

\subsection{\bf Eigenfunction Analysis}

\begin{slide}
\begin{itemize}
\item There is an intuitively obvious solution of this.  

\item Consider the second order expansion term
\[
\left(\Underline{$\alpha$} - \Underline{$\alpha$}_0 \right)^T \mbox{\bf
M} \left(\Underline{$\alpha$} - \Underline{$\alpha$}_0 \right)
\]

\item $\mbox{\bf M}$ is a symmetric positive semi-definite matrix.
\item $\mbox{\bf M}$ can be diagonalized

\vspace*{10pt}
\centerline{\psfig{figure=fig18.phys.ps,clip=true,width=4.5in}}
\vspace*{10pt}

\item $\lambda_j$ are eigenvalues of $M$.  

\item $D$ is a diagonal matrix with diagonal entries $\lambda_j$.

\item {\bf P} is $n\times n$ matrix formed by eigenfunctions of $M$.
\end{itemize}
\end{slide}

