## Homework for Eigenvalues and Eigenvectors

1. (EigenPractice) Lots and lots of practice finding eigenvalues and eigenvectors.

(The solutions to this problem need additional examples from the newest version of the activity.)

2. (Eigenrotation) Straightfoward practice finding eigenvalues and eigenvectors for the particular case of a generic rotation matrix around the $z$-axis. Warning: the eigenvectors and eigenvalues in this case are complex numbers.

3. (SpinMatrix)This problem is a prerequisite for the next one. Students find this problem very strange. It requires them to take the formal dot product of a vector with another vector whose components are matrices. The result is the spin operator for a generic spin $\frac{1}{2}$ system, with spin up in the $\hat n$-direction. This can be a useful problem if the students are going to be covering the content of the Quantum Measurement and Spin Course.

4. (EigenSpinChallenge)This problem requires the previous problem as a prerequisite. It is long and messy. It requires the students to use trigonometric identities and to persist through a messy calculation. In this problem, students find the eigenvalues and eigenvectors for the generic spin $\frac{1}{2}$ matrix in the $\hat n$-direction. Therefore, this can be a useful problem if the students are going to be covering the content of the Quantum Measurement and Spin Course. This problem needs to be updated so the phase conventions agree with Spins conventions, that the first component should be real.

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