Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Spectra of nearly Hermitian random matrices

Sean O’Rourke and Philip Matchett Wood

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Abstract

We consider the eigenvalues and eigenvectors of matrices of the form $\mathbf{M}+\mathbf{P}$, where $\mathbf{M}$ is an $n\times n$ Wigner random matrix and $\mathbf{P}$ is an arbitrary $n\times n$ deterministic matrix with low rank. In general, we show that none of the eigenvalues of $\mathbf{M}+\mathbf{P}$ need be real, even when $\mathbf{P}$ has rank one. We also show that, except for a few outlier eigenvalues, most of the eigenvalues of $\mathbf{M}+\mathbf{P}$ are within $n^{-1}$ of the real line, up to small order corrections. We also prove a new result quantifying the outlier eigenvalues for multiplicative perturbations of the form $\mathbf{S}(\mathbf{I}+\mathbf{P})$, where $\mathbf{S}$ is a sample covariance matrix and $\mathbf{I}$ is the identity matrix. We extend our result showing all eigenvalues except the outliers are close to the real line to this case as well. As an application, we study the critical points of the characteristic polynomials of nearly Hermitian random matrices.

Résumé

Nous considérons les valeurs et les vecteurs propres de matrices de la forme $\mathbf{M}+\mathbf{P}$, où $\mathbf{M}$ est une matrice de Wigner $n\times n$ et $\mathbf{P}$ est une matrice arbitraire déterministe $n\times n$ de rang petit. Nous montrons que, génériquement, aucune des valeurs propres de $\mathbf{M}+\mathbf{P}$ n’est réelle, même quand $\mathbf{P}$ a rang un. Nous montrons aussi que, sauf pour un petit nombre d’exceptions, la plupart des valeurs propres de $\mathbf{M}+\mathbf{P}$ sont à distance au plus $n^{-1}$ de la droite réelle, à des corrections d’ordre petit près. Nous montrons aussi un nouveau résultat qui quantifie les valeurs propres exceptionnelles pour des perturbations multiplicatives de la forme $\mathbf{S}(\mathbf{I}+\mathbf{P})$, où $\mathbf{S}$ est une matrice de covariance empirique et $\mathbf{I}$ est la matrice identité. Nous étendons à ce cas notre résultat montrant que toutes les valeurs propres sauf les valeurs propres exceptionnelles sont proches de la droite réelle. Comme application, nous étudions les points critiques du polynôme caractéristique de matrices aléatoires presque hermitiennes.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 3 (2017), 1241-1279.

Dates
Received: 21 October 2015
Revised: 3 February 2016
Accepted: 17 March 2016
First available in Project Euclid: 21 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1500624041

Digital Object Identifier
doi:10.1214/16-AIHP754

Mathematical Reviews number (MathSciNet)
MR3689967

Zentralblatt MATH identifier
1381.60022

Subjects
Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)

Keywords
Random matrices Perturbation Random eigenvalues Random eigenvectors Wigner matrices Sample covariance matrices Outlier eigenvalues

Citation

O’Rourke, Sean; Wood, Philip Matchett. Spectra of nearly Hermitian random matrices. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 3, 1241--1279. doi:10.1214/16-AIHP754. https://projecteuclid.org/euclid.aihp/1500624041


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