Note, also, that unitary matrices (isometries) are of the form $U=e^{iH}$ for $H$ self-adjoint commuting with $U$. In fact, we can define a $C^\infty$ ( exponential ) map $U:\mathbb{R}\rightarrow\mathbb{R}^{n\times n}$ by $U(t)=e^{itH}$ with derivatives $U^{(n)}(t)=t\,U^*H^n$. The space (a Lie algebra ) of self-adjoint matrices $H$ (or anti-self-adjoint if we drop the