= Real rotation matrices that are Hermitian
A real <rotation matrix> is a <unitary matrix> because it is an <orthogonal matrix>. It is a <Hermitian matrix> precisely when it is symmetric, equivalently when its square is the identity. Its <eigenvalues> are then $\pm1$. In three dimensions a proper rotation is therefore Hermitian only for the identity or a half-turn. A half-turn about a unit vector $n$ has matrix $2nn^T-I$: it fixes $n$ and reverses the perpendicular plane. For every real rotation $R$, the difference $R-R^T$ is <anti-Hermitian>; it is also Hermitian only in the zero case.
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