AIMS Lie Groups: Fall-2022
Practice Exercises 8 : Due 08 Wed 14 Dec

1. New basis for $\mathfrak{so}(3)$ S0 4489S Recall the standard basis \(\{r_x,r_y,r_z\}\) for the Lie algebra \(\mathfrak{so}(3)\), satisfying \([r_x,r_y]=r_z\), etc.
   1. What are the eigenvalues of \(r_z\)?
   2. What are the eigenvectors of \(r_z\)?
   3. What is the matrix representation of \(r_z\) in an eigenbasis, that is, in a basis consisting of eigenvectors? In other words, change the basis from the original \(x,y,z\) components to a new basis consisting of eigenvectors, and express \(r_z\) with respect to this basis.
   4. If time permits, express \(r_x\) and \(r_y\) with respect to this basis.