Why is my singular value decomposition off?
Find a singular value decomposition for $A=\left(\begin{matrix}0 & 3 & 4\\0 & -3 & -4\end{matrix}\right)=U\Sigma V^\intercal$
$AA^\intercal =\left( \begin{matrix}25 & -25 \\ -25 & 25\end{matrix}\right)$ with eigenvectors 50, 0 and eigenvectors $\left(\begin{matrix}1/\sqrt 2\\-1/\sqrt 2\end{matrix}\right)$ and $\left(\begin{matrix}1/\sqrt 2 \\ 1/\sqrt 2\end{matrix}\right)$. So $U=\left(\begin{matrix}1/\sqrt 2 & 1/\sqrt 2\\-1/\sqrt 2 & 1/\sqrt 2\end{matrix}\right)$ and $\Sigma=\left(\begin{matrix}50 & 0 & 0\\0&0&0\end{matrix}\right)$.
$A^\intercal A=\left(\begin{matrix}0&0&0\\0&18&24\\0&24&32\end{matrix}\right)$ with eigenvalues 50,0,0 and eigenvectors $\left(\begin{matrix}0\\3/5\\4/5\end{matrix}\right)$, $\left(\begin{matrix}1\\0\\0\end{matrix}\right)$,$\left(\begin{matrix}0\\4/5\\-3/5\end{matrix}\right)$, so $V=\left(\begin{matrix}0 & 1 & 0\\3/5&0&4/5\\4/5&0&-3/5\end{matrix}\right)$. But $U\Sigma V^\intercal=\left(\begin{matrix}1/\sqrt 2 & 1/\sqrt 2\\-1/\sqrt 2 & 1/\sqrt 2\end{matrix}\right)\left(\begin{matrix}50 & 0 & 0\\0&0&0\end{matrix}\right)\left(\begin{matrix}0 & 1 & 0\\3/5&0&4/5\\4/5&0&-3/5\end{matrix}\right)^\intercal=\left(\begin{matrix}0&15\sqrt 2&20\sqrt 2\\0&-15\sqrt 2&-20\sqrt 2\end{matrix}\right)$, which is off by a factor of $5\sqrt 2$?