% file s6.tex
necessary condition for optimality:
\[
\exists~ \hat{\lambda} \in \Re~~ {\rm s.t.}~~ B-\hat{\lambda} C \geq 0.
\]

{\bf Define}:\\
{\em \underline{regular case}}
or the {\em \underline{positive definite pencil case}}
\[
\exists ~ \hat{\lambda} \in \Re~{\rm s.t.}~ B-\hat{\lambda} C > 0
\]

Ignore the irregular case since:
\begin{itemize}
\item
(i) The set of
$t$ where $B-tC$ is positive definite is an open
interval which is bounded if and only if $C$ is indefinite.
\item
(ii) In  the irregular case with the function $\det(B-tC)$
not identically 0 in $t$,
there is only one value $\hat{\lambda}$ such that 
$B-\hat{\lambda} C \geq 0$ holds.
\end{itemize}