Hello everyone, I think I know how to do the proof but I just wanted to clarify something just in case so I don't screw up (for part b). For part B when it says let $U=(uij)$ be a two-dimensional rotation matrix, that doesn't mean that U is a 2 x 2 rotational matrix, right? Otherwise then it wouldn't make sense to let $B=UAU^{*}$ where A is an n x n matrix (as the multiplication could not be done. I'm pretty sure this is just the Givens Rotation matrix where basically we have a n x n identity matrix with only 4 off diagonal elements being non-zero (that is they are defined to be $cos(\theta)$,$cos(\theta)$,$sin(\theta)$,-$sin(\theta)$. I just wanted to verify whether this was correct so I don't screw up the proof. I was just concerned because it uses the words two-dimensional and I just wanted to make sure.
Yes, you're right; as already stated in a comment, $U$ is an $n\times n$ matrix that performs a rotation through an angle $\theta$ in a two-dimensional plane in $\mathbb R^n$.