Möbius transformations representable as 2x2 matrices

Although there may be some question of whether the implicit or the explicit form of one-to-one and invertible mappings should be called Möbius transformations, that is the name given to this class of transformations. Mappings often form a group. Since the associative law for their composition is a given, the items to be checked are closure, invertibility and the existence of the identity.

To check closure, consider the composite of two maps. First, suppose we have

$$\frac{Aw+B}{Cw+D}$$ t = (29)

$$\frac{az+b}{cz+d}$$ w = (30)

which gives, on substituting,

$$\frac{A\frac{az+b}{cz+d}+B}{C\frac{az+b}{cz+d}+D},$$ t = (31)

and, on simplifying,

$$\frac{(Aa+Bc)z+(Ab+Bd)}{(Ca+Dc)z+(Cb+Dd)},$$ t = (32)

which mimics the matrix product

$$\left[ \begin{array}{cc} A & B \\ C & D \end{array} \right] ... ...{array}{cc} Aa+Bc & Ab+Bd \\ Ca+Dc & Cb+Dd \end{array} \right]$$

From this it is apparent that the $2\times 2$ unimodular matrices form a representation of the group of Möbius transformations; in particular the identity is represented by the unit matrix,

$$\frac{1z+0}{0z+1},$$ z = (33)

and the inverse transformation by the inverse matrix.