  We obtain a tail bound for the least non-zero singular value of $A-z$ when
$A$ is a random matrix and $z$ is an eigenvalue of $A$ in a neighbourhood of a
given point $z_0$ in the bulk of the spectrum. The argument relies on a
resolvent comparison and a tail bound for Gauss-divisible matrices. The latter
can be obtained by the method of partial Schur decomposition. Using this bound
we prove that any finite collection of components of a right eigenvector
corresponding to an eigenvalue uniformly sampled from a neighbourhood of a
point in the bulk is Gaussian. A byproduct of the calculation is an asymptotic
formula for the odd moments of the absolute value of the characteristic
polynomial of real Gauss-divisible matrices.
