
\subsection{\bf Equations for Eigenvalues---I}

\begin{slide}

\begin{itemize}
\item If $A\underline{v} = \lambda \underline{v}$

$( A - \lambda I ) \underline{v} = 0$

\hspace{1em} $\bullet$ $I$ is identity matrix

\item $B = A - \lambda I$.  Then we have
 
\begin{eqnarray*}
B \underline{v} = 0 & \mbox{and} & \underline{v} \neq \left\lbrack
0, \ldots 0 \right\rbrack \\
b_{11} v_1 + b_{12} v_2 & \ldots & + b_{1n} v_n = 0 \\
\ \ \vdots \\
b_{n1} v_1 \ldots & \ldots & b_{nn} v_n = 0
\end{eqnarray*}

\item $n$ linear equations for $n$ unknowns $v_i$.  This will
normally only have solution
\[
v_1 = v_2 = \ldots = v_k = \ldots = v_n = 0
\]
\end{itemize}

\end{slide}

