Spectral Analysis for Non-Hermitian Matrices and Directed Graphs

Abstract: We generalize classical results in spectral graph theory and linear algebra more broadly, from the case where the underlying matrix is Hermitian to the case where it is non-Hermitian. New admissibility conditions are introduced to replace the Hermiticity condition. We prove new variational estimates of the Rayleigh quotient for non-Hermitian matrices. As an application, a new Delsarte-Hoffman-type bound on the size of the largest independent set in a directed graph is developed. Our techniques consist in quantifying the impact of breaking the Hermitian symmetry of a matrix and are broadly applicable.