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The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices. (English) Zbl 1226.15023
Let $$X_n$$ be an $$n\times n$$ Hermitian or symmetric random matrix. Let $$P_n$$ be an $$n\times n$$ Hermitian or symmetric matrix of rank $$r$$. The authors study the behaviour of the eigenvalues and eigenvectors of perturbations of $$X_n$$ by $$P_n$$, namely $$X_n+P_n$$, $$X_n(I_n+P_n)$$, $$(I_n+P_n)^{1/2}X_n(I_n+P_n)^{1/2}$$. Almost sure convergence of the extreme eigenvalues and of the projections of the corresponding eigenvectors on the eigenspaces of $$P_n$$ are proven.
The limiting eigenvalue is shown to depend explicitly on the limiting eigenvalue distribution of $$X_n$$. A threshold is found where the limit as $$n\to\infty$$ of the extreme eigenvalues of the perturbed matrix differ from those of $$X_n$$ if and only if the eigenvalues of $$P_n$$ are above that threshold. An analogous phase transition is found for the eigenvectors.

##### MSC:
 15B52 Random matrices (algebraic aspects) 15A18 Eigenvalues, singular values, and eigenvectors 46L54 Free probability and free operator algebras 60B20 Random matrices (probabilistic aspects) 62H25 Factor analysis and principal components; correspondence analysis 82B26 Phase transitions (general) in equilibrium statistical mechanics
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