= Special orthogonal group in two dimensions
{title2=$\operatorname{SO}(2)\cong U(1)$}
= SO(2)
{c}
{synonym}
Every real orthogonal two-by-two matrix of determinant one has the unique form
$$
R(z)=\begin{pmatrix}a&-b\\b&a\end{pmatrix},\qquad z=a+ib,\qquad a^2+b^2=1.
$$
Indeed the first column is a <unit vector>, the perpendicular second column has one of two signs, and positive determinant selects the displayed one. The map $z\mapsto R(z)$ from the <circle group> is bijective and satisfies $R(zw)=R(z)R(w)$ by direct multiplication, proving a <group isomorphism>. Writing $z=e^{i\theta}$ parametrizes these matrices as planar <rotations>, with the angle defined modulo $2\pi$.
Back to article page
