Let $A$ be a real, symmetric $n \times n$ matrix with eigenvalues $\lambda_1, \lambda_2, \dots, \lambda_n$ where at least two of the eigenvalues are zero. Let $V$ be a real, symmetric $n \times n$ matrix. Consider $A_\varepsilon = A + \varepsilon V$, where $\varepsilon$ is a small number. I am looking to determine the change in the eigenvalues induced by the perturbation $\varepsilon V$.
My question is very similar to this post, except that in my case, I have repeated zero eigenvalues. I have been searching the literature for a method that would work in this case, but I haven't been able to find anything.
Let $\lambda$ be an eigenvalue of $A$ and $P$ the projection onto the associated eigenspace. If $P VP $ has distinct eigenvalues $\tilde{\lambda}_1, \dots ,\tilde{\lambda}_m$ with each eigenspace being one dimensional, then $\lambda$ splits into $m$ eigenvalues $\lambda_1 (\varepsilon) , \dots , \lambda_m(\varepsilon)$ of $A_\varepsilon $ which are given by $$ \lambda_i (\varepsilon) = \lambda +\varepsilon \tilde{\lambda}_i +o(\varepsilon).$$
For a derivation and more general cases, see Chapter 2 (in particular §2 section 3 "the reduction process") in the book "Pertubation Theory For Linear Operators" by Tosio Kato.