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\centerline{\bf Eigenvalues and Eigenvectors}
\bigskip

Each group will be assigned one of the following matrices.\\

$$
A_1\doteq \left(\begin{array}{cc}
0&-1\\
1&0\\ \end{array} \right)
\hs
A_2\doteq \left(\begin{array}{cc}
0&1\\
1&0\\ \end{array} \right)
\hs
A_3\doteq \left(\begin{array}{cc}
-1&0\\
0&-1\\ \end{array} \right)
$$
\medskip
$$
A_4\doteq \left(\begin{array}{cc}
a&0\\
0&d\\ \end{array} \right)
\hs
A_5\doteq \left(\begin{array}{cc}
3&1\\
1&3\\ \end{array} \right)
\hs
A_6\doteq \left(\begin{array}{cc}
0&0\\
0&1\\ \end{array} \right)
\hs
A_7\doteq \left(\begin{array}{cc}
1&1\\
2&2\\ \end{array} \right)
$$
\medskip
$$
A_8\doteq \left(\begin{array}{ccc}
-1&0&0\\
0&-1&0\\
0&0&-1\\ \end{array} \right)
\hs
A_9\doteq \left(\begin{array}{ccc}
-1&0&0\\
0&-1&0\\
0&0&1\\ \end{array} \right)
$$
\medskip
$$
S_x\doteq \frac{\hbar}{2}\left(\begin{array}{cc}
0&1\\
1&0\\ \end{array} \right)
\hs
S_y\doteq \frac{\hbar}{2}\left(\begin{array}{cc}
0&-i\\
i&0\\ \end{array} \right)
\hs
S_z\doteq \frac{\hbar}{2}\left(\begin{array}{cc}
1&0\\
0&-1\\ \end{array} \right)
$$\\

For your matrix:
\begin{enumerate}
\item Find the eigenvalues.
\item Find the (unnormalized) eigenvectors.
\item Normalize your eigenstate.
\item Describe what this transformation does.\\
\end{enumerate}

When you are finished, write your solutions on the board.  \\

If you finish early, try another matrix with a different structure, \emph{i.e.} real vs. complex entries, diagonal vs. non-diagonal, $2\times 2$ vs. $3\times 3$, with vs. without explicit dimensions.

\vfill
\leftline{\textit{ by Corinne Manogue, Kerry Browne, Elizabeth Gire, David McIntyre\hfill}}
\leftline{\copyright 2010 Corinne A. Manogue}
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