Consider the generalized eigenvalue problem $$ Av = \lambda M v$$ where $A$ is symmetric positive semi-definite and $M$ is symmetric.
What happens to $\lambda$ if I add a multiple of the identity to $A$, i.e. if I instead solve $$ (A + \zeta I)v = \tilde{\lambda} M v $$ then how are $\tilde{\lambda}$ and $\lambda$ related?
Context: I need to perturb $A$ (which is symmetric positive semi-definite) to make it positive definite, so that I can use a particular numerical solver on $Av=\lambda Mv$. However, if I add $\zeta I$ to $A$, then I need to know how to correct the resulting eigenvalues. I also wonder whether I could perturb $M$ as well (without breaking its symmetry) to simplify the issue.
The closest question appears to be this one, though it is slightly different and unanswered.
The most I can really show is that $$ \zeta M^{-1}v=(\tilde{\lambda}-\lambda)v $$ which merely says that $M$ does have an effect on the magnitude of the perturbation of $\lambda$.