Proving $A=U_{r}S_{r}V_{r}^{T}$ for a Full SVD

Question

Given the full SVD decomposition of a rank $r$ matrix $A\in\mathbb{R}^{m\times n}$, how can we show that $A=U_{r}S_{r}V_{r}^{T}$ where $U\in\mathbb{R}^{m\times m}$, $V\in\mathbb{R}^{n\times n}$, and $S\in\mathbb{R}^{m\times n}$?

Answer 1

Note that : \begin{align*} A&=U S V^{T}=\left[U_{r} \mid U_{m-r}\right]\left[\begin{array}{c|c} S_{r} & 0_{r, n-r} \\ \hline 0_{m-r, r} & 0_{m-r, n-r} \end{array}\right]\left[\begin{array}{c} V_{r}^{T} \\ \hline V_{n-r}^{T} \end{array}\right]=\left[U_{r} \mid U_{m-r}\right]\left[\frac{S_{r} V_{r}^{T}}{0_{m-r, n}}\right]\\ &=U_{r} S_{r} V_{r}^{T} \end{align*}