= Solution
For an <eigenvector> $\mathbf v=(v_1,v_2,v_3)^{\mathsf T}$, the equation $A\mathbf v=\lambda\mathbf v$ reads
$$
v_3=\lambda v_1,\qquad v_1=\lambda v_2,\qquad v_2=\lambda v_3.
$$
A nonzero solution requires $\lambda^3=1$. Write the three <roots of unity> in the requested <polar form of a complex number>:
$$
\boxed{\lambda_k=e^{2\pi ik/3},\qquad
\mathbf v_k=\begin{pmatrix}1\\e^{4\pi ik/3}\\e^{2\pi ik/3}\end{pmatrix},
\qquad k=0,1,2.}
$$
Indeed $\mathbf v_k=(1,\lambda_k^2,\lambda_k)^{\mathsf T}$ satisfies all three equations. These <eigenvectors> are unnormalized; any nonzero scalar multiples are equally valid.
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