\begin{enumerate}
\item
$x \in R^{n-1}$, with $x^{t}x = 1$, is
a minimum (maximum) point of $\mu (\cdot)$ over $S_{n-1}$ \underline{if 
and only if} there
exists a scalar $\lambda$ such that $x$ and $\lambda$ together
satisfy the Lagrange equation (stationarity condition), with the matrix
$B - \lambda I$ being \underline{positive (negative) semidefinite}.
(no gap in nec \& suff)
\item
The set $\Lambda$ of Lagrange multipliers
of $\mu (\cdot)$ with respect to  $S_{n-1}$ is finite.  
Let $\Lambda$ be given
by
\[ \lambda_1 > \lambda_2 > \ldots > \lambda_k ,\]
and let $x^{\lambda_i} \in S_{\mu}(\lambda_i),~\; i=1,2,\ldots,k$.  Then
\[\mu(x^{\lambda_1}) > \mu(x^{\lambda_2}) > \ldots > \mu(x^{\lambda_k}).\]
In particular, the minimum (maximum) of $\mu (\cdot)$ over $S_{n-1}$ is
attained
at any stationary point associated with $\lambda_k$ ($\lambda_1$).
\end{enumerate}
