Orbits of $\mathbb{GL}_2$ acting on the set of singular matrices Given that the group $\mathbb{GL}_2$ , which acts on the set of all 2x2 singular matrices with real entries, and the action on the set of singular matrices $X$ is defined by conjugation (i.e. $g \circ A  = g A g^{-1}$ where $g \in \mathbb{GL}_2$ and $A \in X$.). How many orbits are there?
I have been trying to figure out this question for over a week and I am still stuck. I considered using Jordan Forms, but Jordan Normal Forms will not necessarily have all real entries and this question comes from an Algebra course where knowing Jordan Normal Forms is not a pre-requisite.
 A: I think it makes more sense to say "classify the orbits" than "count the orbits".
A singular 2-by-2 real matrix has only real eigenvalues since one eigenvalue is zero and the trace is real. Thus the minimal polynomial splits over the the reals. Hence  the Jordan form already exists over the reals, i.e. the Jordan form is already in the orbit obtained by conjugation by real invertible matrices.  So go ahead and use Jordan form.
Edit: You also can do it without (formally) knowing Jordan form.  Such a matrix is either zero, non-zero but nilpotent, or diagonalizable and non-zero.
A: $gAg^{-1}$ is a similar matrix to $A$  (has the same eigenvalues as $A$)
Since $A$ is singular (at least) one eigenvalue $= 0$
In the set of matrices with real values, if an eigenvalue is complex, its conjugates is also and eigenvalue.  If one eigenvalue is 0, then the other eigenvalue must be real.
In the set of orbits, we have one equivalence class for each value of the non-zero eigenvalue,  and one for the non-diagonalizable matrices.
