= Two-dimensional complex orthogonal matrix
{title2=$a^2+b^2=1$}
Every two-by-two <complex orthogonal matrix> has form $A=\begin{pmatrix}a&b\\-\varepsilon b&\varepsilon a\end{pmatrix}$, where $\varepsilon=\det A\in\{+1,-1\}$ and $a^2+b^2=1$. This follows by equating $A^{-1}$ to $A^T$ in the two-by-two inverse formula, and direct multiplication proves the converse. A complete parametrization is $a=\cosh t$, $b=i\sinh t$ with complex $t$; restricting $t$ to real values selects only a subfamily.
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