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!

I assume you mean uniqueness up to similarity transformations (i.e. change of basis).  Suppose that the linear transformation $T$ is represented by the matrix $\left(\begin{array}{cc}1 & 0 \\ 0 & 2 \end{array} \right)$ with respect to basis $\{e_1,e_2\}$.  In other words, $T$ maps any vector $(a,b)=ae_1+be_2$ to $(a,2b)=ae_1+2be_2$. Then, with respect to basis $\{e_2,e_1\}$ (note that the order of the basis elements is different), the linear transformation $T$ is represented by the matrix  $\left(\begin{array}{cc}2 & 0 \\ 0 & 1 \end{array} \right)$.  Hence, there is a similarity transformation that takes one matrix to the other.  So, the diagonal matrices that represent a linear transformation are not unique.  But the diagonal matrices will be unique up to some permutation of the diagonal entries. 
