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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

An inequality for selfadjoint operators on a Hilbert space


Author: Herbert J. Bernstein
Journal: Proc. Amer. Math. Soc. 100 (1987), 319-321
MSC: Primary 47A30; Secondary 65F15, 65J10
Corrigendum: Proc. Amer. Math. Soc. 101 (1987), 394.
MathSciNet review: 884472
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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

$\displaystyle {\left\vert {(g,{e_\lambda })} \right\vert^2} \leq \frac{{{{\left... ...ht\Vert}^2} - {{(g,Ag)}^2}}}{{{{\left\Vert {(A - \lambda I)g} \right\Vert}^2}}}$

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 [Enhancements On Off] (What's this?)

  • [1] G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge Univ. Press, Cambridge, 1934.
  • [2] H. Rutishauser, Handbook Series Linear Algebra: Simultaneous iteration method for symmetric matrices, Numer. Math. 16 (1970), no. 3, 205–223. MR 1553979, 10.1007/BF02219773
  • [3] J. H. Wilkinson, The algebraic eigenvalue problem, Clarendon Press, Oxford, 1965. MR 0184422

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DOI: https://doi.org/10.1090/S0002-9939-1987-0884472-8
Article copyright: © Copyright 1987 American Mathematical Society