Is a diagonalization of a matrix unique?

The diagonal matrix is unique up to a permutation of the entries (assuming we use a similarity transformation to diagonalize). If we diagonalize a matrix $M = U\Lambda U^{-1}$, the $\Lambda$ are the eigenvalues of $M$, but they can appear in any order.

Assume A is some diagnalizable matrix. Then, we can write A = P D$\ P^{-1}$. But, then, we can change the order of our eigenvalues along the diagnal in our matrix D, to produce some other matrix G. But, this corresponds to a change in the order of the eigenvectors in P, which again produces another matrix Q. So, in conclusion , we have A = Q G$\ Q^{-1}$. So, this implies we do NOT have uniqueness.