }$ is also an eigenvector of $A$.References
- G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge Univ. Press, Cambridge, 1934.
- H. Rutishauser, Handbook Series Linear Algebra: Simultaneous iteration method for symmetric matrices, Numer. Math. 16 (1970), no. 3, 205–223. MR 1553979, DOI 10.1007/BF02219773
- J. H. Wilkinson, The algebraic eigenvalue problem, Clarendon Press, Oxford, 1965. MR 0184422
Additional Information
- © Copyright 1987 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 100 (1987), 319-321
- MSC: Primary 47A30; Secondary 65F15, 65J10
- DOI: https://doi.org/10.1090/S0002-9939-1987-0884472-8
- MathSciNet review: 884472