Linear fractional transformation

In mathematics, a linear fractional transformation is, roughly speaking, an invertible transformation of the form

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

linear

.

In the most basic setting, $a, b, c, d$ , and $z$ are

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).^[1]

In the most general setting, the $a, b, c, d$ and $z$ are elements of a

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\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 $A 2$ , the units $u$ determine an equivalence relation $(z,t)\sim (uz,ut).$ An

projective coordinates

. 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 $z \to U [z : 1]$ , 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 $\operatorname {PGL} _{2}(A).$

The group $\operatorname {PGL} _{2}(\mathbb {Z} )$ 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

In the

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

SU(1, 1) where the linear fractional transformations are "special unitary", and for the upper half-plane the isometry group is

PSL(2, R)

, a projective linear group of linear fractional transformations with real entries and determinant equal to one.^[2]

Use in higher mathematics

Möbius transformations commonly appear in the theory of

Anosov flow

for a worked example of the fibration: in this example, the geodesics are given by the fractional linear transform

(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)

with $a$ , $b$ , $c$ and $d$

parabolic transformations

, 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

damped harmonic oscillator. Another elementary application is obtaining the Frobenius normal form, i.e. the companion matrix

of a polynomial.

Conformal property

The commutative rings of

group of units

(U, \times )

:^[5]

\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.

Linear fractional transformations are shown to be conformal maps by consideration of their generators: multiplicative inversion $z \to 1/ z$ and affine transformations $z \to az + b$ . Conformality can be confirmed by showing the generators are all conformal. The translation $z \to z + b$ is a change of origin and makes no difference to angle. To see that $z \to az$ is conformal, consider the polar decomposition of $a$ and $z$ . In each case the angle of $a$ is added to that of $z$ resulting in a conformal map. Finally, inversion is conformal since $z \to 1/ z$ sends $\exp(yb)\mapsto \exp(-yb),\quad b^{2}=1,0,-1.$

References

Linear Algebra and its Applications
56:251–90

ISBN 0-471-79080 X

^ John Doyle, Andy Packard, Kemin Zhou, "Review of LFTs, LMIs, and mu", (1991) Proceedings of the 30th Conference on Decision and Control [1]

^ Juan C. Cockburn, "Multidimensional Realizations of Systems with Parametric Uncertainty" [2]

MR 2977041
.

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,
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,
MR14282

Isaak Yaglom (1968) Complex Numbers in Geometry, page 130 & 157, Academic Press

Retrieved from "https://en.wikipedia.org/w/index.php?title=Linear_fractional_transformation&oldid=1293378563"

[1] Linear Algebra and its Applications
56:251–90

[2] ISBN 0-471-79080 X

[3] John Doyle, Andy Packard, Kemin Zhou, "Review of LFTs, LMIs, and mu", (1991) Proceedings of the 30th Conference on Decision and Control [1]

[4] Juan C. Cockburn, "Multidimensional Realizations of Systems with Parametric Uncertainty" [2]

[5] MR 2977041
.

[1]

[2]

[5]