I will discuss a method of studying the joint probability density (JPD) of an eigenvalue and the associated 'non-orthogonality overlap factor' (also known as the 'eigenvalue condition number') of the left and right eigenvectors for non-selfadjoint Gaussian random matrices of size N x N. I will first derive the general finite N expression for the JPD of a real eigenvalue and the associated non-orthogonality factor in the real Ginibre ensemble, and then analyze its 'bulk' and 'edge' scaling limits. I will also discuss ongoing work on real elliptic ensembles. The ensuing distribution is maximally heavy-tailed, so that all integer moments beyond normalization are divergent. A similar calculation for the associated non-orthogonality factor in the complex Ginibre ensemble yields a distribution with the finite first moment complementing recent studies by P. Bourgade and G. Doubach. Its 'bulk' scaling limit yields a distribution whose first moment reproduces the well-known result of Chalker and Mehlig (1998), and I will provide the 'edge' scaling distribution for this case as well.
The presentation will be mainly based on the paper: Y.V. Fyodorov, Commun. Math. Phys. 363 (2), 579-603 (2018)