HAL: hal-00575203, version 2

arXiv: 1103.2221

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v1 (2011-03-11)

v2 (2012-01-26)

The singular values and vectors of low rank perturbations of large rectangular random matrices

Florent Benaych-Georges 1, 2, Raj Rao Nadakuditi 3

(2011)

In this paper, we consider the singular values and singular vectors of finite, low rank perturbations of large rectangular random matrices. Specifically, we prove almost sure convergence of the extreme singular values and appropriate projections of the corresponding singular vectors of the perturbed matrix. As in the prequel, where we considered the eigenvalue aspect of the problem, the non-random limiting value is shown to depend explicitly on the limiting singular value distribution of the unperturbed matrix via an integral transforms that linearizes rectangular additive convolution in free probability theory. The large matrix limit of the extreme singular values of the perturbed matrix differs from that of the original matrix if and only if the singular values of the perturbing matrix are above a certain critical threshold which depends on this same aforementioned integral transform. We examine the consequence of this singular value phase transition on the associated left and right singular eigenvectors and discuss the finite $n$ fluctuations above these non-random limits.

1:	Laboratoire de Probabilités et Modèles Aléatoires (LPMA)
	CNRS : UMR7599 – Université Pierre et Marie Curie (UPMC) - Paris VI – Université Paris VII - Paris Diderot
2:	Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP)
	Polytechnique - X – CNRS : UMR7641
3:	Department of Electrical Engineering and Computer Science (EECS)
	University of Michigan

Subject

:

Mathematics/Probability

Keyword(s): Random matrices – Haar measure – free probability – phase transition – random eigenvalues – random eigenvectors – random perturbation – sample covariance matrices

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From: Florent Benaych-Georges <>

Submitted on: Thursday, 26 January 2012 14:47:28

Updated on: Thursday, 26 January 2012 16:44:24