%Eigenvalues and eigenvectors
%If you modify this file, please indicate here and in the footnote
%Math410
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{\Large
Eigenvalues and Eigenvectors\footnote{Copyright \copyright 2002 Steve Chapin and Todd Young.
All rights reserved. Please address comments to young@math.ohiou.edu.}}
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\begin{enumerate} \item Try the following commands \begin{enumerate}
\item \verb&digits(4)&
\item \verb&A = sym([1,1; 0,1])&
\item \verb&E = eig(A)&
\item \verb&[V,E] = eig(A)&
\end{enumerate}
Find the eigenvalues and eigenvectors for this matrix by hand
and interpret the output.
\item Input the \textbf{symbolic} matrix (use \verb&sym& as above):
$$
\textsf{B = }\left[
{\begin{array}{rrr}
3 & -1 & -1 \\
-1 & 0 & 2 \\
1 & 1 & -3
\end{array}}
\right]
$$ and try the commands:
\begin{enumerate}
\item \verb&SE = eig(B)& \dotfill Finds eigenvalues symbolically.
\item \verb&NE = eig(vpa(B))& \dotfill \verb&vpa& changes from symbolic to numeric.
\item \verb&[SV,SE] = eig(B)&
\item \verb&[NV,NE] = eig(vpa(B))&
\end{enumerate}
\item Create a matrix using the command: \verb& C = sym(hilb(5))&\\
and repeat the process in the previous part.
\item What are your observations about symbolic vs. numerical computations
from the last two parts?
\item Using complete sentences and standard mathematical notation, write a brief report.
Show your hand calculations and answer all the questions.
\end{enumerate}
\vfill
\noindent
\textsf{In the first
example students must consider multiplicities. The last part should
lead to a discussion of the fact that polynomials of degree 5 or
higher cannot in general be solved symbolically and so exact symbolic
eigenvalues cannot be found for 5 by 5 matrices. They should also
notice that symbolic solutions are sometimes too complicated to be
useful.}
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