  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.
