MAIN RESULTS - preliminary notation

\[
A=\left(
\begin{array}{cc}
B & -\alpha\\
\beta^t & t
\end{array}
\right),
\]

$P$ orthogonal matrix which diagonalizes $B$ 
\[
\hat{P} =
\left(
\begin{array}{cc}
P & 0\\
0 & 1
\end{array}
\right).
\]

\[
\hat{A}:=\hat{P}^{t} A \hat{P}=
\left(
\begin{array}{cc}
D & -\hat{\alpha}\\
\hat{\beta}^t & t
\end{array}
\right),
\]
where
\[\hat{\alpha}=P^{t}\alpha,~~~\hat{\beta}=P^{t}\beta.\]
 
assumption
\[
\hat{\alpha}_i\hat{\beta}_i \geq 0,~ \forall i=1,2, \ldots,n-1.
\]
\newpage
without loss of generality assume that
\[
\hat{A}=
\left(
\begin{array}{ccc}
\tilde{D}&0&-\tilde{\alpha}\\
0&\bar{D}&-\bar{\alpha}\\
\tilde{\beta}^t&\bar{\beta}^t&t
\end{array}
\right),
\]
where
\begin{eqnarray*}
\tilde{D}& =&\rm{diag}(\tilde{\gamma}_1,\tilde{\gamma}_2,\ldots,
\tilde{\gamma}_{\tilde{n}}),\\
\bar{D}&=&\rm{diag}(\bar{\gamma}_1,\bar{\gamma}_2,\ldots,\bar{\gamma}_{\bar{n}
}),\\
\tilde{\alpha}_i\tilde{\beta}_i& =&0,~ \forall i=1,2,\ldots,\tilde{n},\\
\bar{\alpha}_i\bar{\beta}_i& >& 0,~ \forall i=1,2,\ldots,\bar{n},
\end{eqnarray*}
and
\[ \tilde{n} + \bar{n}=n-1.\]
and the ordering
\[ \bar{\gamma}_1 \geq \bar{\gamma}_2 \geq \ldots \geq \bar{\gamma}_{\bar{n}}
\]
\newpage
(`easy case') submatrix of $\hat{A}$ given by
\[
\bar{A}=
\left(
\begin{array}{cc}
\bar{D}& -\bar{\alpha}\\
\bar{\beta}^t&t
\end{array}
\right).
\]
associated quadratic function
 \[
\bar{\mu}(\bar{x})=\bar{x}^t\bar{D}\bar{x}-
2\sum_{i=1}^{\bar{n}}(\bar{\alpha}_i\bar{\beta}_i)^{1/2}\bar{x}_i.
\]
associated secular antiderivative function 
\[
g_{\bar{\mu}}(\lambda) =
\lambda -
\sum_{i=1}^{\bar{n}}\left(\frac{\bar{\alpha}_i\bar{\beta}_i}{\bar{\gamma}_i-
\lambda}\right).
\]
characteristic polynomial of
$A$ is
\[
p(\lambda)=\bar{p}(\lambda)\prod_{i=1}^{\tilde{n}}(\tilde{\gamma}
_i -\lambda),
\label{cp}
\]
where
\[
\bar{p}(\lambda)=\det(\bar{A}-\lambda \bar{I}).
\]
To characterize the spectrum of $A$, we need only consider the spectrum
of $\bar{A}$, 'easy case'.