Improper rotation ($R^{T}R=I,\det R=-1$)

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 $\mathrm{tr}R=-1+2\cos \theta$. A plane reflection is the case $\theta =0$, while inversion through the origin is the case $\theta =\pi$.