The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices

We consider the eigenvalues and eigenvectors of finite, low rank perturbations of random matrices. Specifically, we prove almost sure convergence of the extreme eigenvalues and appropriate projections of the corresponding eigenvectors of the perturbed matrix for additive and multiplicative perturbation models. The limiting non-random value is shown to depend explicitly on the limiting eigenvalue distribution of the unperturbed random matrix and the assumed perturbation model via integral transforms that correspond to very well-known objects in free probability theory that linearize non-commutative free additive and multiplicative convolution. Furthermore, we uncover a phase transition phenomenon whereby the large matrix limit of the extreme eigenvalues of the perturbed matrix differs from that of the original matrix if and only if the eigenvalues of the perturbing matrix are above a certain critical threshold. Square root decay of the eigenvalue density at the edge is sufficient to ensure that this threshold is finite. This critical threshold is intimately related to the same aforementioned integral transforms and our proof techniques bring this connection and the origin of the phase transition into focus. Consequently, our results extend the ✩ F.B.G.’s work was partially supported by the Agence Nationale de la Recherche grant ANR-08-BLAN-0311-03. R.R.N.’s research was partially supported by an Office of Naval Research postdoctoral fellowship award and grant N00014-07-1-0269. R.R.N. thanks Arthur Baggeroer for his feedback, support and encouragement. We thank Alan Edelman for feedback and encouragement and for facilitating this collaboration by hosting F.B.G.’s stay at M.I.T. We gratefully acknowledge the Singapore-MIT alliance for funding F.B.G.’s stay. * Corresponding author. E-mail addresses: [email protected] (F. Benaych-Georges), [email protected] (R.R. Nadakuditi). URLs: http://www.cmapx.polytechnique.fr/~benaych/ (F. Benaych-Georges), http://www.eecs.umich.edu/~rajnrao/ (R.R. Nadakuditi). 0001-8708/$ – see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.aim.2011.02.007 F. Benaych-Georges, R.R. Nadakuditi / Advances in Mathematics 227 (2011) 494–521 495 class of ‘spiked’ random matrix models about which such predictions (called the BBP phase transition) can be made well beyond the Wigner, Wishart and Jacobi random ensembles found in the literature. We examine the impact of this eigenvalue phase transition on the associated eigenvectors and observe an analogous phase transition in the eigenvectors. Various extensions of our results to the problem of non-extreme eigenvalues are discussed. © 2011 Elsevier Inc. All rights reserved. MSC: 15A52; 46L54; 60F99