Determine the eigenvalues, eigenvectors and eigenspaces for a plane rotation ( cos α − sin α sin α cos α ) {\displaystyle {}{\begin{pmatrix}\operatorname {cos} \,\alpha &-\operatorname {sin} \,\alpha \\\operatorname {sin} \,\alpha &\operatorname {cos} \,\alpha \end{pmatrix}}} , with rotation angle α {\displaystyle {}\alpha } , 0 ≤ α < 2 π {\displaystyle {}0\leq \alpha <2\pi } , over R {\displaystyle {}\mathbb {R} } .
