Consider an $m \times n$ real matrix $M$ whose SVD is $U \Sigma V^\top$ and an $n \times r$ matrix $A$ with orthonormal columns.
If $r = n$ then the SVD of $M A$ is simply $U \Sigma \left( A^\top V \right)^\top$. But what can be said about the SVD of $M A$ in the case $r < n$? In particular, is there a relation between the (left and right) singular vectors of $M$ (i.e., $U$ and $V$), and those of $M A$?