Perturbation theory without closure relation

Question

I have learnt how to do perturbation theory in both degenerate and non-degenerate cases. However, since I have been doing quantum mechanics, and the operators (in finite dimensional situations, they are simply matrices) I have been playing with they all admit a complete basis formed by eigenvectors (what we call a closure relation in quantum mechanics). However, I am not sure what we can do if the closure relation no longer holds. I have never seen any text illustrating how to do that. For instance, if we want to solve the perturbation problem $$A+\epsilon B$$ with $$A=\begin{bmatrix}1&0\\0&0\end{bmatrix}$$ and $$B=\begin{bmatrix}0&1\\1&0\end{bmatrix}$$ There is only one non-zero eigenvector of $A$ and so we cannot form a basis with eigenvectors (the closure relation fails). What can we do in this situation? Of course, I am asking for a general solution instead of this $2\times 2$ example, which can be directly and easily solved.