<s>[INST] Showing two matrices are similar I have to show that each of the following matrices 
$$ \frac{1}{\sqrt{2}}
\begin{pmatrix}
0&1&0\\
1&0&1\\
0&1&0
\end{pmatrix}\quad , 
\frac{1}{\sqrt{2}}
\begin{pmatrix}
0&-i&0\\
i&0&-i\\
0&i&0
\end{pmatrix}
 , 
\begin{pmatrix}
1&0&0\\
0&0&0\\
0&0&-1
\end{pmatrix}$$
are equivalent to one of the following 
$$\begin{pmatrix}0&0&0\\0&0&1\\0&-1&0\end{pmatrix},\begin{pmatrix}0&0&-1\\0&0&0\\1&0&0\end{pmatrix},\begin{pmatrix}0&1&0\\-1&0&\\0&0&0\end{pmatrix}$$
It is a relatively simple but time consuming problem to construct a similarity transformation. I have nine real independent variables and 9 equations to check for each matrix. I want to know if there is any process or software that could help me do this calculation? 
 [/INST] Two matrices are similar if their traces are equal. 
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