We define $A_1 \sim A_2 $ in $M_n(\mathbb{R})$ if there is $G \in Gl_n(\mathbb{R})$ such that $A_1 = G A_2 $. Find a distinguished element in each equivalence class associated to the equivalence relation $\sim$.
It is easy to show that the relation is equivalent. I have done that. Now what about the next? Clearly if two matrices $A_1$ and $A_2 $ are invertible, they are in same equivalence class. What about other cases?