  We extend the so-called "single ring theorem"[1], also known as the
Haagerup-Larsen theorem[2], by showing that in the limit when the size of the
matrix goes to infinity a particular correlator between left and right
eigenvectors of the relevant non-hermitian matrix $X$, being the spectral
density weighted by the squared eigenvalue condition number, is given by a
simple formula involving only the radial spectral cumulative distribution
function of $X$. We show that this object allows to calculate the conditional
expectation of the squared eigenvalue condition number. We give examples and we
provide cross-check of the analytic prediction by the large scale numerics.
