If we let $S$ be the the set of real $m \times n$ matrices and take G to be the direct product $G=GL(m,\mathbb{R})\times GL(n,\mathbb{R})$ and consider the action of G on S as follows
$(P,Q) \star A=PAQ^{-1}$
how is S decomposed into G orbits?
Having verified that this is indeed an action on the set, I think I'm looking to find the equivalence classes of $\sim$ where $A \sim B$ if $B$ can be expressed in the form $PAQ^{-1}=B$ for some matrices $P$, $Q$ of the required dimension.
Intuitively, I imagine we call always find invertible matrices $P$, $Q$ that satisfy the condition, though I'm not sure if this is the case. Any help would be much appreciated. Best.