Linear fractional transformation

When A is a commutative ring, then a linear fractional transformation has the familiar form 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

Then linear fractional transformations act on the right of an element of P(A): 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.

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.

To construct models of the hyperbolic plane the unit disk and the upper half-plane are used to represent the points.

Linear fractional transformations leave cross ratio invariant, so any linear fractional transformation that leaves the unit disk or upper half-planes stable is an isometry of the hyperbolic plane metric space.

[2] Möbius transformations commonly appear in the theory of continued fractions, and in analytic number theory of elliptic curves and modular forms, as they describe automorphisms of the upper half-plane under the action of the modular group.

See Anosov flow for a worked example of the fibration: in this example, the geodesics are given by the fractional linear transform with a, b, c and d real numbers, with ad − bc = 1.

Linear fractional transformations are widely used in control theory to solve plant-controller relationship problems in mechanical and electrical engineering.

[3][4] The general procedure of combining linear fractional transformations with the Redheffer star product allows them to be applied to the scattering theory of general differential equations, including the S-matrix approach in quantum mechanics and quantum field theory, the scattering of acoustic waves in media (e.g. thermoclines and submarines in oceans, etc.)

Perhaps the simplest example application of linear fractional transformations occurs in the analysis of the damped harmonic oscillator.

Another elementary application is obtaining the Frobenius normal form, i.e. the companion matrix of a polynomial.