Eigenvectors of a matrix under random perturbation

TL;DR

This paper develops a perturbative expansion method to analyze how eigenvectors of large Hermitian matrices are affected by small random perturbations, providing insights into spectral measure changes.

Contribution

It introduces a perturbative expansion approach for eigenvector coordinates under random perturbations, based on elementary computations and spectral measures.

Findings

Provides explicit formulas for eigenvector perturbations

Analyzes spectral measure changes due to random noise

Applicable to large Hermitian matrices with independent entries

Abstract

In this text, based on elementary computations, we provide a perturbative expansion of the coordinates of the eigenvectors of a Hermitian matrix of large size perturbed by a random matrix with small operator norm whose entries in the eigenvector basis of the first one are independent, centered, with a variance profile. This is done through a perturbative expansion of spectral measures associated to the state defined by a given vector.