To prove the existence of SVD is no trivial task, but it turns out that it’s not too difficult either. Looks like one needs a few ingredients (hence the title), but once we know them and understand the overall idea, the proof is not too difficult. Below we list the basic ingredients needed to prove the existence of SVD. The URV decomposition $\mathbf{A} = \mathbf{URV} = \mathbf{U}\begin{pmatrix}\mathbf C & \mathbf 0 \\ \mathbf 0 & \mathbf 0\end{pmatrix}\mathbf{V}$ $||\mathbf{A}||_2 = ||\mathbf{URV}||_2 =…