Find Orbit of element $m \in M, M = M(2, \mathbb{R})$ under action of group $G = GL(2, \mathbb{R})$ mapping $m$ to $g^{-1}mg$, $g \in G$.
The element $m$ is $ \left( \begin{matrix} 2 & 1 \\ 0 & 2 \end{matrix} \right) $.
The exact question I have about it: for a given matrix how to learn for sure whether it belongs to the Orbit?
I can tell it must be 2x2 matrix with $tr = det = 4$, as the action is essentially a change of coordinates, the element is obviously in Jordan normal form and suppose that other attributes are from linear algebra as well, but I'm out of clues. Maybe it's some other invariant?
For this particular case, by considering the minimal polynomial and characteristic polynomial of $m$, one may conclude that the orbit of $m$ is the set of matrices $A$ such that $A-2I\ne0$ and $(A-2I)^2=0$. Yet, for a general $m$, the orbit is the set of matrices similar to $m$ and I don't think one can find a much simpler characterisation than using the Jordan normal form.