How to compute the Eigen values of diagonal plus a rank one matrix? [duplicate]

I'm trying to find information on the eigenvalues of an n×n matrix $A$ such that

$A=D+J$ Where $D$ is some complex valued diagonal matrix, and $J$ is a rank one matrix, $J = uu^T$. How to compute the eigen values of $A$ in this case?

• Perhaps this might be useful: Bunch–Nielsen–Sorensen formula? Apr 29, 2018 at 15:33
• Just to avoid possible missunderstandings: do you assume the perturbation $J$ to be a general rank-$1$ matrix (which would be of the form $J = uv^T$ rather than $J = uu^T$), or do you really wish $J$ to be of the form $J = uu^T$ (and thus symmetric)? Apr 29, 2018 at 19:35
• Yes, I assume $J = uu^T$ Jul 26, 2018 at 10:19