[QUE/QM-23011] Time Independent Perturbation Theory

\(\newcommand{\ket}[1]{\vert#1\rangle}\)
\(\newcommand{\bra}[1]{\langle#1\vert}\)In an orthonormal basis consisting of three elements $\{\ket{1},\ket{2}, \ket{3} \}$, the Hamiltonian of a system is given by \begin{equation} H = \ket{1}\bra{1} + i\epsilon \ket{1}\bra{2} - i\epsilon \ket{2}\bra{1} + 2 \ket{2}\bra{2} + 2\epsilon \ket{2}\bra{3} + 2 \epsilon\ket{3}\bra{2} + \ket{3}\bra{3} \end{equation} Find the eigenvalues and eigenvectors of the total Hamiltonian \(H\) upto first order in \(\epsilon\). [Hint: First construct the matrix representation for $H$ in the given basis.]