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