I have read about perturbation theory for symmetric matrices. Now suppose that $A$ is a symmetric real matrix, and $B$ is a real matrix (not necessarily symmetric). Suppose we know the eigenvalues of $A$, and are interested in the eigenvalues of $A + \epsilon B$, where $\epsilon > 0$ is a small quantity.
I want to apply the classical perturbation theory formulae to this problem. Are there any pitfalls I should be aware of, due to the fact that $B$ is non-symmetric?