Spectral Theorem for Unitary Operators

Theorem

1. Let $U\in \mathcal{L}(V)$ be Unitary
2. Then, a Orthonormal Basis $\beta$ of unit vectors $V$ exists such that $[U{]}_{\beta }=diag(e^{i\theta },\dots ,e^{i\theta n})$, where $e^{i\theta },\dots ,e^{i\theta n}$ are the Eigenvalues of $U$