Inertia laws and localization of real eigenvalues for generalized indefinite eigenvalue problems
This work provides theoretical bounds for eigenvalue localization in indefinite generalized eigenvalue problems, benefiting researchers in numerical linear algebra and applications requiring eigenvalue counts in intervals.
The paper investigates how the inertia (counts of positive, negative, zero eigenvalues) of a Hermitian matrix pencil changes under congruence transformations in the indefinite case, deriving sharp bounds that show the inertia of individual matrices still provides useful eigenvalue information, especially when one matrix is almost definite. This enables estimation of the number of real eigenvalues in an interval for generalized indefinite and nonlinear eigenvalue problems.
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.