Editing Möbius transformation (section)

==== 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.