COROLLARY  Let $B$ be an $(n-1)\times(n-1)$ real symmetric matrix, and
consider
the perturbation of $B$ given by
\[
  A=\left(
\begin{array}{cc}
B & -\alpha\\
\alpha^t & t
\end{array}
\right),
\]
where $\alpha$ is a real $(n-1)$-vector.  Define
\[
\mu(x) = x^tBx-2\alpha^tx.
\]
  Let
\[
 \mu_1 = \max\{\mu(x)~:~x^tx = 1\}
\label{mu1}
\]
and
\[
 \mu_k = \min\{\mu(x)~:~x^tx = 1\}.
 \label{mu2}
 \]
Then either of the conditions
\[
t \geq \mu_1
\label{geek}
\]
or
\[
t \leq \mu_k
\]
are sufficient for the spectrum of $A$ to be real.