Complex orthogonal matrix
= Complex orthogonal matrix
{title2=$A^TA=I$}
An invertible matrix over the <complex numbers> preserving the symmetric bilinear form $v^Tw$: $A^TA=I$. The transpose here does not include <complex conjugation>, unlike the defining condition for a <unitary matrix>. For example, $\begin{pmatrix}\cosh t&i\sinh t\\-i\sinh t&\cosh t\end{pmatrix}$ is complex orthogonal for real $t$, but is not unitary when $t\ne0$.