Improper rotation
= Improper rotation
{title2=$R^TR=I,\quad\det R=-1$}
An orientation-reversing <orthogonal transformation>. In three dimensions it is a rotation about an axis followed by a <reflection> perpendicular to that axis. Its <eigenvalues> are $-1,e^{i\theta},e^{-i\theta}$, giving $\operatorname{tr}R=-1+2\cos\theta$. A plane <reflection> is the case $\theta=0$, while inversion through the origin is the case $\theta=\pi$.