Editing Linear fractional transformation

{{Use American English|date = January 2019}}
{{Short description|Möbius transformation generalized to rings other than the complex numbers}}

In [[mathematics]], a '''linear fractional transformation''' is, roughly speaking, an [[inverse function|invertible]] transformation of the form
: <math>z \mapsto \frac{az + b} {cz + d}.</math>

The precise definition depends on the nature of {{math|''a'', ''b'', ''c'', ''d''}}, and {{mvar|z}}. In other words, a linear fractional transformation is a ''[[transformation (function)|transformation]]'' that is represented by a ''fraction'' whose numerator and denominator are ''[[linear polynomial|linear]]''.

In the most basic setting, {{math|''a'', ''b'', ''c'', ''d''}}, and {{mvar|z}} are [[complex number]]s (in which case the transformation is also called a [[Möbius transformation]]), or more generally elements of a [[field (mathematics)|field]]. The invertibility condition is then {{math|''ad'' – ''bc'' ≠ 0}}. Over a field, a linear fractional transformation is the [[restriction (mathematics)|restriction]] to the field of a [[projective transformation]] or [[homography]] of the [[projective line]].

When {{math|''a'', ''b'', ''c'', ''d''}} are [[integer|integers]] (or, more generally, belong to an [[integral domain]]), {{math|''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 {{math|''ad'' – ''bc''}} must be a [[unit (ring theory)|unit]] of the domain (that is {{math|1}} or {{math|−1}} in the case of integers).<ref>N. J. Young (1984) [http://www.sciencedirect.com/science/article/pii/0024379584901319 "Linear fractional transformations in rings and modules"], [[Linear Algebra and its Applications]] 56:251–90</ref>

In the most general setting, the {{math|''a'', ''b'', ''c'', ''d''}} and {{math|''z''}} are elements of a [[ring (mathematics)|ring]], such as [[square matrices]]. An example of such linear fractional transformation is the [[Cayley transform]], which was originally defined on the {{nowrap|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#Over a ring|homography]] of {{math|P(''A'')}}, the [[projective line over a ring]] {{math|''A''}}. When {{math|''A''}} is a [[commutative ring]], then a linear fractional transformation has the familiar form
: <math>z \mapsto \frac{az + b} {cz + d},</math>
where {{math|''a'', ''b'', ''c'', ''d''}} are elements of {{math|''A''}} such that {{math|''ad'' – ''bc''}} is a [[unit (ring theory)|unit]] of {{math|''A''}} (that is {{math|''ad'' – ''bc''}} has a [[multiplicative inverse]] in {{math|''A''}}).

In a non-commutative ring {{math|''A''}}, with {{math|(''z'', ''t'')}} in {{math|''A''<sup>2</sup>}}, the units {{math|''u''}} determine an [[equivalence relation]] <math>(z,t) \sim (uz,ut) .</math> An [[equivalence class]] in the projective line over ''A'' is written {{math|''U''[''z'' : ''t'']}}, where the brackets denote [[projective coordinates]]. Then linear fractional transformations act on the right of an element of {{math|P(''A'')}}:
: <math>U[z:t] \begin{pmatrix}a & c \\ b & d \end{pmatrix} = U[za + tb:\ zc + td] \sim U[(zc + td)^{-1}(za + tb):\ 1].</math>
The ring is embedded in its projective line by {{math|''z'' → ''U''[''z'' : 1]}}, so {{math|1=''t'' = 1}} recovers the usual expression. This linear fractional transformation is well-defined since {{math|''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 {{math|''A''}} form a [[group (mathematics)|group]], the [[projective linear group]] denoted <math>\operatorname{PGL}_2(A). </math>

The group <math>\operatorname{PGL}_2(\Z)</math> 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 ==
{{main|Hyperbolic geometry}}
In the [[complex plane]] a [[generalized circle]] is either a line or a circle. When completed with the point at infinity, the generalized circles in the plane correspond to circles on the surface of the [[Riemann sphere]], an expression of the complex projective line. Linear fractional transformations permute these circles on the sphere, and the corresponding finite points of the generalized circles in the complex plane.

To construct models of the hyperbolic plane the [[unit disk]] and the [[upper half-plane]] are used to represent the points. These subsets of the complex plane are provided a [[metric (mathematics)|metric]] with the [[Cayley–Klein metric]]. Then the distance between two points is computed using the generalized circle through the points and perpendicular to the boundary of the subset used for the model. This generalized circle intersects the boundary at two other points. All four points are used in the [[cross ratio]] which defines the Cayley–Klein metric. 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]]. Since [[Henri Poincaré]] explicated these models they have been named after him: the [[Poincaré disk model]] and the [[Poincaré half-plane model]]. Each model has a [[group (mathematics)|group]] of isometries that is a subgroup of the [[Mobius group]]: the isometry group for the disk model is {{math|[[SU(1,&nbsp;1)]]}} where the linear fractional transformations are "special unitary", and for the upper half-plane the isometry group is {{math|PSL(2, '''R''')}}, a [[projective linear group]] of linear fractional transformations with real entries and [[determinant]] equal to one.<ref>[[C. L. Siegel]] (A. Shenitzer & M. Tretkoff, translators) (1971) ''Topics in Complex Function Theory'', volume 2, Wiley-Interscience {{ISBN|0-471-79080 X}}</ref>

== Use in higher mathematics ==
Möbius transformations commonly appear in the theory of [[continued fraction]]s, and in [[analytic number theory]] of [[elliptic curve]]s and [[modular form]]s, as they describe automorphisms of the upper half-plane under the action of the [[modular group]]. They also provide a canonical example of [[Hopf fibration]], where the [[geodesic flow]] induced by the linear fractional transformation decomposes complex projective space into [[stable manifold|stable and unstable manifolds]], with the [[horocycle]]s 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
:<math>(i \exp(t),\ 1 )\begin{pmatrix} a & c \\ b & d \end{pmatrix}\ =\ (ai\exp(t)+b, \ ci\exp(t)+d) \thicksim \left(\frac{ai\exp(t)+b}{ci\exp(t)+d},\ 1\right)</math>
with {{math|''a''}}, {{math|''b''}}, {{math|''c''}} and {{math|''d''}} [[real number]]s, with {{math|1=''ad'' − ''bc'' = 1}}.  Roughly speaking, the [[center manifold]] is generated by the [[parabolic transformation]]s, the unstable manifold by the hyperbolic transformations, and the stable manifold by the elliptic transformations.

== Use in control theory == 
Linear fractional transformations are widely used in [[control theory]] to solve plant-controller relationship problems in [[mechanical engineering|mechanical]] and [[electrical engineering]].<ref>John Doyle, Andy Packard, Kemin Zhou, "Review of LFTs, LMIs, and mu", (1991) ''Proceedings of the 30th Conference on Decision and Control'' [http://www.cds.caltech.edu/~doyle/wiki/images/7/70/CDC1991.pdf]</ref><ref>Juan C. Cockburn, "Multidimensional Realizations of Systems with Parametric Uncertainty" [https://www.math.ucsd.edu/~helton/MTNSHISTORY/CONTENTS/2000PERPIGNAN/CDROM/articles/SI20A_4.pdf]</ref> 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 {{nowrap|3 × 3}} 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.

== Conformal property ==
[[File:Planar_rotations.png|500px|center|alt=Planar rotations with complex, hyperbolic and dual numbers.]]
The commutative rings of [[split-complex number]]s and [[dual number]]s join the ordinary [[complex number]]s as rings that express angle and "rotation". In each case the [[exponential map (Lie theory)|exponential map]] applied to the imaginary axis produces an [[group isomorphism|isomorphism]] between [[one-parameter group]]s in {{math|(''A'', + )}} and in the [[group of units]] {{math|(''U'', × )}}:<ref>{{cite book |last=Kisil |first=Vladimir V. |year=2012 |title=Geometry of Möbius transformations. Elliptic, parabolic and hyperbolic actions of SL(2,R) | location=London |publisher=Imperial College Press|page=xiv+192 |isbn=978-1-84816-858-9 | mr=2977041 | doi=10.1142/p835}}</ref>
: <math>\exp(y j) = \cosh y + j \sinh y, \quad j^2 = +1 ,</math>
: <math>\exp(y \epsilon) = 1 + y \epsilon, \quad \epsilon^2 = 0 ,</math>
: <math>\exp(y i) = \cos y + i \sin y, \quad i^2 = -1 .</math>
The "angle" {{math|''y''}} is [[hyperbolic angle]], [[slope]], or [[angle|circular angle]] according to the host ring.

Linear fractional transformations are shown to be [[conformal map]]s by consideration of their [[generator (mathematics)|generator]]s: [[multiplicative inverse|multiplicative inversion]] {{math|''z'' → 1/''z''}} and [[affine transformation]]s {{math|''z'' → ''az'' + ''b''}}. Conformality can be confirmed by showing the generators are all conformal. The translation {{math|''z'' → ''z'' + ''b''}} is a change of origin and makes no difference to angle. To see that {{math|''z'' → ''az''}} is conformal, consider the [[polar decomposition#Alternative planar decompositions|polar decomposition]] of {{math|''a''}} and {{math|''z''}}. In each case the angle of {{math|''a''}} is added to that of {{math|''z''}} resulting in a conformal map. Finally, inversion is conformal since {{math|''z'' → 1/''z''}} sends <math>\exp(y b) \mapsto \exp(-y b), \quad b^2 = 1, 0, -1 .</math>

== See also==
* [[Laguerre transformations]]
* [[Linear-fractional programming]]
* [[H-infinity methods in control theory]]

== References ==
{{Reflist}}
* 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) [http://projecteuclid.org/euclid.pja/1195578674 Gemeinsame Behandlungsweise der elliptischen konformen, hyperbolischen konformen und parabolischen konformen Differentialgeometrie, 2], [[Japan Academy|Proceedings of the Imperial Academy]] 17(8): 330–8, link from [[Project Euclid]], {{mr|id=14282}}
* [[Isaak Yaglom]] (1968) ''Complex Numbers in Geometry'', page 130 & 157, [[Academic Press]]

[[Category:Rational functions]]
[[Category:Conformal mappings]]
[[Category:Projective geometry]]