I have a symmetric matrix $\Sigma$ and a diagonal matrix $D$. I know that eigenvalue decomposition of $\Sigma = Q\Lambda Q^T$. Now I would like to find the eigenvalues and the eigenvectors of the following matrix: $$ \Sigma_v = \Sigma-D $$ I understand that if $D=nI$, things are trivial. But in a general case, what is the best approach to understand how much the eigenvalues and vector change from that of $\Sigma$, due to the diagonal perturbation $D$?
Unfortunately, there isn't really anything special about the case where $D$ is diagonal (as opposed to an arbitrary symmetric perturbation). In particular, for any $P = UDU^T$, the matrix $\Sigma - P$ is similar to $U^T \Sigma U - D$.
As far as eigenvalues go, we have Weyl's inequalities, which I think is as much as you can get without more constraints on these matrices.
For eigenvectors, the inequalities are tricky, but do exist. A good reference for that information is chapter 7 of Bhatia's Matrix Analysis.
Note that all I've really given you here are inequalities. If you want the precise eigenvalues and eigenvectors of the updated matrix, there isn't any method that's going to give that you significantly more quickly than just computing them from scratch.