Is a diagonalization of a matrix unique? I was solving problems of diagonalization of matrices and I wanted to know if a diagonalization of a matrix is always unique? but there's nothing about it in the books nor the net. 
I was trying to look for counter examples but I found none.
Any hint would be much appreciated
Thanks!

Actually it is not unique. Even without changing the order of eigenvalues. For example, the matrix $$A=\begin{pmatrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0&0&1\end{pmatrix}=\begin{pmatrix}1&0& -1 \\ 1 & 0 & -1 \\ 0&1&0\end{pmatrix}\begin{pmatrix}1&0& 0 \\ 0 & 1 & 0 \\ 0&0&-1\end{pmatrix}\begin{pmatrix}1&1& 0 \\ 0 & 0 & 1 \\ -1&1&0\end{pmatrix}=\begin{pmatrix}\frac{1}{\sqrt{2}}&0& -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & 0 & -\frac{1}{\sqrt{2}} \\ 0&1&0\end{pmatrix}\begin{pmatrix}1&0& 0 \\ 0 & 1 & 0 \\ 0&0&-1\end{pmatrix}\begin{pmatrix}\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}& 0 \\ 0 & 0 & 1 \\ -\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}&0\end{pmatrix}.$$
