Inversions come in two kinds: inversions at circles and reflections at lines. Since the two have very similar properties, we combine them and talk about inversions at generalized circles.
Given any three distinct points in the extended plane, there exists precisely one generalized circle that passes through the three points.
The extended plane of inversive geometry can be identified with the extended complex plane, so that equations of complex numbers can be used to describe lines, circles and inversions.
Two such invertible hermitian matrices specify the same generalized circle if and only if they differ by a real multiple.