Singular Value Decomposition Example

Hello. I was wondering if someone could perform a singular value decomposition with every single step laid out for me? I found an example of an SVD online, but some steps are skipped on this document.

https://www.d.umn.edu/~mhampton/m4326svd_example.pdf

If $A$ is a matrix, then I understand how to find the eigenvalues and singular values of $AA^T$ . However, the following row-reduction and other following steps confuse me due to my lacking linear algebra skills. 

If the link doesn't work, the problem requests one to find the SVD, $A\Sigma V^T$, of $\begin{pmatrix} 3 & 2 & 2 \\ 2 & 3 & -2 \end{pmatrix} $