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 , and . 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, , and are
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s (in which case the transformation is also called a
Möbius transformation), or more generally elements of a
field. The invertibility condition is then . Over a field, a linear fractional transformation is the
restriction
Restriction, restrict or restrictor may refer to:
Science and technology
* restrict, a keyword in the C programming language used in pointer declarations
* Restriction enzyme, a type of enzyme that cleaves genetic material
Mathematics and logi ...
to the field of a
projective transformation
In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In general, ...
or
homography of the
projective line
In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a ''point at infinity''. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; ...
.
When are
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
(or, more generally, belong to an
integral domain
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...
), is supposed to be a
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rat ...
(or to belong to the
field of fractions
In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
of the integral domain. In this case, the invertibility condition is that must be a
unit
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* ''Unit'' (a ...
of the domain (that is or in the case of integers).
In the most general setting, the and are
square matrices
In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied.
Square matrices are often ...
, 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
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
,
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...
(they are used, for example, in
Wiles's proof of Fermat's Last Theorem
Wiles's proof of Fermat's Last Theorem is a proof by British mathematician Andrew Wiles of a special case of the modularity theorem for elliptic curves. Together with Ribet's theorem, it provides a proof for Fermat's Last Theorem. Both Ferma ...
),
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
,
control theory
Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a ...
.
General definition
In general, a linear fractional transformation is a
homography of P(''A''), the
projective line over a ring
In mathematics, the projective line over a ring is an extension of the concept of projective line over a field. Given a ring ''A'' with 1, the projective line P(''A'') over ''A'' consists of points identified by projective coordinates. Let ''U ...
''A''. When ''A'' is a
commutative ring, then a linear fractional transformation has the familiar form
:
where are elements of ''A'' such that is a
unit
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* ''Unit'' (a ...
of ''A'' (that is has a
multiplicative inverse
In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/ ...
in ''A'')
In a non-commutative ring ''A'', with (''z,t'') in ''A''
2, the units ''u'' determine an
equivalence relation An
equivalence class in the projective line over ''A'' is written U
'z : t''where the brackets denote
projective coordinates. 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 over ''A'' form a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
, denoted
The group
of the linear fractional transformations is called the
modular group. It has been widely studied because of its numerous applications to
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...
, which include, in particular,
Wiles's proof of Fermat's Last Theorem
Wiles's proof of Fermat's Last Theorem is a proof by British mathematician Andrew Wiles of a special case of the modularity theorem for elliptic curves. Together with Ribet's theorem, it provides a proof for Fermat's Last Theorem. Both Ferma ...
.
Use in 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
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers ...
, 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
In mathematics, the upper half-plane, \,\mathcal\,, is the set of points in the Cartesian plane with > 0.
Complex plane
Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds to ...
are used to represent the points. These subsets of the complex plane are provided a
metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathem ...
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
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' me ...
of the hyperbolic plane
metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
. 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
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
of isometries that is a subgroup of the
Mobius group: the isometry group for the disk model is
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
In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space ''V'' on the associate ...
of linear fractional transformations with real entries and
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
equal to one.
Use in higher mathematics
Möbius transformations commonly appear in the theory of
continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer ...
s, and in
analytic number theory of
elliptic curve
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
s 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
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
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
In mathematics, more particularly in the fields of dynamical systems and geometric topology, an Anosov map on a manifold ''M'' is a certain type of mapping, from ''M'' to itself, with rather clearly marked local directions of "expansion" and "contr ...
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
Linear fractional transformations are widely used in
control theory
Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a ...
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
In physics, the ''S''-matrix or scattering matrix relates the initial state and the final state of a physical system undergoing a scattering process. It is used in quantum mechanics, scattering theory and quantum field theory (QFT).
More forma ...
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 In linear algebra, the Frobenius companion matrix of the monic polynomial
:
p(t)=c_0 + c_1 t + \cdots + c_t^ + t^n ~,
is the square matrix defined as
:C(p)=\begin
0 & 0 & \dots & 0 & -c_0 \\
1 & 0 & \dots & 0 & -c_1 \\
0 & 1 & \dots & 0 & -c_2 ...
of a polynomial.
Conformal property
The commutative rings of
split-complex numbers and
dual number
In algebra, the dual numbers are a hypercomplex number system first introduced in the 19th century. They are expressions of the form , where and are real numbers, and is a symbol taken to satisfy \varepsilon^2 = 0 with \varepsilon\neq 0.
Du ...
s join the ordinary
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s as rings that express angle and "rotation". In each case the
exponential map applied to the imaginary axis produces an
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
between
one-parameter groups in (''A'', + ) and in the
group of units (''U'', × ):
:
:
:
The "angle" ''y'' is
hyperbolic angle
In geometry, hyperbolic angle is a real number determined by the area of the corresponding hyperbolic sector of ''xy'' = 1 in Quadrant I of the Cartesian plane. The hyperbolic angle parametrises the unit hyperbola, which has hyperbolic function ...
,
slope
In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is use ...
, 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'' → 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
In mathematics, the polar decomposition of a square real or complex matrix A is a factorization of the form A = U P, where U is an orthogonal matrix and P is a positive semi-definite symmetric matrix (U is a unitary matrix and P is a positive se ...
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'' → 1/''z'' sends
See also
*
Laguerre transformations
*
Linear-fractional programming
*
H-infinity methods in control theory ''H''∞ (i.e. "''H''-infinity") methods are used in control theory to synthesize controllers to achieve stabilization with guaranteed performance. To use ''H''∞ methods, a control designer expresses the control problem as a mathematical optimiz ...
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
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Originally founded in 1842 ...
.
* 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
The University of British Columbia (UBC) is a public research university with campuses near Vancouver and in Kelowna, British Columbia. Established in 1908, it is British Columbia's oldest university. The university ranks among the top thre ...
.
* 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
''Advances in Applied Clifford Algebras'' is a peer-reviewed scientific journal that publishes original research papers and also notes, expository and survey articles, book reviews, reproduces abstracts and also reports on conferences and workshops ...
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 Project Euclid is a collaborative partnership between Cornell University Library and Duke University Press which seeks to advance scholarly communication in theoretical and applied mathematics and statistics through partnerships with independent an ...
, {{mr, id=14282
*
Isaak Yaglom (1968) ''Complex Numbers in Geometry'', page 130 & 157,
Academic Press
Academic Press (AP) is an academic book publisher founded in 1941. It was acquired by Harcourt, Brace & World in 1969. Reed Elsevier bought Harcourt in 2000, and Academic Press is now an imprint of Elsevier.
Academic Press publishes referen ...
Rational functions
Conformal mappings
Projective geometry