Inertia indices and eigenvalue inequalities for Hermitian matrices

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...

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Published inLinear & multilinear algebra Vol. 70; no. 8; pp. 1543 - 1552
Main Authors Zheng, Sai-Nan, Chen, Xi, Liu, Lily Li, Wang, Yi
Format Journal Article
LanguageEnglish
Published Abingdon Taylor & Francis 24.05.2022
Taylor & Francis Ltd
Subjects
(Hermitian) normalized Laplacian
eigenvalue
Eigenvalues
Graph theory
Hermitian matrix
Inequalities
Inertia
inertia index
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Summary: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.
ISSN:0308-1087
1563-5139
DOI:10.1080/03081087.2020.1765957