Perturbation methods play a very important role wherever linear operators are deployed, often allowing the calculation of eigenvalues and eigenvectors in otherwise difficult situations. Therefore it behooves us to thoroughly investigate and understand the simplest cases, and so we ask, as both a pragmatic and illustrative example,
How may we find second order perturbation of the eigenvalues of matrix \begin{equation} \begin{pmatrix} E_1 & 0 \\ 0 & E_2 \end{pmatrix} + \begin{pmatrix} 0 & \omega \\ -\omega & 0 \end{pmatrix}, \end{equation} for both small $\omega$ and for large $\omega$?
The answer should explicate the principles and techniques involved in calculating these eigenvalues. The engaged reader may take extra time and discuss finding the eigenvectors as well.