On the sensitivities of multiple eigenvalues

Publication: Research - peer-reviewJournal article – Annual report year: 2011

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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.
Original languageEnglish
JournalStructural and Multidisciplinary Optimization
Publication date2011
Volume44
Issue4
Pages583-587
ISSN1615-147X
DOIs
StatePublished
CitationsWeb of Science® Times Cited: 0

Keywords

  • Multiple eigenvalues, Sensitivity analysis, Symmetric polynomials
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