Inertia laws and localization of real eigenvalues for generalized indefinite eigenvalue problems

Yuji Nakatsukasa, Vanni Noferini*

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review


Sylvester's law of inertia states that the number of positive, negative and zero eigenvalues of Hermitian matrices is preserved under congruence transformations. The same is true of generalized Hermitian definite eigenvalue problems, in which the two matrices are allowed to undergo different congruence transformations, but not for the indefinite case. In this paper we investigate the possible change in inertia under congruence for generalized Hermitian indefinite eigenproblems, and derive sharp bounds that show the inertia of the two individual matrices often still provides useful information about the eigenvalues of the pencil, especially when one of the matrices is almost definite. A prominent application of the original Sylvester's law is in finding the number of eigenvalues in an interval. Our results can be used for estimating the number of real eigenvalues in an interval for generalized indefinite and nonlinear eigenvalue problems.

Original languageEnglish
Pages (from-to)272-296
Number of pages25
JournalLinear Algebra and Its Applications
Publication statusPublished - 1 Oct 2019
MoE publication typeA1 Journal article-refereed


  • Congruence transformation
  • Generalized indefinite eigenvalue problem
  • Nonlinear eigenvalue problems
  • Number of eigenvalues in an interval
  • Sylvester's law of inertia

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