I have a nearly degenerate matrix $A$ and a perturbation $\epsilon B$, where $\epsilon$ is a small positive number. I know the eigenvalues and eigenvectors of $A$, but as I said, some of the eigenvalues of $A$ can be very close to each other.
I need to find an approximate analytical expression for the eigenvalues of $A + \epsilon B$. Conventional perturbation theory does not work, because of the small denominators. Moreover, I think degenerate perturabtion theory does not apply because the matrix $A$ is not really degenerate (it almost is).
Is there a theory that I can apply in this case?