Not sure how to do vectors using this so I apologise for the bad notation in advance ;-).
How do you go about finding the orbits of a given vector. I can find the orbits of eigen vectors but other than that it feels very laborious to check other vectors and even then you cannot check all vectors, obviously.
Eg let $H$ be the subgroup $\begin{pmatrix}a&b\\b&a\end{pmatrix}$with $a^2-b^2=1$ them $H$ acts on $R^2$ by matrix multiplication. Investigate the orbits of this action.
The eigen vectors are the lines $y=x$ and $y=-x$. These get scaled by their respective eigen values. What else can be said of the orbits?