Inertia indices and eigenvalue inequalities for Hermitian matrices

TL;DR

This paper introduces a unified approach to eigenvalue inequalities for Hermitian matrices using inertia indices, covering classical results and generalizations for graph Laplacians and digraphs.

Contribution

It provides a novel characterization of eigenvalue inequalities via inertia indices and unifies classical and new inequalities for various Hermitian matrices.

Findings

Unified proof of classical eigenvalue inequalities

Generalization to Laplacian matrices of graphs and digraphs

Applicable to Hermitian matrices of the second kind of digraphs

Abstract

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.