Take $U \in SU(n)$ and $X \in \mathfrak{su}(n)$. What is known about
\begin{align} \text{Tr} (UX) \end{align}
In particular
- Are there any useful identities that apply here?
- When does $\text{Tr} (UX)$ vanish?
- If we can't answer the previous questions, can we bound the expression? But then what is known about $\text{Tr} (X^2)$?