\begin{equation}
{\hat{h}_{bd}} =
\mathbf{U}^{\dagger}
\left[\begin{array}{cc}
\hat{h}_{LL} & \hat{h}_{LS} \\
\hat{h}_{SL} & \hat{h}_{SS} \end{array}\right]
\mathbf{U} =
\left[\begin{array}{cc}
\hat{h}_{++} & 0 \\
     0           & \hat{h}_{--} \end{array}\right],
\end{equation}