Structural and Multidisciplinary Optimization

, Volume 44, Issue 4, pp 583–587 | Cite as

On the sensitivities of multiple eigenvalues

Brief Note


We consider the generalized symmetric eigenvalue problem where matrices depend smoothly on a parameter. It is well known that in general individual eigenvalues, when sorted in accordance with the usual ordering on the real line, do not depend smoothly on the parameter. Nevertheless, symmetric polynomials of a number of eigenvalues, regardless of their multiplicity, which are known to be isolated from the rest depend smoothly on the parameter. We present explicit readily computable expressions for their first derivatives. Finally, we demonstrate the utility of our approach on a problem of finding a shape of a vibrating membrane with a smallest perimeter and with prescribed four lowest eigenvalues, only two of which have algebraic multiplicity one.


Multiple eigenvalues Sensitivity analysis Symmetric polynomials 


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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Jens Gravesen
    • 1
  • Anton Evgrafov
    • 1
  • Dang Manh Nguyen
    • 1
  1. 1.Department of MathematicsTechnical University of DenmarkKgs. LyngbyDenmark

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