Let the Matrix $U \in \mathbb{C}^{N \times N}$ is Unitary. Define $\widetilde{U} \in \mathbb{R}^{2N \times 2N}$ as:

$$ \widetilde{U}=\begin{bmatrix}

Re(U) & -Im(U)\\

\\

Im(U)& Re(U)

\end{bmatrix}$$

Prove that $\widetilde{U}$ is Orthonormal

$$ \widetilde{U}=\begin{bmatrix}

Re(U) & -Im(U)\\

\\

Im(U)& Re(U)

\end{bmatrix}$$

Prove that $\widetilde{U}$ is Orthonormal

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