In mathematics, a linear fractional transformation is, roughly speaking, an invertible transformation of the form
z ↦ a z + b c z + d . {\displaystyle z\mapsto {\frac {az+b}{cz+d}}.}
The precise definition depends on the nature of a, b, c, d, and z. In other words, a linear fractional transformation is a transformation that is represented by a fraction whose numerator and denominator are linear. In the most basic setting, a, b, c, d, and z are complex numbers (in which case the transformation is also called a Möbius transformation), or more generally elements of a field. The invertibility condition is then ad – bc ≠ 0. Over a field, a linear fractional transformation is the restriction to the field of a projective transformation or homography of the projective line. When a, b, c, d are integers (or, more generally, belong to an integral domain), z is supposed to be a rational number (or to belong to the field of fractions of the integral domain). In this case, the invertibility condition is that ad – bc must be a unit of the domain (that is 1 or −1 in the case of integers). In the most general setting, the a, b, c, d and z are elements of a ring, such as square matrices. An example of such linear fractional transformation is the Cayley transform, which was originally defined on the 3 × 3 real matrix ring. Linear fractional transformations are widely used in various areas of mathematics and its applications to engineering, such as classical geometry, number theory (they are used, for example, in Wiles's proof of Fermat's Last Theorem), group theory, control theory.
General definition In general, a linear fractional transformation is 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
z ↦ a z + b c z + d , {\displaystyle z\mapsto {\frac {az+b}{cz+d}},}
where a, b, c, d are elements of A such that ad – bc is a unit of A (that is ad – bc has a multiplicative inverse in A). In a non-commutative ring A, with (z, t) in A2, the units u determine an equivalence relation ( z , t ) ∼ ( u z , u t ) . {\displaystyle (z,t)\sim (uz,ut).} An equivalence class in the projective line over A is written U[z : t], where the brackets denote projective coordinates. Then linear fractional transformations act on the right of an element of P(A):
U [ z : t ] ( a c b d ) = U [ z a + t b : z c + t d ] ∼ U [ ( z c + t d ) − 1 ( z a + t b ) : 1 ] . {\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 over A form a group, the projective linear group denoted PGL 2 ( A ) . {\displaystyle \operatorname {PGL} _{2}(A).}
The group PGL 2 ( Z ) {\displaystyle \operatorname {PGL} _{2}(\mathbb {Z} )} of the linear fractional transformations is called the modular group. It has been widely studied because of its numerous applications to number theory, which include, in particular, Wiles's proof of Fermat's Last Theorem.
Use in hyperbolic geometry
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