I would like to perform diagonalization of a Hermitian matrix $A$ and I know the steps but at the end I am not getting diagonal matrix with eigenvalues on the main diagonal, can anyone help me why?
Assume I choose one of the eigenvectors to be $[0 1 0]'$ and I found two other orthonormal vectors as $[1 0 0]'$ and $[0 0 1]'$. I would like to know why my final matrix $T$ such that $T'AT$ is not diagonal?
$$A=\begin{pmatrix}0&0&\sqrt{2}\\0&1&0\\\sqrt{2}&0&-1\end{pmatrix}$$