Showing two matrices are similar I have to show that each of the following matrices 

$$ \frac{1}{\sqrt{2}}
\begin{pmatrix}
0&amp;1&amp;0\\
1&amp;0&amp;1\\
0&amp;1&amp;0
\end{pmatrix}\quad , 
\frac{1}{\sqrt{2}}
\begin{pmatrix}
0&amp;-i&amp;0\\
i&amp;0&amp;-i\\
0&amp;i&amp;0
\end{pmatrix}
 , 
\begin{pmatrix}
1&amp;0&amp;0\\
0&amp;0&amp;0\\
0&amp;0&amp;-1
\end{pmatrix}$$

are equivalent to one of the following 

$$\begin{pmatrix}0&amp;0&amp;0\\0&amp;0&amp;1\\0&amp;-1&amp;0\end{pmatrix},\begin{pmatrix}0&amp;0&amp;-1\\0&amp;0&amp;0\\1&amp;0&amp;0\end{pmatrix},\begin{pmatrix}0&amp;1&amp;0\\-1&amp;0&amp;\\0&amp;0&amp;0\end{pmatrix}$$

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?
Since you mentioned a "similarity transform", I presume your "equivalent" means "similar"; also I presume this is over $\mathbb C$.  Well, the first thing I would do is find the eigenvalues of each matrix; the second (if necessary) is the Jordan canonical form.  As for software, Maple will handle this quite easily; I imagine
most other CAS's will also.

EDIT: Oops, all of your first three matrices are similar to each other, and none is similar to any of the last three.  Are you sure you quoted the question right?
Two matrices are similar if their traces are equal.