Proper rotation
= Proper rotation
{title2=$R^TR=I,\quad\det R=1$}
An orientation-preserving <orthogonal transformation>. In three dimensions it has an invariant axis and acts as a planar rotation through angle $\theta$ perpendicular to that axis. Its <eigenvalues> are $1,e^{i\theta},e^{-i\theta}$, so $\operatorname{tr}R=1+2\cos\theta$. In higher dimensions a <proper rotation> can involve several mutually orthogonal planes and need not fix an axis.