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 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.