Eigenvector Delocalization for Non-Hermitian Random Matrices and Applications

TL;DR

This paper proves that eigenvectors of certain non-Hermitian random matrices are delocalized, meaning their mass is evenly spread out, with results matching simulations and applicable to complex and real entries.

Contribution

It establishes optimal delocalization bounds for eigenvectors of independent-entry random matrices, extending previous results and including new bounds for normal vectors to random hyperplanes.

Findings

Eigenvectors are delocalized with high probability.

Bounds match numerical simulations, indicating optimality.

Results apply to both complex and real matrices.

Abstract

Improving upon results of Rudelson and Vershynin, we establish delocalization bounds for eigenvectors of independent-entry random matrices. In particular, we show that with high probability every eigenvector is delocalized, meaning any subset of its coordinates carries an appropriate proportion of its mass. Our results hold for random matrices with genuinely complex as well as real entries. In both cases, our bounds match numerical simulations, up to lower order terms, indicating the optimality of our results. As an application of our methods, we also establish delocalization bounds for normal vectors to random hyperplanes. The proofs of our main results rely on a least singular value bound for genuinely complex rectangular random matrices, which generalizes a previous bound due to the first author, and may be of independent interest.