However if you have to find the similarity matrices, you have to do find the eigen vectors and you have to do the following:
Call , v is a cube root of unity.
Then observe that . This means x is an eigenvector associated with v^2. But v^2 is also a cube root of unity,so replacing that in the place of v,we get the associated eigen vector Thus . Also 1 is a cube root of unity. Thus associated with that is and .
Thus x,y,z are the required eigenvectors. As a side note they are clearly linearly independent since the matrix formed by x,z and y will be Vandermonde.