I have two matrices, both are $n \times m$ where $m < n$. These two matrices are $A$ and $B$.
I also know the singular value decomposition of $AB^{\top} = U \Sigma V^{\top}$.
Is there anything I can say about the SVD of $A$ or $B$? (for example, can I say that the left singular vectors of $A$ are $U$?) Are there any conditions under which I could say something like that?
Thanks.