
\subsection{\bf Eigenvectors and Eigenvalues}

\begin{slide}

\begin{itemize}
\item Suppose $A \underline{v} = \lambda \underline{v}$
($\underline{v}$ is not all zeros), or $A \vert v > = \lambda \vert
v >$, where $\underline{v}$ is an eigenvector of $A$ with
eigenvalue $\lambda$.

\item Suppose $\underline{e}_1 = ( 1, 0, 0 \ldots 0 )$ is an
eigenvector of $A$ with eigenvalue $\lambda_1$.
\begin{eqnarray*}
A \underline{e}_1 = \lambda_1 \underline{e}_1 & = & \sum_l \left(
\lambda_1 \delta_{l1} \right) \underline{e}_l \\
\lambda_1 \delta_{l1} & = & \sum_k A_{l1} \delta_{k1} \\
A & = & \left\lbrack \begin{array}{l} 
\lambda_1 \\
0 \\
0 \hspace{2em} \mbox{ANY} \\
0 \hspace{2em} \mbox{VALUE} \\
0 \\
\vdots \end{array} \right\rbrack \\
\end{eqnarray*}
\end{itemize}

\end{slide}

