A non-Hermitian \( \overline{H_N} \) has eigenvalues \( E_\alpha \) corresponding to left \( \langle L_\alpha\vert \) and right \( \vert R_\alpha \rangle \) eigenstates. Thus
$$ \begin{align} \overline{H_N} = \sum_\alpha \vert R_\alpha\rangle E_\alpha \langle L_\alpha \vert \tag{14} \end{align} $$with bi-orthonormal \( \langle L_\alpha\vert R_\beta\rangle = \delta_\alpha^\beta \).