Rotation Matrices (Cont.)

All of these rotation matrices have the same eigen-values:
$$
\lambda_{i}\in\{1,\pm i\}
$$
though they have different eigen-vectors:
$$
\mathbf{X}^{\left(\pm\right)}_{\hat{n}}=\frac{e^{\frac{i\pi}{4}}}{\sqrt{2}}
\begin{pmatrix}
e^{\frac{-i\pi}{4}}  & e^{\frac{-i\pi}{4}} &  0 \\
\mp e^{\frac{i\pi}{4}}  & \pm e^{\frac{i\pi}{4}}  & 0 \\
0  & 0 &  \sqrt{2}e^{\frac{-i\pi}{4}}
\end{pmatrix} \\
$$
$$
\mathbf{X}^{\left(\pm\right)}_{\parallel}=\frac{e^{\frac{i\pi}{4}}}{\sqrt{2}}
\begin{pmatrix}
0  & 0 &  \sqrt{2}e^{\frac{-i\pi}{4}} \\
\pm e^{\frac{-i\pi}{4}}  & \pm e^{\frac{-i\pi}{4}} &  0 \\
- e^{\frac{i\pi}{4}}  &  e^{\frac{i\pi}{4}}  & 0
\end{pmatrix} \\
$$
$$
\mathbf{X}^{\left(\pm\right)}_{\perp}=\frac{e^{\frac{i\pi}{4}}}{\sqrt{2}}
\begin{pmatrix}
e^{\frac{-i\pi}{4}}  & e^{\frac{-i\pi}{4}} &  0 \\
0  & 0 &  \sqrt{2}e^{\frac{-i\pi}{4}} \\
\pm e^{\frac{i\pi}{4}}  & \mp e^{\frac{i\pi}{4}}  & 0
\end{pmatrix}
$$
What's interesting is assigning some sort of coordinate system to these eigen-vectors. For example, one could say that the third eigen-vector of the set of eigen-vectors for \(\mathbf{X}^{\left(\pm\right)}_{\hat{n}}\) corresponds to \(\hat{z}\), that is to say:
$$
\mathbf{X}^{\left(\pm\right)}_{\hat{n}}= \begin{pmatrix}
0\\
0\\
1
\end{pmatrix}=\hat{z}
$$
$$
\mathbf{X}^{\left(\pm\right)}_{\perp}= \begin{pmatrix}
0\\
1\\
0
\end{pmatrix}=\hat{y}
$$
$$
\mathbf{X}^{\left(\pm\right)}_{\parallel}= \begin{pmatrix}
1\\
0\\
0
\end{pmatrix}=\hat{x}
$$
But, you may ask yourself, we have other eigen-vectors, what about those? What do they correspond to? Or, more importantly, what can I use them for?

Well, that's the part that's interesting. I've run out of time and will continue this later tonight.
P.S. Finding the eigen-vectors and eigen-values is easily verified with qtoctave or other linear algebra software.

A=[0,-1,0;1,0,0;0,0,1]
det(A)
eig(A)
[EVECT,EVAL]=eig(A)
A=[0,1,0;-1,0,0;0,0,1]
det(A)
eig(A)
[EVECT,EVAL]=eig(A)
A=[1,0,0;0,0,-1;0,1,0]
det(A)
eig(A)
[EVECT,EVAL]=eig(A)
A=[1,0,0;0,0,1;0,-1,0]
det(A)
eig(A)
[EVECT,EVAL]=eig(A)
A=[0,0,1;0,1,0;-1,0,0]
det(A)
eig(A)
[EVECT,EVAL]=eig(A)
A=[0,0,-1;0,1,0;1,0,0]
det(A)
eig(A)
[EVECT,EVAL]=eig(A)