# Linear fractional transformation

In mathematics, the phrase linear fractional transformation usually refers to a Möbius transformation, which is a homography on the complex projective line P(C) where C is the field of complex numbers.

More generally in mathematics, C may be replaced by another ring (A, +, ×).[1] For example, the Cayley transform is a linear fractional transformation originally defined on the 3 x 3 real matrix ring.

In general, a linear fractional transformation refers to a homography of P(A), the projective line over a ring A. When A is a commutative ring, then a linear fractional transformation has the familiar form

${\displaystyle z\mapsto {\frac {az+b}{cz+d}},}$

where a, b, c, d are elements of A such that acbd is a unit of a (that is acbd has a multiplicative inverse in A)

In a non-commutative ring (A, +, ×), with (z,t) in A2, the units u determine an equivalence relation ${\displaystyle (z,t)\sim (uz,ut).}$ An equivalence class in the projective line over A is written U(z,t). Then linear fractional transformations act on the right of an element of P(A):

${\displaystyle U(z,t){\begin{pmatrix}a&c\\b&d\end{pmatrix}}=U(za+tb,zc+td)\sim U((zc+td)^{-1}(za+tb),1).}$

The ring is embedded in its projective line by z → U(z,1), so t = 1 recovers the usual expression. This linear fractional transformation is well-defined since U(za + tb, zc + td) does not depend on which element is selected from its equivalence class for the operation.

The linear fractional transformations form a group, denoted ${\displaystyle \operatorname {PGL} _{1}(A).}$

The group ${\displaystyle \operatorname {PGL} _{1}(\mathbb {Z} )}$ of the linear fractional transformations is called the modular group. It has been widely studied because its numerous applications to number theory, which include, in particular, Wiles's proof of Fermat's Last Theorem.

## Conformal property

The commutative rings of split-complex numbers and dual numbers join the ordinary complex numbers as rings that express angle. In each case the exponential map applied to the imaginary axis produces an isomorphism between one-parameter groups in (A, + ) and in the group of units (U, × ):

${\displaystyle \exp(yj)=\cosh y+j\sinh y,\quad j^{2}=+1,}$
${\displaystyle \exp(y\epsilon )=1+y\epsilon ,\quad \epsilon ^{2}=0,}$
${\displaystyle \exp(yi)=\cos y+i\sin y,\quad i^{2}=-1.}$

The "angle" y is hyperbolic angle, slope, or circular angle according to the host ring.

A linear fractional transformation can be generated by multiplicative inversion z → 1/z and affine transformations za z + b. Conformality can be confirmed by showing the generators are all conformal. The translation zz + b is a change of origin and makes no difference to angle. To see that zaz is conformal, consider the polar decomposition of a and z. In each case the angle of a is added to that of z giving a conformal map. Finally, inversion is conformal since z → 1/z sends ${\displaystyle \exp(yb)\mapsto \exp(-yb),\quad b^{2}=1,0,-1.}$