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We consider matrices formed by a random $N\times N$ matrix drawn from theGaussian Orthogonal Ensemble (or Gaussian Unitary Ensemble) plus a rank-oneperturbation of strength $\theta$, and focus on the largest eigenvalue, $x$,and the component, $u$, of the corresponding eigenvector in the directionassociated to the rank-one perturbation. We obtain the large deviationprinciple governing the atypical joint fluctuations of $x$ and $u$.Interestingly, for $\theta>1$, in large deviations characterized by a smallvalue of $u$, i.e. $u<1-1/\theta$, the second-largest eigenvalue pops out fromthe Wigner semi-circle and the associated eigenvector orients in the directioncorresponding to the rank-one perturbation. We generalize these results to theWishart Ensemble, and we extend them to the first $n$ eigenvalues and theassociated eigenvectors.

Large deviations for the largest eigenvalues and eigenvectors of spiked random matrices