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.
Acknowledgments
The authors thank the anonymous referee for his/her careful reading and valuable suggestions. This work was supported partially by the National Natural Science Foundation of China (Nos. 11601062, 11771065, 11871304), the Natural Science Foundation of Shandong Province of China (No. ZR2017MA025), the Fundamental Research Funds for the Central Universities (No. DUT18RC(4)068), and the Young Talents Invitation Program of Shandong Province.
Disclosure statement
No potential conflict of interest was reported by the author(s).