
\subsection{\bf Polynomial Equations for Eigenvalues}

\begin{slide}

\begin{itemize}
\item det $(A-\lambda I) = $
\[
\mbox{det}\ \left\lbrack \begin{array}{llll}
a_{11} - \lambda & a_{12} \\
a_{21} & a_{22} - \lambda \\
& a_{32} \\
\\
\\
&&&a_{nn} - \lambda \end{array} \right\rbrack
\]

\item so det $( A - \lambda I ) = 0$ is a polynomial of degree $n$
in $\lambda$.  Coefficient of $\lambda^n$ is $(-1)^n$, i.e., always
nonzero.

\item On general principles, we will ALWAYS have $n$ solutions
\[
\lambda = \lambda_1, \lambda_2, \ldots \lambda_n
\]
of det $(A - \lambda I) = 0$
\end{itemize}

\end{slide}

