This is a small group activity for groups of 3-4. The students will be given one of 10 matrices. The students are then instructed to find the eigenvectors and eigenvalues for this matrix and record their calculations on their medium-sized whiteboards. In the class discussion that follows students report their finding and compare and contrast the properties of the eigenvalues and eigenvectors they find. Two topics that should specifically discussed are the case of repeated eigenvalues (degeneracy) and complex eigenvectors, e.g., in the case of some pure rotations, special properties of the eigenvectors and eigenvalues of hermitian matrices, common eigenvectors of commuting operators.
Each group will be assigned one of the following matrices.
\[
A_1\doteq
\begin{pmatrix}
0&-1\\
1&0\\
\end{pmatrix}
\hspace{2em}
A_2\doteq \begin{pmatrix}
0&1\\
1&0\\ \end{pmatrix}
\hspace{2em}
A_3\doteq \begin{pmatrix}
-1&0\\
0&-1\\ \end{pmatrix}
\]
\[
A_4\doteq \begin{pmatrix}
a&0\\
0&d\\ \end{pmatrix}
\hspace{2em}
A_5\doteq \begin{pmatrix}
3&-i\\
i&3\\ \end{pmatrix}
\hspace{2em}
A_6\doteq \begin{pmatrix}
0&0\\
0&1\\ \end{pmatrix}
\hspace{2em}
A_7\doteq \begin{pmatrix}
1&2\\
1&2\\ \end{pmatrix}
\]
\[
A_8\doteq \begin{pmatrix}
-1&0&0\\
0&-1&0\\
0&0&-1\\ \end{pmatrix}
\hspace{2em}
A_9\doteq \begin{pmatrix}
-1&0&0\\
0&-1&0\\
0&0&1\\ \end{pmatrix}
\]
\[
S_x\doteq \frac{\hbar}{2}\begin{pmatrix}
0&1\\
1&0\\ \end{pmatrix}
\hspace{2em}
S_y\doteq \frac{\hbar}{2}\begin{pmatrix}
0&-i\\
i&0\\ \end{pmatrix}
\hspace{2em}
S_z\doteq \frac{\hbar}{2}\begin{pmatrix}
1&0\\
0&-1\\ \end{pmatrix}
\]
For your matrix:
If you finish early, try another matrix with a different structure, i.e. real vs. complex entries, diagonal vs. non-diagonal, \(2\times 2\) vs. \(3\times 3\), with vs. without explicit dimensions.