I have this eigenvalue problem $[\varepsilon -\varepsilon_iI]n_i=0$, and I want to find $$\frac{\partial}{\partial \varepsilon}n_i\otimes n_i $$, where $n_i$ is eigenvector and $\varepsilon$ is symmetric matrix 3x3. I know that i will have three case depends on relation beween $\varepsilon_i$, first case is when we have distinct $\varepsilon_i$ or $\varepsilon_1 \neq \varepsilon_2 \neq \varepsilon_3$, second when we have two the same principal values $\varepsilon_1 = \varepsilon_2 \neq \varepsilon_3$, and third one is when $\varepsilon_1 = \varepsilon_2 = \varepsilon_3$.
Is there any book or paper which shows how this can be done in detail.
Yes, the first chapter of Kato's perturbation theory of linear operators will tell you more than you ever wanted to know.