direct comparison of mixing angles is not exactly valid.

Third family single right-handed neutrino dominance (SRHND) [@KingSRND; @SFKing1; @SFKing2] is a mechanism that can explain the large atmospheric ($\theta_{23}$) and the solar LMA MSW ($\theta_{12}$) neutrino mixing angles and a small $\theta_{13}$. SRHND relies on the possibility that the neutrino mass matrix ($m_{LL}$) is dominated by the contributions coming solely from a single right-handed neutrino (for example $\nu^c_\tau$.) In this scheme a maximal $\theta_{23}$ angle arises when the tau right-handed neutrino $\nu^c_\tau$ couples to the left-handed muon $\nu_\mu$ and tau neutrino $\nu_\tau$ with equal strength. Similarly, if $\nu^c_\mu$ couples to $\nu_e$ and to $\nu_\mu$ with comparable strength then $\theta_{12}$ is large. The role of the (sub-dominant) muon neutrino is also important since it provides small perturbations to the $m_{LL}$ matrix (which otherwise has one heavy and two massless eigenstates), thus leading to a neutrino mass splitting $\Delta m^2_{12}=|m^2_{\nu_2}-m^2_{\nu_1}|$ compatible with experiment. In this section we summarize the theory behind SRHND and review the analytic results presented in Ref. [@SFKing2] for the case of the diagonal dominated right-handed neutrino mass matrix $M_{RR}$.

The see-saw formula for the left-handed neutrino matrix in Eq.  depends explicitly on $M_{RR}$. Although $M_{RR}$ might have a 