Problem with Singular Value Decomposition I have a very trivial SVD Example, but I'm not sure what's going wrong.
The typical way to get an SVD for a matrix $A = UDV^T$ is to compute the eigenvectors of $A^TA$ and $AA^T$. The eigenvectors of $A^TA$ make up the columns of $U$ and the eigenvectors of $AA^T$ make up the column of $V$. From what I've read, the singular values in $D$ are square roots of eigenvalues from $AA^T$ or $A^TA$, and must be non-negative.
However, for the simple example $A = \left[\begin{matrix} 1 & 0 \\ 0 & -1 \end{matrix}\right]$, $A^TA$ and $AA^T$ are both the identity, and thus have eigenvectors $\bigg\lbrace \left[\begin{matrix} 1 \\ 0 \end{matrix}\right]\ ,\ \left[\begin{matrix} 0 \\ 1 \end{matrix}\right]\bigg\rbrace$. Clearly, the eigenvalues are 1 and 1, so our decomposition ought to be:
\begin{align}
 \left[\begin{matrix} 1 & 0 \\ 0 & -1 \end{matrix}\right] = \left[\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}\right]\left[\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}\right]\left[\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}\right]
\end{align}
What has gone wrong?

To directly answer your question, the problem is that the "typical way" you describe for getting an SVD is incorrect:

The typical way to get an SVD for a matrix $A = UDV^T$ is to compute the eigenvectors of $A^TA$ $\color{red}{and}$ $AA^T$. The eigenvectors of $A^TA$ make up the columns of $U$ and the eigenvectors of $AA^T$ make up the column of $V$. 

One should either give $U$ then solve $V$, or give $V$ then solve $U$ by using the relation $A=UDV^T$. 
On the other hand, you wrote that 

... and thus have eigenvectors $\bigg\lbrace \left[\begin{matrix} 1 \\ 0 \end{matrix}\right]\ ,\ \left[\begin{matrix} 0 \\ 1 \end{matrix}\right]\bigg\rbrace$

But one could equally have $\bigg\lbrace \left[\begin{matrix} -1 \\ 0 \end{matrix}\right]\ ,\ \left[\begin{matrix} 0 \\ 1 \end{matrix}\right]\bigg\rbrace$, or 
$\bigg\lbrace \left[\begin{matrix} 1 \\ 0 \end{matrix}\right]\ ,\ \left[\begin{matrix} 0 \\ -1 \end{matrix}\right]\bigg\rbrace$, or
$\bigg\lbrace \left[\begin{matrix} -1 \\ 0 \end{matrix}\right]\ ,\ \left[\begin{matrix} 0 \\ -1 \end{matrix}\right]\bigg\rbrace$. 
Any one of these sets consists of independent eigenvectors (with unit length) of the identity matrix. If one chooses from these sets arbitrarily to form $U$ and $V$, there is no reason to expect that
$$
A=UDV^T.
$$
