Let $A = U\Sigma V^*$ by the compact SVD where rank$(A)=r$ and $\Sigma$ is $r\times r$. If $A$ is Hermitian, then $U=V$.
Let us form another matrix $A_k = UK\Sigma V^*$, where $K\ne I$ is positive diagonal i.e. diag$(K)$ is a weight vector on the singular values.
Is there any application which makes use of this concept i.e. re-scales or weighs the singular values for a desired $A_k$ or a property of $A_k$? I did search but couldn't find.
Thanks in advance.