
\subsection{\bf Structure of Matrix}

\begin{slide}

\begin{itemize}
\item If $A$ is also symmetric (Hermitean)
\[
A = \left\lbrack \begin{array}{ccccccc}
\lambda_1 & 0 & 0 & 0 & 0 & \ldots & 0 \\
0 \\
0 \\
0 && A&N&Y \\
0 && V&A&L&U&E \\
0 \end{array} \right\rbrack \\
\]

\item If $\underline{e}_2 \ldots \underline{e}_n$ are also
eigenvectors with eigenvalues $\lambda_2 \ldots \lambda_n$,
\begin{eqnarray*}
\underline{e}_2 & = & ( 0, 1, 0, \ldots 0 ) \\
\underline{e}_n & = & ( 0, 0, \ldots 1 ) 
\end{eqnarray*}

Then
\[
A = \left\lbrack \begin{array}{ccccc}
\lambda_1 \\
& \lambda_2 && 0 \\
&& \lambda_3 \\
& 0 && \ddots \\
&&&& \lambda_n \end{array} \right\rbrack
\]
\end{itemize}

$A$ is diagonal with diagonal entries $\lambda_i$.

\end{slide}

