 Solution

For an eigenvector $\mathbf{v}=(v_1,v_2,v_3{)}^{\mathsf{T}}$, the equation $A\mathbf{v}=\lambda \mathbf{v}$ readsA nonzero solution requires ${\lambda }^3=1$. Write the three roots of unity in the requested polar form of a complex number: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.