The problem is too easy if all you have to find is the diagonal matrix. This is because we see that A^3 = I and thus the characteristic polynomial must divide x^3 - 1. But A is a linear transformation on a vec. space of dimension 3, thus the characteristic poly must be a monic of a degree 3. Thus x^3 - 1 is the char poly. The roots are cube roots of unity. Since the roots are distinct, the lin. transformation is diagonalizable. And thus is similar to diag(1, v , v^2) where v^3 = 1.

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.