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. It generally takes the form

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

namely, it takes the form of a fraction, with both the numerator and the denominator being linear. In the most basic setting (the Möbius transformation) the a, b, c, d are complex numbers. In the general setting, they are matrices, or, more generally, elements of a ring.

Linear fractional transformations are widely used in control theory to solve plant-controller relationship problems in mechanical and electrical engineering. 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.) and the general analysis of scattering and bound states in differential equations. Here, the 3x3 matrix components refer to the incoming, bound and outgoing states. 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.

In mathematics, the most basic setting for linear fractional transforms is the Möbius transformation, which commonly appears in the theory of continued fractions, and in the analytic number theory of elliptic curves and modular forms, as it describes the automorphisms of the upper half-plane under the action of the modular group. It also provides a canonical example of Hopf fibration, where the geodesic flow induced by the linear fractional transformation decomposes complex projective space into stable and unstable manifolds, with the horocycles appearing perpendicular to the geodesics. See Anosov flow for a worked example of the fibration: in this example, the geodesics are given by the fractional linear transform

${\begin{pmatrix}a&b\\c&d\end{pmatrix}}\cdot i\exp(t)={\frac {ai\exp(t)+b}{ci\exp(t)+d}}$

with a, b, c and d real, with $ad-bc=1$.

More generally in mathematics, the linear fractional transformation can be formulated on any ring. For example, the Cayley transform is a linear fractional transformation originally defined on the 3 x 3 real matrix ring.

On rings

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

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

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 $(z,t)\sim (uz,ut).$ 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):

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

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 $\operatorname {PGL} _{1}(A).$

The group $\operatorname {PGL} _{1}(\mathbb {Z} )$ 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, × ):

$\exp(yj)=\cosh y+j\sinh y,\quad j^{2}=+1,$

$\exp(y\epsilon )=1+y\epsilon ,\quad \epsilon ^{2}=0,$

$\exp(yi)=\cos y+i\sin y,\quad i^{2}=-1.$

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 z → a z + b. Conformality can be confirmed by showing the generators are all conformal. The translation z → z + b is a change of origin and makes no difference to angle. To see that z → az 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 $\exp(yb)\mapsto \exp(-yb),\quad b^{2}=1,0,-1.$

References

• B.A. Dubrovin, A.T. Fomenko, S.P. Novikov (1984) Modern Geometry — Methods and Applications, volume 1, chapter 2, §15 Conformal transformations of Euclidean and Pseudo-Euclidean spaces of several dimensions, Springer-Verlag ISBN 0-387-90872-2.
• Geoffry Fox (1949) Elementary Theory of a hypercomplex variable and the theory of conformal mapping in the hyperbolic plane, Master’s thesis, University of British Columbia.
• P.G. Gormley (1947) "Stereographic projection and the linear fractional group of transformations of quaternions", Proceedings of the Royal Irish Academy, Section A 51:67–85.
• A.E. Motter & M.A.F. Rosa (1998) "Hyperbolic calculus", Advances in Applied Clifford Algebras 8(1):109 to 28, §4 Conformal transformations, page 119.
• Tsurusaburo Takasu (1941) Gemeinsame Behandlungsweise der elliptischen konformen, hyperbolischen konformen und parabolischen konformen Differentialgeometrie, 2, Proceedings of the Imperial Academy 17(8): 330–8, link from Project Euclid, MR 0014282
• Isaak Yaglom (1968) Complex Numbers in Geometry, page 130 & 157, Academic Press