Determine all eigenvalues of the matrix $A=\begin{bmatrix}0&0&1\\1&0&0\\0&1&0\end{bmatrix}$ and then determine a base for each eigenspace.
It's easy to compute $\chi_A(z)=z^3-1$ so my roots (and therefore eigenvalues) are $z_1=1, z_2=\cos(2\pi/3)+i \sin(2\pi/3)$ and $z_3=\cos(4\pi/3)+i \sin(4\pi/3)$.
Usually I would determine the eigenspaces by $E_\lambda=\ker(A-I_n\lambda)$, but having the solution to this problem shows that the result should be
$E_1=\left\langle\begin{bmatrix}1\\1\\1\end{bmatrix}\right\rangle,\qquad\qquad E_{z_2}=\left\langle\begin{bmatrix}z_3\\z_2\\1\end{bmatrix}\right\rangle,\qquad\qquad E_{z_3}=\left\langle\begin{bmatrix}1\\z_2\\z_3\end{bmatrix}\right\rangle.$
The first one is obvious, but I don't see where the trick is to quickly compute the other two eigenspaces / eigenvectors! Any help would be appreciated!