Complex orthogonal matrix ($A^{T}A=I$)

An invertible matrix over the complex numbers preserving the symmetric bilinear form $v^{T}w$: $A^{T}A=I$. The transpose here does not include complex conjugation, unlike the defining condition for a unitary matrix. For example, $\left(\begin{array}{cc}\cosh t& i\sinh t\\ -i\sinh t& \cosh t\end{array}\right)$ is complex orthogonal for real $t$, but is not unitary when $t\ne 0$.

Every two-by-two complex orthogonal matrix has form $A=\left(\begin{array}{cc}a& b\\ -\epsilon b& \epsilon a\end{array}\right)$, where $\epsilon =\det A\in \{+1,-1\}$ 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.