  We present a characterization of eigenvalue inequalities between two
Hermitian matrices by means of inertia indices. As applications, we deal with
some classical eigenvalue inequalities for Hermitian matrices, including the
Cauchy interlacing theorem and the Weyl inequality, in a simple and unified
approach. We also give a common generalization of eigenvalue inequalities for
(Hermitian) normalized Laplacian matrices of simple (signed, weighted,
directed) graphs. Our approach is also suitable for Hermitian matrices of the
second kind of digraphs recently introduced by Mohar.
