I have looked for a source on the relationship between elementary matrix row operations and singular values, but I can't find a good, compact set of information.
I'm really only interested in interchanging rows. If two rows switch position in a matrix M, will the singular values of M change?
My hunch is that they will not, and simple experimentation with matrix calculators backs this up, but I am looking for a proof of this.
On
Changing rows is the same as multiplying by a permutation (which is a rotation, and perhaps a reflection)in the codomain. That means that the SVD before and after look like $$ M = U D V^t \\ M' = P U D V^t $$ In the case where there's no reflection ($det P > 0$), using $PU$ as $U'$, you get an SVD for $M'$. The singular values are evidently the same. I'll elt you handle the case with the negative determinant.
Permuting the rows of $M$ gives you $P M$ where $P$ is a permutation matrix. The singular values of $PM$ are the square roots of the (nonzero) eigenvalues of $(PM)^* PM = M^* P^* P M = M^* M$, so they are the same as the singular values of $M$.