Is the diagonalizing matrix for a matrix with non-repeated eigenvalues unique?

Question

In presence of degeneracy in the eigenvalues, the diagonalizing matrix is not unique. Is the converse also true? Suppose $A$ is an arbitrary (say, $3\times 3$) matrix with non-degenerate eigenvalues $a, b$ and $c$. Is the diagonalizing matrix unique? If not, a counterexample will answer my question. If yes, I would prefer a proof.

Note When I say unique, I mean apart from reshuffling of normalized eigenvectors among the columns.

Answer 1

It is never unique. If $P$ is such that$$P^{-1}.A.P=\begin{pmatrix}a&0&0\\0&b&0\\0&0&c\end{pmatrix}$$then, if you replace each column of $P$ by itself times a non-zero scalar, then the new matrix will work too.