= Orthogonal involutions of the plane
{title2=$A^TA=I,\quad A^2=I$}
For a real orthogonal involution, $A^{-1}=A=A^T$, so it is a <symmetric matrix>. Its <eigenvalues> are $\pm1$. In two dimensions the possibilities are the identity, the half-turn $-I$, and reflections
$$
A=\begin{pmatrix}\cos\theta&\sin\theta\\\sin\theta&-\cos\theta\end{pmatrix}.
$$
The reflection fixes the line making angle $\theta/2$ with the first coordinate axis and negates its perpendicular direction. This follows either from the two corresponding <eigenvectors> or by rotating $\operatorname{diag}(1,-1)$. A nonidentity element of order two in a <faithful representation> must be either the half-turn or a reflection; the identity matrix is excluded by faithfulness.
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