I'm reading this article.
I don't understand the last sentence of the section Proof (optional):
Now, we just solve U, V and S for $A = USV^T$ and prove the theorem.
It seems to me that this article suddenly jumped to the conclusion.
Do we have to calculate further?
I think I understand both $A^TAV = VS^2$ and $AA^TU = US^2$ well because I calculated them with a concrete example, like: $$ A = \begin{pmatrix} 3 & 3 & 2 \\ 2 & 3 & -2 \\ \end{pmatrix} $$
How can we utilize the information from $A^TAV = VS^2$ and $AA^TU = US^2$ to prove the theorem?
Please help me. Thank you in advance.
That last sentence is poorly worded but is actually a summary. The few lines before have demonstrated how to find $V$, by solving for the eigen-vectors of $A^TA$, find $U$ from the eigen-vectors of $AA^T$. You know $S$ up to a sign ( since you have actually have $S^2$) and these signs are the only unknown, which you can decide by demanding $A=USV^T$