Equivalence relation on matrices

Consider $$M_{n\times n}$$ the $$n\times n$$ matrices over some field $$F$$. Define an equivalence relation $$A\sim B$$ if there is invertible $$C$$ such that $$A=CB$$. What are the equivalence classes of $$\sim$$ in $$M_{n\times n}$$?

This question has to do with echelon form I think. My intuition is that the equivalence classes are just the matrices of a particular the rank, although I'm not sure of this. Clearly every matrix is in the same equivalence class as its echelon form, so we can consider only matrices in echelon form, where to go from here?

• The equivalence classes of $\sim$ are matrices with the same row space. No need to play with echelon forms. – user10354138 May 19 at 15:21

When you multiply $$A$$ by $$C$$ you get a new matrix whose rows are linear combinations of the rows of $$A$$. Actually, when $$C$$ is invertible $$A$$ and $$CA$$ have the same row space. Therefore, $$A\sim B$$ if and only if the rows of $$A$$ and of $$B$$ span the same space.
• So the equivalence classes are one to one paired with subspaces of $F^n$, yes? – Joshua Tilley May 19 at 15:29