I want to construct a matrix $\textbf{A} \in \mathbb{F}^{m \times n}$ such that $\textbf{A} = \textbf{US}\textbf{V}^{\ast}$.
My goal is to do reverse SVD after I select the desired singular values for my matrix. During my research I have encountered this technique but the matrices that they choose for $\textbf{U}$,$\textbf{V}$ are $\textbf{I}_{m}$ and $\textbf{I}_{n}$ respectively.
This doesn't really make sense to me because you are basically constructing a diagonal matrix $\textbf{A}=\textbf{S}$.
Is there another way to construct the matrix $\textbf{A}$ without it being diagonal?