I have a matrix $\Phi A \Phi^T$, where $A$ is a diagonal matrix that changes constantly and $\Phi$ is a $n\times m$ constant matrix ($n<m$) whose singular value decomposition is $UDV^T$.
Question is what is the SVD of the product $\Phi A \Phi^T$.
Side Question, if $A$ is a sparse diagonal, how would this affect the factorisation?