Least non-zero singular value and the distribution of eigenvectors of non-Hermitian random matrices

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 neighborhood of a given point z0 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 neighborhood 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.