Linear fractional transformation

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

which has an inverse. 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 to a field. The invertibility condition is then adbc ≠ 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 integer (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 adbc 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 square matrices, or, more generally, elements of a ring. An example of such linear fractional transformation is the Cayley transform, which was originally defined on the 3 x 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[edit]

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

where a, b, c, d are elements of A such that adbc is a unit of a (that is adbc has a multiplicative inverse in A)

In a non-commutative ring A, with (z,t) in A2, the units u determine an equivalence relation 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):

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

The group 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.

Use in higher mathematics[edit]

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 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

with a, b, c and d real, with . Roughly speaking, the center manifold is generated by the parabolic transformations, the unstable manifold by the hyperbolic transformations, and the stable manifold by the elliptic transformations.

Use in control theory[edit]

Linear fractional transformations are widely used in control theory to solve plant-controller relationship problems in mechanical and electrical engineering.[2][3] 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.

Conformal property[edit]

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, × ):

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 za z + b. Conformality can be confirmed by showing the generators are all conformal. The translation zz + b is a change of origin and makes no difference to angle. To see that zaz 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

See also[edit]

References[edit]

  1. ^ N. J. Young (1984) "Linear fractional transformations in rings and modules", Linear Algebra and its Applications 56:251–90
  2. ^ John Doyle, Andy Packard, Kemin Zhou, "Review of LFTs, LMIs, and mu", (1991) Proceedings of the 30th Conference on Decision and Control [1]
  3. ^ Juan C. Cockburn, "Multidimensional Realizations of Systems with Parametric Uncertainty" [2]
  • 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, MR14282
  • Isaak Yaglom (1968) Complex Numbers in Geometry, page 130 & 157, Academic Press