V  V^* =  \begin{bmatrix} 0 & 0 & \sqrt{0.2} & 0 & \sqrt{0.8}\\ 1 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0\\ 0 & 0 & \sqrt{0.8} & 0 & -\sqrt{0.2} \end{bmatrix} \cdot \begin{bmatrix} 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0\\ \sqrt{0.2} & 0 & 0 & 0 & \sqrt{0.8}\\ 0 & 0 & 0 & 1 & 0\\ \sqrt{0.8} & 0 & 0 & 0 & -\sqrt{0.2}\end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 1\end{bmatrix}  \equiv I_5.
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