Universality for Diagonal Eigenvector Overlaps of non-Hermitian Random Matrices

We prove the universality of the joint distribution of an eigenvalue and the corresponding diagonal eigenvector overlap, in the bulk and at the edge, for eigenvalues of complex matrices and real eigenvalues of real matrices. As part of the proof we obtain a bound for the least non-zero singular value of $X-z$ when $z$ is an edge eigenvalue and a bound for the inner product between left and right singular vectors of $X-z$ when $\mathrm{\mid }z\mathrm{\mid }=1+O(N^{-1\mathrm{/}2})$.