I am interested in exploring the structure of the quotient space formed by the action of the orthogonal group $O(n)$ on the space of $n\times m$ real matrices $\mathbb{R}^{n\times m}$ and $m>n$. Specifically, I am considering the action defined by left multiplication:
$$ Q\cdot A = QA, for\ Q\in O(n), A\in \mathbb{R}^{n\times m}. $$
I would like to understand the geometric and algebraic properties of the quotient space $\mathbb{R}^{n\times m}/O(n)$, where the equivalence classes are formed by the orbits of this action.
Question: What is the explicit structure of the quotient space $\mathbb{R}^{n\times m}/O(n)$ or is there any method to obtain it?
Simple Attempt: My previous thought is doing by QR decomposition so that quotient space could be described by a set of upper triangular matrices. However, since $m>n$, the decomposition to those upper triangular matrices is not unique.
Any insights, references, or suggestions for further reading would be greatly appreciated.
You can do this using singular value decomposition. Left multiplication by orthogonal matrices lets you arbitrarily shuffle around the left singular vectors but leaves the singular values and the right singular vectors unaffected.
More precisely, the orbit of a matrix $A$ is exactly determined by the decomposition of $\mathbb{R}^m$ into a direct sum of the orthogonal eigenspaces of the symmetric matrix $A^T A$ (these are a completely unique version of the right singular vectors) together with the eigenvalues of $A^T A$ acting on each of these (these are the squares of the singular values).
Edit: Ah, there's an easier way to say this: the orbit of $A$ is just determined by $A^T A$, full stop.