MAIN RESULT

THEOREM
The following hold:
\begin{enumerate}
\item
There exist $n-2$ real eigenvalues $\{\delta_i\}_{i=1}^{n-2}$ of $A$,
including all the eigenvalues of $\tilde{D}$ and $\bar{n}-1$ eigenvalues of
$\bar{A}$,
which interlace the $n-1$ ordered eigenvalues $\{\gamma_i\}_{i=1}^{n-1}$ of
$B$; that is,
\[
\gamma_1 \geq \delta_1 \geq \gamma_2 \geq  \ldots \geq \gamma_{n-2}
\geq \delta_{n-2} \geq \gamma_{n-1}.
\label{interlace}
\]
\newpage
\item
The remaining two eigenvalues of $A$ (which are eigenvalues of $\bar{A}$), say
$\bar{\delta}_a$ and $\bar{\delta}_b$, are real if and only if
\[
t \in \{\bar{\Lambda}'\}\cup\{I_1\}\cup\{I_m\}\cup
\{\bigcup_{i=1}^vI''_i\}.
\label{realness}
\]
\item
Furthermore, $\bar{\delta}_a$ and $\bar{\delta}_b$ are real and distinct if
and only if $t$ is in the interior of one of the $v+2$ intervals.
In this case, the $\bar{n}+1$ eigenvalues of $\bar{A}$ are
real and distinct.
\newpage
\item
In the real case, we have the following relations
involving $\bar{\delta}_a$ and $\bar{\delta}_b$, where we assume
$\bar{\delta}_a \leq \bar{\delta}_b$:
 
 
~~(a)~~$t>\bar{\mu}_1 \Longrightarrow \bar{\gamma}_1 <
\bar{\delta}_a<\bar{\lambda}_1<\bar{\delta}_b \leq \bar{\mu}_1$.
 
~~(b)~~$t=\bar{\mu}_1 \Longrightarrow \bar{\gamma}_1<
\bar{\delta}_a=\bar{\lambda}_1=\bar{\delta}_b \leq \bar{\mu}_1$.
 
~~(c)~~$t=\bar{\mu}''_{2i-1} \Longrightarrow \bar{\lambda}''_{2i}
< \bar{\delta}_a = \bar{\lambda}''_{2i-1} = \bar{\delta}_b$.
 
~~(d)~~$t\in (\bar{\mu}''_{2i},\bar{\mu}''_{2i-1}) \Longrightarrow
      \bar{\lambda}''_{2i} < \bar{\delta}_b <
\bar{\lambda}''_{2i-1} < \bar{\delta}_a$
 
~~or $\bar{\delta}_b < \bar{\lambda}''_{2i} < \bar{\delta}_a <
\bar{\lambda}''_{2i}$.
 
~~(e)~~$t=\bar{\mu}''_{2i} \Longrightarrow \bar{\lambda}''_{2i}
= \bar{\delta}_a = \bar{\lambda}''_{2i}= \bar{\delta}_b <
\bar{\lambda}''_{2i-1}$.
 
~~(f)~~$t=\bar{\mu}'_i \Longrightarrow \bar{\delta}_a =\bar{\mu}'_i
=\bar{\delta}_b$.
 
~~(g)~~$t=\bar{\mu}_m \Longrightarrow
\bar{\mu}_m \leq \bar{\delta}_a=\bar{\lambda}_m=\bar{\delta}_b
< \bar{\gamma}_{\bar{n}}$.
 
~~(h)~~$t<\bar{\mu}_m \Longrightarrow
\bar{\mu}_m \leq \bar{\delta}_a<\bar{\lambda}_m <\bar{\delta}_b
<\bar{\gamma}_{\bar{n}}$.
\end{enumerate}
