Find the eigenvalues of block matrix $C$ 
Find the eigenvalues of $$C = \begin{bmatrix}\begin{array}{c|c} 0 & A\\ \hline A^T & 0\end{array}\end{bmatrix}$$ where $$A = \begin{bmatrix}
0&0&0&1&1&1&1\\0& 1& 1& 0& 1& 0& 1\\0 &1 &1 &1 &0 &1 &0\\1& 0& 1& 0& 0& 1& 1\\1 &0 &1& 1& 1& 0& 0\\1 &1 &0& 0& 1& 1& 0\\1 &1& 0& 1& 0& 0 &1
\end{bmatrix}$$

The characteristic polynomial of $C$ is $$\det(xI-C)=x^2-A^2$$
Will the eigenvalues of $A$ be related to those of $C$ in some way? 
I am unable to proceed here.

Something you have written is wrong but can be corrected. We have$$xI-C=\begin{bmatrix}
\begin{array}{c|c}
xI & -A \\
\hline
-A^T & xI
\end{array}
\end{bmatrix}$$therefore$$|xI-C|=|x^2I-A^TA|$$therefore $$x=\pm|\lambda|$$where $\lambda$ is an eigenvalue of $A$ therefore the eigenvalues of $C$ are the $\pm$ singular values of $A$
