Now provide a qualitative description of the eigenstructure
of the border perturbed  matrix $A$  when $\alpha_i \beta_i \geq 0$.

Realness of the spectrum of $A$ is characterized in terms of
the graph of $g_{\bar{\mu}}(\cdot)$, and in particular, in terms of the
stationary values of the quadratic function $\bar{\mu}(\cdot)$.  Should the
spectrum be real, the
interlacing of the eigenvalues of $A$ and $B$ is described.
\newpage
Let $\bar{\lambda} \in \bar{\Lambda}$. $\bar{\lambda}$ is a
{\em type-1 critical point} of $g_{\bar{\mu}}(\cdot)$ if it is a critical
point which is also an inflection.  Otherwise $\bar{\lambda}$ a {\em
type-2 critical point} of $g_{\bar{\mu}}(\cdot)$.

denote the sets of type-1 and
type-2 critical
points of $g_{\bar{\mu}}(\cdot)$ by $\bar{\Lambda}'$ and $\bar{\Lambda}''$,
respectively.  
\[ \bar{\Lambda}=\bar{\Lambda}'\cup \bar{\Lambda}''.\]
$\bar{\Lambda}'$ is
$ \bar{\lambda}'_1 > \bar{\lambda}'_2 > \ldots > \bar{\lambda}'_w,$\\
$\bar{\Lambda}''$ is
$ \bar{\lambda}_1 > \bar{\lambda}''_1 > \bar{\lambda}''_2 > \ldots >
\bar{\lambda}''_{2v} > \bar{\lambda}_{\bar{n}}$\\
disjoint closed
intervals:
\begin{eqnarray*}
I_1=[\bar{\mu}_1,\infty), ~
I_m=(-\infty,\bar{\mu}_m], ~
I''_i = [\bar{\mu}''_{2i},\bar{\mu}''_{2i-1}]
\end{eqnarray*}