Equivalence classes in $M_n(\mathbb{R})$

by ChakSayantan   Last Updated August 11, 2018 07:20 AM

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?

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