Find the eigenvalues of block matrix $C$

Question

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.

Answer 1

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 $A$

Answer 2

Since $\det(xI-C)=\det(x^2I-AA^T)$, the eigenvalues of $C$ are the singular values of $A$ and their negatives.

Now, note that $AA^T=2I+2ee^T$, where $e$ denotes the all-one vector. Hence the set of eigenvalues of $AA^T$ consists of a simple eigenvalue $2+2\times7=16$ and the eigenvalue $2$ of multiplicity $6$. It follows that the singular values of $A$ are $4$ and six copies of $\sqrt{2}$.

Hence the eigenvalues of $C$ are $+4,-4$, six copies of $\sqrt{2}$ and six copies of $-\sqrt{2}$.