= Three-dimensional improper orthogonal transformation
A real <orthogonal matrix> $T$ of size three with $\det T=-1$ has an <eigenvector> $n$ of <eigenvalue> $-1$. Its <orthogonal complement> is invariant, and the restriction there is an orientation-preserving plane rotation. Thus $T$ is a rotation about the axis $\mathbb Rn$ composed with a <reflection (mathematics)> across the plane perpendicular to $n$. Writing the plane angle as $\phi$, $\operatorname{tr}T=-1+2\cos\phi$. A pure plane <reflection (mathematics)> is the case $\phi=0$; if $\det(T-I)\ne0$, it cannot be a pure plane <reflection (mathematics)>.
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