  We prove that with high probability, every eigenvector of a random matrix is
delocalized in the sense that any subset of its coordinates carries a
non-negligible portion of its $\ell_2$ norm. Our results pertain to a wide
class of random matrices, including matrices with independent entries,
symmetric and skew-symmetric matrices, as well as some other naturally arising
ensembles. The matrices can be real and complex; in the latter case we assume
that the real and imaginary parts of the entries are independent.
