Chapter 6.5, Problem 10E

Let A be an $n\times n$ matrix with singular value decomposition $U{\displaystyle \sum V^T}$ and let
$B=\left(\begin{array}{cc}O& A^T\\ A& O\end{array}\right)$
Show that if
$x_i=\left(\begin{array}{c}{v}_i\\ u_i\end{array}\right),y_i=\left(\begin{array}{c}-v_i\\ u_i\end{array}\right),i=1,...,n$
then the $x_i\text{'}s$ and $y_i\text{'}s$ are eigenvectors of B. How do the eigenvalues of B relate to the singular values