Suppose I have a (psd) matrix $A$ and a diagonal matrix $\Sigma$. I want to know how the eigenvectors and eigenvalues of $A-\Sigma$ behave. The elements of $\Sigma$ are very small (compared to the eigenvalues of $A$).
I know that when $\Sigma$ is a multiple of identity, it only shifts the spectrum of $A$. Such a thing does not hold for other diagonal matrices. But are their results from matrix perturbation which tells how the eigenvectors change (upper and lower bounds) on such perturbation to the matrix?