I've got some questions about an exercise. We consider $\mathcal{M}_{m \times n}(\mathbb{R})$ and $GL(m,\mathbb{R}) \times GL(n,\mathbb{R})$, and the action of group given by : $(P,Q).A = PAQ^{-1}$.
The first question is to determine the orbits. So, obviously, if $A=0$, then $\mathcal{O}_A = \{0 \}$. Now, I wanted to show that if $A$, $B$ has the same rank, then $B \in \mathcal{O}_A$. I try to think with the linear functions and determine two linear function which would permit to conclude, but I didn't succeed to do that. Actually, I got lost on my calculations...
Then, we suppose $m \leq n$, and we want to determine the stabilizer of $(\; I_m \, | \, 0 \;)$. And again, I get lost on my calculations, mainly because I think I should use block matrix multiplication, and I'm really bad when it's start with such things...
Someone could help me, please ?
Thank you !