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The textbook says that since $U$ is a unitary matrix, its eigenvalue should be of the form $e^{2 \pi i \theta}$. The thing I don't understand is why it's not $e^{i \theta}$ because it also lies on the unit circle and has a magnitude of 1. Are they the same? If yes, can you please show how?

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in case of $e^{2\pi\cdot i\cdot \theta}$ the values of $\theta\in [0,1]$, in the case of $e^{i\cdot \theta}$ the values of $\theta \in [0,2\pi]$. it's just a different convention but they are equivalent (simple bijection between the two)

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