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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.

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
Full Text: DOI
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