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

TL;DR

This paper investigates how the inertia of matrices in generalized indefinite eigenvalue problems changes under congruence transformations, providing bounds and insights useful for estimating real eigenvalues.

Contribution

It extends Sylvester's law of inertia to indefinite cases, deriving bounds that relate matrix inertia to eigenvalue localization in generalized problems.

Findings

Inertia bounds for generalized indefinite eigenproblems

Useful estimates for counting real eigenvalues

Insights into eigenvalue localization in nonlinear problems

Abstract

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.