  We study the eigenvalues and the eigenvectors of $N\times N$ structured
random matrices of the form $H = W\tilde{H}W+D$ with diagonal matrices $D$ and
$W$ and $\tilde{H}$ from the Gaussian Unitary Ensemble. Using the supersymmetry
technique we derive general asymptotic expressions for the density of states
and the moments of the eigenvectors. We find that the eigenvectors remain
ergodic under very general assumptions, but a degree of their ergodicity
depends strongly on a particular choice of $W$ and $D$. For a special case of
$D=0$ and random $W$, we show that the eigenvectors can become critical and are
characterized by non-trivial fractal dimensions.
