Generalised Circle - Representation By Hermitian Matrices

Representation By Hermitian Matrices

The data defining the equation of a generalized circle

$A z \bar z + B z + C \bar z + D = 0$

can be usefully put into the form of an invertible hermitian matrix

$\mathfrak C = \begin{pmatrix}A & B \\ C & D \end{pmatrix} = \mathfrak C ^\dagger.$

Two such invertible hermitian matrices specify the same generalized circle if and only if they differ by a real multiple.

To transform a generalized circle described by by the Möbius transformation, you simply do

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