Eigenvalues of a rotation

MediumLinear Algebra~4m

What are the eigenvalues of the 2D rotation matrix $\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$ when $\theta$ is not a multiple of $\pi$ ?

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The characteristic polynomial is $\lambda^2 - 2\cos\theta\,\lambda + 1$ .
Roots are complex of modulus 1.

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$e^{\pm i\theta} = \cos\theta \pm i\sin\theta$

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Eigenvalues of a rotation

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