  We consider eigenvectors of the Hamiltonian $H_0$ perturbed by a generic
perturbation $V$ modelled by a random matrix from the Gaussian Unitary Ensemble
(GUE). Using the supersymmetry approach we derive analytical results for the
statistics of the eigenvectors, which are non-perturbative in $V$ and valid for
an arbitrary deterministic $H_0$. Further we generalise them to the case of a
random $H_0$, focusing, in particular, on the Rosenzweig-Porter model. Our
analytical predictions are confirmed by numerical simulations.
