[math] \newcommand{\mathds}{\mathbb}[/math]

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Prove that the symmetry and projection with respect to the [math]Ox[/math] axis rotated by an angle [math]t/2\in\mathbb R[/math] are given by the matrices

[[math]] S_t=\begin{pmatrix}\cos t&\sin t\\ \sin t&-\cos t\end{pmatrix} [[/math]]

[[math]] P_t=\frac{1}{2}\begin{pmatrix}1+\cos t&\sin t\\ \sin t&1-\cos t\end{pmatrix} [[/math]]

and then diagonalize these matrices, and if possible without computations.