If $A$ is a $n\times m$ matrix and $B$ a $m \times k$ matrix (both real), can then something be said about how the singular value of $A$ and $B$ are related to the singular values of the product $AB$?
I'd be interested ideally in a formula that gives the singular values of $AB$ in terms of the singular values of $A$ and $B$; or, if that is not possible, at least in an upper bound of the singular values of $AB$ in terms of the singular values of $A$ and $B$.
I have spent some time googling know and all I could find where research articles describing the singular value decomposition of $AB$, but extracting what this means for the singular values is actually non-trivial because a ton of notation is introduced and the theorems are straightforward to read, so I was wondering whether perhaps more "readable" reference or answer could be given here? I'm guessing since I'm only after the singular values, perhaps there is a more straightforward answer?