  We investigate the evolution of a given eigenvector of a symmetric
(deterministic or random) matrix under the addition of a matrix in the Gaussian
orthogonal ensemble. We quantify the overlap between this single vector with
the eigenvectors of the initial matrix and identify precisely a "Cauchy-flight"
regime. In particular, we compute the local density of this vector in the
eigenvalues space of the initial matrix. Our results are obtained in a non
perturbative setting and are derived using the ideas of [O. Ledoit and S.
P\'ech\'e, Prob. Th. Rel. Fields, {\bf 151} 233 (2011)]. Finally, we give a
robust derivation of a result obtained in [R. Allez and J.-P. Bouchaud, Phys.
Rev. E {\bf 86}, 046202 (2012)] to study eigenspace dynamics in a
semi-perturbative regime.
