SYMMETRIC BORDER PERTURBATIONS  (well-known properties) 

the eigenvalues $ \delta_{1} \geq \delta_{2} \ldots \geq \delta_{n}$ of
the $n \times n$ matrix
\begin{eqnarray*}
A= \left( \begin{array}{cc}
B & \eta \\
\eta ^{t} & t
\end{array} \right)
\end{eqnarray*}
interlace the eigenvalues of $B$
\[
 \delta_1 \geq \gamma_1 \geq \delta_2 \geq \gamma_2 \geq \ldots \geq
\gamma_{n-1}
\geq \delta_n 
\]