Let $T=\begin{pmatrix}5 & 0 & 0 \\ 0 & 2 & i\\ 0 & -i & 2 \end{pmatrix}$.
I found the eigenvalues and eigenvectors already and they are $1,3,5$ and $\begin{pmatrix}0\\-i\\1\end{pmatrix}$,$\begin{pmatrix}0\\i\\1\end{pmatrix}$, and $\begin{pmatrix}1\\0\\0\end{pmatrix}$.
According to mathematica, their orthoganalized form will be $\left(
\begin{array}{ccc}
1 & 0 & 0 \\
0 & \frac{i}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\
0 & -\frac{i}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\
\end{array}
\right)$
But I am getting wildly different answers when I try to find the normalized eigenvectors using the Gram-Schmidt process.
Best Answer
You may be right. There is no unique orthonormal basis. Just check that whatever you are getting are eigenvectors, are orthogonal and have norm $1$ each.