Suppose I have a large $N$-by-$N$ random, asymmetric, square matrix, $X$ (e.g. Gaussian i.i.d. elements with variance $\frac{1}{N}$). And I add a low-rank matrix, say for simplicity, rank one, $\lambda u v^T$: $$ J=\lambda u v^T +X$$
Suppose for example, that $\lambda$ is sufficiently large that there is an outlier eigenvalue in $J$. What is the overlap of the associated eigenvector of $J$ with $u$ and $v$?
I know the answer for the symmetric (Hermitian) case, following Baik-Ben Arous-Peche, and presented for example in Potters and Bouchaud A First Course in Random Matrix Theory Ch 14. I also know the relationship between the singular vectors of $J$ and $u$/$v$, following results by Benayach-Georges and Nadakuditi 2012. Are there analogous results for the eigenvectors of non-Hermitian matrices?