Editing Möbius transformation (section)

=== Isomorphism between the Möbius group and {{nowrap|PGL(2, C)}} ===
The natural [[Group action (mathematics)|action]] of {{nowrap|PGL(2, '''C''')}} on the [[complex projective line]] '''CP'''<sup>1</sup> is exactly the natural action of the Möbius group on the Riemann sphere
==== Correspondance between the complex projective line and the Riemann sphere ====
Here, the projective line '''CP'''<sup>1</sup> and the Riemann sphere are identified as follows:
<math display="block">[z_1 : z_2]\ \thicksim \frac{z_1}{z_2}.</math>

Here [''z''<sub>1</sub>:''z''<sub>2</sub>] are [[homogeneous coordinates]] on '''CP'''<sup>1</sup>; the point [1:0] corresponds to the point {{math|∞}} of the Riemann sphere. By using homogeneous coordinates, many calculations involving Möbius transformations can be simplified, since no case distinctions dealing with {{math|∞}} are required.

==== Action of PGL(2,&nbsp;C) on the complex projective line ====
Every [[invertible matrix|invertible]] complex 2×2 matrix
<math display="block">\mathfrak H = \begin{pmatrix} a & b \\ c & d \end{pmatrix}</math>
acts on the projective line as
<math display="block">z = [z_1:z_2]\mapsto w = [w_1:w_2],</math>
where
<math display="block"> \begin{pmatrix}w_1\\w_2\end{pmatrix}=\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} z_1 \\ z_2 \end{pmatrix}= \begin{pmatrix}az_1 + bz_2\\ cz_1 + dz_2\end{pmatrix}.</math>

The result is therefore 
<math display="block"> w = [w_1:w_2] = [az_1 + bz_2 : cz_1 + dz_2] </math>

Which, using the above identification, corresponds to the following point on the Riemann sphere : 

<math display="block"> w = [az_1 + bz_2 : cz_1 + dz_2] \thicksim \frac{az_1 + bz_2}{cz_1 + dz_2} = \frac{a\frac{z_1}{z_2} + b}{c\frac{z_1}{z_2} + d}. </math>

==== Equivalence with a Möbius transformation on the Riemann sphere ====
Since the above matrix is invertible if and only if its [[determinant]] {{math|''ad'' − ''bc''}} is not zero, this induces an identification of the action of the group of Möbius transformations with the action of {{nowrap|PGL(2, '''C''')}} on the complex projective line. In this identification, the above matrix <math>\mathfrak H</math> corresponds to the Möbius transformation <math>z\mapsto \frac{az+b}{cz+d}.</math>

This identification is a [[group isomorphism]], since the multiplication of <math>\mathfrak H</math> by a non zero scalar <math>\lambda</math> does not change the element of {{nowrap|PGL(2, '''C''')}}, and, as this multiplication consists of multiplying all matrix entries by <math>\lambda,</math> this does not change the corresponding Möbius transformation.