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| Title: | Eigenvalues and eigenvectors of finite, low rank perturbation of large random matrices |
| Authors: | Benaych-Georges, Florent
Nadakuditi, Raj Rao |
| Keywords: | random matrices
eigenvectors
eigenvalues |
| Issue Date: | 15-Oct-2009 |
| Abstract: | In this paper, 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 spectral
measure 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. Moreover, we uncover a remarkable 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. This critical threshold is
intimately related to the same aforementioned integral transforms.
We examine the consequence of this eigenvalue phase transition on the associated
eigenvectors and generalize our results to examine the singular values and vectors of finite,
low rank perturbations of rectangular random matrices. The analysis brings into sharp
focus the analogous connection with rectangular free probability. Various extensions of
our results are discussed. |
| Appears in Collections: | Electrical Engineering and Computer Science, Department of (EECS)
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Size | Format | |
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| r-dim_lambda_max.pdf | Main article | 605Kb | Adobe PDF | View/Open |
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