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Linear fractional transformation

A linear fractional transformation, also known as a Möbius transformation, is a function f: \hat{\mathbb{C}} \to \hat{\mathbb{C}} of the form f(z) = \frac{az + b}{cz + d}, where a, b, c, d \in \mathbb{C} are complex coefficients satisfying ad - bc \neq 0, and \hat{\mathbb{C}} = \mathbb{C} \cup \{\infty\} denotes the extended complex plane.^[1]^[2] These transformations are rational functions of degree one, analytic everywhere except at the pole z = -d/c (if c \neq 0), and they extend continuously to the point at infinity.^[1] Linear fractional transformations form a group under composition, isomorphic to the projective linear group \mathrm{PGL}(2, \mathbb{C}), as the associated $2 \times 2 matrices \begin{pmatrix} a & b \\ c & d \end{pmatrix} (up to nonzero scalar multiples) multiply to yield the coefficients of the composite transformation.^[2] They preserve the cross-ratio of four points, a fundamental invariant in projective geometry, and map generalized circles—circles or straight lines in the complex plane—to other generalized circles.^[1]^[3] This mapping property arises from their decomposition into elementary transformations: translations, rotations, scalings, and inversions.^[1] Given any three distinct points in the extended complex plane, there exists a unique linear fractional transformation mapping them to any other three distinct points, preserving orientation.^[2] Each non-identity transformation has either one or two fixed points, solutions to f(z) = z, which classify them as parabolic, elliptic, hyperbolic, or loxodromic based on the trace of the matrix.^[1] In complex analysis, these transformations are essential for conformal mappings, such as stereographic projections and mappings between domains like the unit disk and upper half-plane.^[3] They also play key roles in hyperbolic geometry, where subgroups act as isometries of the hyperbolic plane, and in broader applications including Kleinian groups and Riemann surface theory.^[1]

Definition and Algebraic Structure

General Form and Notation

A linear fractional transformation, also known as a Möbius transformation, is a function of the form f(z) = \frac{az + b}{cz + d}, where a, b, c, d are complex numbers and the coefficients satisfy the condition ad - bc \neq 0.^[4] This form arises naturally in complex analysis as a ratio of linear polynomials, providing a flexible way to map points in the complex plane. The condition ad - bc \neq 0 ensures that the transformation is not constant; if ad - bc = 0, the numerator would be a scalar multiple of the denominator, resulting in a constant function.^[5] This determinant, often denoted as \Delta = ad - bc, plays a crucial role in preserving the non-degeneracy of the map.^[6] The coefficients a, b, c, d are defined up to a common scalar multiple, meaning that f(z) remains unchanged if a, b, c, d are multiplied by the same nonzero complex number k.^[7] This ambiguity allows for normalization, such as scaling so that ad - bc = 1, which associates the transformation with matrices in the special linear group \mathrm{SL}(2, \mathbb{C}).^[8] In certain contexts, other normalizations like dividing by \sqrt{ad - bc} may be used, but the unit determinant is standard for group-theoretic studies.^[6] Simple examples illustrate the versatility of these transformations. A translation is given by f(z) = z + b with b \neq 0, corresponding to a = d = 1, c = 0.^[9] A scaling (or dilation) takes the form f(z) = a z where a \neq 0, with b = c = 0, d = 1.^[9] An inversion, such as f(z) = 1/z, has a = d = 0, b = c = 1, and maps the origin to infinity.^[10] These transformations are defined on the complex plane \mathbb{C} excluding poles, where a pole occurs at z = -d/c if c \neq 0; if c = 0, the map is entire (holomorphic everywhere in \mathbb{C}).^[4] The range is the extended complex plane \hat{\mathbb{C}} = \mathbb{C} \cup \{\infty\}, with f(z) \to \infty as z approaches a pole, and \infty mapping to a/c if c \neq 0.^[5] This setup allows linear fractional transformations to act bijectively on \hat{\mathbb{C}}.

Matrix Representation

Linear fractional transformations can be represented using 2×2 matrices over the complex numbers. Specifically, the transformation f(z) = \frac{az + b}{cz + d}, where ad - bc \neq 0, corresponds to the matrix \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm{GL}(2, \mathbb{C}). This association arises from the action on homogeneous coordinates: the matrix acts on the vector \begin{pmatrix} z \\ 1 \end{pmatrix} to yield \begin{pmatrix} az + b \\ cz + d \end{pmatrix}, and the resulting transformation is obtained by normalizing the first component by the second, provided the denominator is nonzero.^[4] Matrices representing the same linear fractional transformation are equivalent up to scalar multiplication: if \lambda \in \mathbb{C}^\times, then \lambda \begin{pmatrix} a & b \\ c & d \end{pmatrix} defines the identical map f(z). This equivalence identifies matrices differing by a nonzero scalar multiple, leading to the projective general linear group \mathrm{PGL}(2, \mathbb{C}) = \mathrm{GL}(2, \mathbb{C}) / \mathbb{C}^\times I, where I is the identity matrix; the group of all linear fractional transformations is isomorphic to \mathrm{PGL}(2, \mathbb{C}).^[4] A special case occurs when the matrices have determinant 1, forming the special linear group \mathrm{SL}(2, \mathbb{C}). The corresponding transformations are a subgroup, and quotienting \mathrm{SL}(2, \mathbb{C}) by its center \{\pm I\} yields the projective special linear group \mathrm{PSL}(2, \mathbb{C}), which is isomorphic to \mathrm{PGL}(2, \mathbb{C}) over the complexes since any matrix in \mathrm{GL}(2, \mathbb{C}) can be scaled to have determinant 1.^[11] To compute the inverse transformation, solving w = \frac{az + b}{cz + d} for z in terms of w involves rearranging to z = \frac{dw - b}{-cw + a}, which corresponds to the action of the inverse matrix \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} on homogeneous coordinates for w. This matrix inversion approach highlights the linear algebraic structure underlying the transformations.^[12]

Group Operations

Linear fractional transformations (LFTs), also known as Möbius transformations, form a group under the operation of composition. This group structure arises because the composition of two LFTs is again an LFT, and the set is closed under this operation. Specifically, if f(z) = \frac{az + b}{cz + d} with ad - bc \neq 0 and g(z) = \frac{\alpha z + \beta}{\gamma z + \delta} with \alpha \delta - \beta \gamma \neq 0, then the composition is

f \circ g(z) = \frac{(a\alpha + b\gamma)z + (a\beta + b\delta)}{(c\alpha + d\gamma)z + (c\beta + d\delta)},

which corresponds to the matrix product \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} \alpha & \beta \\ \gamma & \delta \end{pmatrix} up to scalar multiple.^[4]^[12] The inverse of an LFT f(z) = \frac{az + b}{cz + d} exists and is given by

f^{-1}(w) = \frac{dw - b}{-cw + a},

provided ad - bc \neq 0; this formula corresponds to the inverse of the associated matrix, scaled by $1/(ad - bc). The identity element of the group is the transformation f(z) = z, represented by the matrix \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}.^[4]^[12]^[13] The group of all LFTs with complex coefficients is isomorphic to the projective special linear group \mathrm{PSL}(2, \mathbb{C}), obtained as the quotient \mathrm{SL}(2, \mathbb{C}) / \{\pm I\}, where \mathrm{SL}(2, \mathbb{C}) consists of $2 \times 2 complex matrices with determinant 1. For LFTs with real coefficients, the group is isomorphic to the subgroup \mathrm{PSL}(2, \mathbb{R}).^[12]^[13] This group is generated by the LFTs corresponding to translations z \mapsto z + b, rotations and dilations z \mapsto e^{i\theta} z (or more generally scalings z \mapsto k z for k \neq 0), and inversion z \mapsto 1/z. These generators suffice to produce any LFT through composition.^[4]^[13]

Analytic and Geometric Properties

Conformal Mapping

Linear fractional transformations, also known as Möbius transformations, are holomorphic functions on the complex plane except at their poles, where they map the extended complex plane to itself in a bijective manner.^[4] A key property is their conformality: at points where they are defined and differentiable, they preserve oriented angles between intersecting curves, achieved through local rotation and scaling by the complex multiplier given by their derivative.^[14] For a transformation f(z) = \frac{az + b}{cz + d} with ad - bc \neq 0, the derivative is f'(z) = \frac{ad - bc}{(cz + d)^2}, which is nonzero wherever f is defined, ensuring the mapping is conformal at every finite, non-pole point.^[4] The conformality arises from the fact that holomorphic functions with nonzero derivative locally resemble multiplication by a complex constant, which rotates and scales vectors without distortion.^[14] A proof sketch uses the chain rule: if curves parameterized by \gamma(t) and \delta(t) intersect at z_0 with angle \theta, their images under f have tangent vectors transformed by f'(z_0) \gamma'(t_0) and f'(z_0) \delta'(t_0), so the angle between them is multiplied by \arg(f'(z_0)) but remains \theta in magnitude and orientation.^[14] This property holds via the Cauchy-Riemann equations, which ensure the Jacobian matrix of the real mapping is a similarity transformation, preserving angles.^[14] Conformality fails at the poles of f, where the transformation is undefined and sends points to infinity, preventing local angle preservation.^[14] However, it holds elsewhere in the domain, including at fixed points where f(z) = z provided f'(z) \neq 0, which is always true for non-constant LFTs. In complex analysis, this angle-preserving nature facilitates solving boundary value problems by conformally mapping complicated regions, such as polygons or annuli, to standard domains like the unit disk or upper half-plane, simplifying the application of harmonic function theory or integral representations.^[14]^[4] The conformal properties of linear fractional transformations were recognized in the 19th century, notably by Bernhard Riemann in his 1851 doctoral thesis, where he established foundational links to uniformization and the mapping of multiply connected domains.^[15] Building on this, Henri Poincaré in 1882 further highlighted their role in modeling non-Euclidean geometries through conformal equivalences, solidifying their importance in the uniformization theorem.^[15]

Fixed Points and Trace Classification

A fixed point of a linear fractional transformation f(z) = \frac{az + b}{cz + d} is a point \alpha \in \hat{\mathbb{C}} satisfying f(\alpha) = \alpha. To find such points, solve the equation \frac{az + b}{cz + d} = z, which rearranges to the equation cz^2 + (d - a)z - b = 0 (when c \neq 0) assuming ad - bc \neq 0.^[16] Non-identity transformations have exactly two fixed points in the extended complex plane, counting multiplicity; this corresponds to one or two distinct fixed points. When c = 0, the equation reduces to a linear one, but the count remains two in the extended plane (e.g., translations fix \infty with multiplicity two). The discriminant of the quadratic is (d - a)^2 + 4bc = \operatorname{tr}^2(A) - 4 \det(A), where A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} and \operatorname{tr}(A) = a + d; a zero discriminant indicates a repeated root (parabolic case).^[17] For matrices with \det(A) = 1, the linear fractional transformations can be classified based on the trace \operatorname{tr}(A). If \operatorname{tr}(A) is real and |\operatorname{tr}(A)| < 2, the transformation is elliptic, corresponding to a rotation around the fixed points; if \operatorname{tr}(A) = \pm 2, it is parabolic, acting like a translation with a single fixed point; if \operatorname{tr}(A) is real and |\operatorname{tr}(A)| > 2, it is hyperbolic, resembling a boost along the axis joining the two fixed points. If \operatorname{tr}(A) is not real, the transformation is loxodromic. This classification arises from the eigenvalues of A, which satisfy \lambda_1 \lambda_2 = 1 and \lambda_1 + \lambda_2 = \operatorname{tr}(A), leading to cases where the eigenvalues are complex conjugates on the unit circle (elliptic), equal to \pm 1 (parabolic), real with product 1 and absolute values not equal to 1 (hyperbolic), or complex conjugates not on the unit circle (loxodromic).^[18] At a fixed point \alpha, the multiplier is defined as k = f'(\alpha), where f'(z) = \frac{ad - bc}{(cz + d)^2}. For elliptic, parabolic, and loxodromic transformations (in the normalized case with \det(A) = 1), wait no—for elliptic and parabolic, the multiplier satisfies |k| = 1, indicating neutral behavior such as rotation or translation; for hyperbolic and loxodromic transformations, |k| \neq 1, with hyperbolic having real positive k \neq 1 (leading to attracting and repelling fixed points) and loxodromic having complex k with |k| \neq 1. The multiplier is invariant under conjugation by other Möbius transformations and provides a local measure of the transformation's action near the fixed point.^[19] The trace classification corresponds to the Jordan canonical form of the matrix A. Elliptic transformations have A similar to a rotation matrix \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} with \theta \neq 0, \pi; parabolic ones are similar to a Jordan block \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}; hyperbolic transformations are similar to a diagonal matrix \begin{pmatrix} \lambda & 0 \\ 0 & 1/\lambda \end{pmatrix} with \lambda > 0, \lambda \neq 1. Loxodromic transformations are similar to a diagonal matrix with complex eigenvalues \lambda, 1/\lambda where |\lambda| \neq 1. This eigenvalue and Jordan form analysis underpins the global geometric interpretation of the transformation on the extended complex plane.^[18]

Action on the Extended Complex Plane

The extended complex plane, denoted \hat{\mathbb{C}} or \mathbb{C} \cup \{\infty\}, is the Riemann sphere, a compactification of the complex plane \mathbb{C} by adjoining a point at infinity, \infty. This construction equips \hat{\mathbb{C}} with the topology of a sphere via stereographic projection.^[20] Linear fractional transformations (LFTs), also known as Möbius transformations, act as holomorphic automorphisms of the Riemann sphere, defined for z \in \mathbb{C} by

f(z) = \frac{az + b}{cz + d},

where a, b, c, d \in \mathbb{C} and ad - bc \neq 0.^[21] To extend this to \hat{\mathbb{C}}, evaluate f(\infty) = a/c if c \neq 0, while f(z) \to \infty as z \to -d/c when c \neq 0; if c = 0, then f(\infty) = \infty. If d = 0, the pole is at \infty. This ensures f is well-defined and continuous on the entire sphere.^[22]^[20] LFTs are bijective mappings of \hat{\mathbb{C}} onto itself, as they are holomorphic and non-constant on the compact Riemann sphere, hence surjective by the open mapping theorem, and invertible with the inverse also an LFT corresponding to the matrix inverse.^[21]^[22] The group of all such transformations under composition is isomorphic to the projective special linear group \mathrm{PSL}(2, \mathbb{C}), acting transitively on \hat{\mathbb{C}}.^[21] A defining geometric property is that LFTs map generalized circles—circles in \mathbb{C} or straight lines (which are circles passing through \infty)—to other generalized circles. This follows from the fact that the equation of a generalized circle can be written in the form \alpha |z|^2 + \beta \overline{z} + \overline{\beta} z + \gamma = 0 with \alpha, \gamma \in \mathbb{R} and \alpha + \gamma \neq 0, and substitution under an LFT preserves this form up to scaling.^[21]^[22] LFTs preserve the cross-ratio of four distinct points z_1, z_2, z_3, z_4 \in \hat{\mathbb{C}}, defined by

(z_1, z_2; z_3, z_4) = \frac{(z_1 - z_3)/(z_1 - z_4)}{(z_2 - z_3)/(z_2 - z_4)},

with appropriate limits for points at \infty. This quantity remains invariant under the action of any LFT, making it a complete invariant for the equivalence of quadruples up to Möbius transformation.^[20] Consequently, given any three distinct points in \hat{\mathbb{C}}, there exists a unique LFT mapping them to any other prescribed triple of distinct points, reflecting the three-dimensional nature of the transformation group.^[21]^[22]^[20]

Applications in Geometry

Hyperbolic Geometry

Linear fractional transformations play a central role in the study of hyperbolic geometry, particularly as isometries of the hyperbolic plane. The hyperbolic plane can be modeled by the upper half-plane \mathbb{H} = \{ z \in \mathbb{C} \mid \operatorname{Im}(z) > 0 \}, equipped with the hyperbolic metric ds = \frac{|dz|}{\operatorname{Im}(z)}. The group \mathrm{PSL}(2, \mathbb{R}), consisting of linear fractional transformations with real coefficients, acts on \mathbb{H} by isometries, preserving this metric and thus the hyperbolic distance.^[23]^[24]^[25] The isometries induced by \mathrm{PSL}(2, \mathbb{R}) on \mathbb{H} are classified based on their fixed points and the trace of the corresponding matrix in \mathrm{SL}(2, \mathbb{R}), mirroring the trace classification discussed earlier. Elliptic isometries fix a point inside \mathbb{H} and correspond to rotations around that point; parabolic isometries fix exactly one point on the boundary \mathbb{R} \cup \{\infty\} and act as translations along horocycles; hyperbolic isometries fix two points on the boundary and translate along the unique geodesic connecting them. This classification determines the type of motion in the hyperbolic plane.^[25]^[26]^[24] Another common model is the Poincaré disk \mathbb{D} = \{ z \in \mathbb{C} \mid |z| < 1 \}, which is conformally equivalent to \mathbb{H} via the Cayley transform z \mapsto \frac{z - i}{z + i}, mapping \mathbb{H} bijectively to \mathbb{D}. Under this correspondence, Möbius transformations preserving \mathbb{D} form the group \mathrm{PSL}(2, \mathbb{R}) acting via linear fractional transformations, again as isometries of the hyperbolic metric ds = \frac{2 |dz|}{1 - |z|^2}.^[26]^[27]^[28] In both models, geodesics are represented as arcs of circles (or straight lines) orthogonal to the boundary at their endpoints, and linear fractional transformations preserve this orthogonality, mapping geodesics to geodesics. This action facilitates the study of hyperbolic structures, such as tessellations. For instance, the modular group \mathrm{PSL}(2, \mathbb{Z}), a discrete subgroup of \mathrm{PSL}(2, \mathbb{R}), acts on \mathbb{H} to produce a fundamental domain consisting of the region \{ z \in \mathbb{H} \mid |z| \geq 1, \, |\operatorname{Re}(z)| \leq 1/2 \}, which tiles the hyperbolic plane via group translations, underlying applications in modular forms and cusp expansions.^[9]^[29]^[30]^[31]

Projective Geometry

Linear fractional transformations arise naturally in projective geometry as automorphisms of the projective line. The real projective line \mathbb{RP}^1 consists of points represented by homogeneous coordinates [x : y], where (x, y) \in \mathbb{R}^2 \setminus \{(0,0)\} and scalar multiples are identified. Similarly, the complex projective line \mathbb{CP}^1 identifies with the Riemann sphere \hat{\mathbb{C}} = \mathbb{C} \cup \{\infty\} via the map [x : y] \mapsto z = x/y for y \neq 0, with [1 : 0] corresponding to \infty. This identification equips \mathbb{CP}^1 with a compact topology, where linear fractional transformations act as biholomorphic maps preserving the spherical structure.^[32] The action on projective space is induced by the general linear group \mathrm{GL}(2, \mathbb{C}), where a matrix \begin{pmatrix} a & b \\ c & d \end{pmatrix} with ad - bc \neq 0 transforms homogeneous coordinates (z_1, z_2) to (w_1, w_2) = (a z_1 + b z_2, c z_1 + d z_2), yielding the map z = z_1/z_2 \mapsto w = w_1/w_2. This defines the projective linear group \mathrm{PGL}(2, \mathbb{C}), isomorphic to the group of linear fractional transformations under composition. In the real case, \mathrm{PGL}(2, \mathbb{R}) acts on \mathbb{RP}^1. These transformations preserve incidence relations: points lie on lines if and only if their images do. In the projective plane \mathbb{RP}^2, linear fractional transformations extend to collineations—bijections preserving lines and their incidences—via the fundamental theorem of projective geometry, which states that every collineation of a projective space over a field (of dimension at least 2) is induced by a semilinear transformation of the underlying vector space.^[32]^[4]^[33] Projective duality interchanges points and hyperplanes (lines in the plane), revealing symmetries in linear fractional transformations. Under inversion, a specific transformation z \mapsto 1/z, poles (where the denominator vanishes) and zeros (where the numerator vanishes) swap roles, as the pole at 0 maps to infinity and vice versa. More generally, duality preserves harmonic divisions: four collinear points form a harmonic set if their cross-ratio is -1, a property invariant under collineations. This invariance stems from the cross-ratio's definition in homogeneous coordinates, which remains unchanged under projective maps.^[34] The historical development of these ideas traces to 19th-century pioneers. Jean-Victor Poncelet, in his 1822 Traité des propriétés projectives des figures, introduced projective properties and transformations, emphasizing invariance under perspective projections and laying foundations for synthetic projective geometry. Karl Georg Christian von Staudt advanced this in works like Geometrie der Lage (1847), developing a purely synthetic framework for projective transformations, including collineations and harmonic properties, without coordinates. These contributions connected linear fractional transformations to broader projective invariance, influencing later analytic formulations.^[35]^[36] Extensions to higher dimensions generalize linear fractional transformations to projective transformations in \mathbb{RP}^n, induced by \mathrm{GL}(n+1, \mathbb{R}) acting on homogeneous coordinates in \mathbb{R}^{n+1}. The group \mathrm{PGL}(n+1, \mathbb{R}) consists of invertible (n+1) \times (n+1) matrices modulo scalars, preserving incidences and cross-ratios in higher-dimensional projective spaces. For n=1, this recovers \mathrm{PGL}(2, \mathbb{R}) and linear fractional transformations on the line.^[37]

Applications in Other Fields

Control Theory

In control theory, linear fractional transformations (LFTs), also known as Möbius transformations, play a crucial role in analyzing and designing dynamical systems, particularly for stability assessment and discretization of continuous-time models into discrete-time equivalents. These transformations map regions of the complex plane, such as the left half-plane (associated with stability in continuous systems) to the unit disk (for discrete systems), enabling engineers to leverage familiar tools like the Routh-Hurwitz criterion in digital contexts. Developed primarily in the mid-20th century amid advances in signal processing and feedback control, LFTs facilitated the transition from analog to digital implementations, with early applications in filter design by figures like Stephen Butterworth, whose 1930 work on maximally flat filters laid groundwork for later digital adaptations using LFT-based mappings. A prominent application is the bilinear transformation, which discretizes continuous-time transfer functions by mapping the s-plane to the z-plane via the relation

z = \frac{1 + \frac{sT}{2}}{1 - \frac{sT}{2}},

where T is the sampling period. This LFT preserves stability by sending the imaginary axis in the s-plane to the unit circle in the z-plane, though it introduces frequency warping that requires prewarping for accurate cutoff frequencies in filter design. For instance, to convert a continuous-time low-pass Butterworth filter to its discrete counterpart, the analog prototype is first designed, then substituted into the bilinear formula, yielding a stable infinite impulse response (IIR) digital filter suitable for real-time control systems. This method, introduced by Arnold Tustin in 1947 for analyzing servo mechanisms, became a cornerstone of digital signal processing by the 1970s, as detailed in foundational texts on the subject.^[38] LFTs also underpin the Jury stability criterion, an algebraic test for ensuring all roots of a discrete-time characteristic polynomial lie inside the unit disk. By applying a specific LFT (a Möbius transformation) to map the unit disk to the left half-plane, the problem reduces to checking Hurwitz stability on the transformed polynomial using Routh-Hurwitz arrays, avoiding direct root solving for high-order systems. This approach is particularly useful in digital control for verifying closed-loop stability without simulation, and it extends classical continuous-time methods to sampled-data systems. The criterion, formalized by Eliahu I. Jury in the 1960s, relies on tabular conditions derived from the transformed coefficients, providing necessary and sufficient stability guarantees.^[39] In network synthesis, LFTs facilitate impedance transformations while preserving the positive real (PR) property of driving-point impedances, essential for realizing passive networks from rational functions. A function Z(s) is PR if it is analytic in the right half-plane, maps reals to non-negative reals, and satisfies \operatorname{Re} Z(s) \geq 0 for \operatorname{Re} s > 0, corresponding to lossless or dissipative physical networks. LFTs of the form Z'(s) = \frac{a Z(s) + b}{c Z(s) + d} with a, b, c, d real and ad - bc > 0 maintain PR-ness, enabling synthesis procedures like continued fraction expansions for ladder networks or Richards transformations for periodic structures. This invariance, rooted in early 20th-century work by Otto Brune and others, allows systematic realization of PR functions as RLC circuits, with applications in analog filter design and passivity-based control. For robust control, LFTs model interconnected systems with uncertainties, forming the basis of \mu-synthesis for designing controllers that guarantee stability and performance under structured perturbations. An uncertain plant is represented as an LFT M \star \Delta, where M is the nominal generalized plant (including weights for performance) and \Delta is a block-diagonal uncertainty matrix; the structured singular value \mu(M) quantifies the smallest perturbation destabilizing the interconnection. \mu-synthesis solves a non-convex optimization via D-K iteration, alternating between \mu-analysis (computing upper/lower bounds on \mu) and optimal H_\infty control synthesis on the LFT framework. Pioneered by John C. Doyle and colleagues in the late 1980s, this method addresses real-world issues like parameter variations in aerospace and process control, offering superior robustness over classical H_\infty designs.^[40]

Algebraic Number Theory

Linear fractional transformations play a key role in algebraic number theory, particularly in generating solutions to Diophantine equations through their matrix representations. For instance, the solutions to Pell's equation x^2 - d y^2 = 1, where d is a positive square-free integer, can be obtained by computing powers of matrices associated with the continued fraction expansion of \sqrt{d}. These matrices correspond to linear fractional transformations that map rational approximations to better ones, yielding the fundamental unit in the ring \mathbb{Z}[\sqrt{d}] and its powers, which generate all positive solutions.^[41]^[42] A concrete example arises for d=2, where the continued fraction of \sqrt{2} is [1; \overline{2}], and the convergents 1/1, 3/2, 7/5, 17/12, ... satisfy p_n^2 - 2 q_n^2 = (-1)^{n+1}. The even-indexed convergents provide solutions to the Pell equation, with the fundamental solution (3,2) generating further solutions such as (17,12) via powers of the associated unit matrix \begin{pmatrix} 3 & 4 \\ 2 & 3 \end{pmatrix}. This method leverages the automorphism group of the quadratic form x^2 - 2 y^2, where units act via linear fractional transformations to preserve the equation.^[41]^[43] The modular group \mathrm{SL}(2,\mathbb{Z}) acts on the upper half-plane \mathcal{H} = \{ z \in \mathbb{C} \mid \operatorname{Im}(z) > 0 \} via linear fractional transformations \gamma z = \frac{az + b}{cz + d} for \gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm{SL}(2,\mathbb{Z}), preserving the hyperbolic metric and defining equivalence classes. The fundamental domain \mathcal{D} = \{ z \in \mathcal{H} \mid |z| \geq 1, -\frac{1}{2} \leq \operatorname{Re}(z) \leq \frac{1}{2} \} tiles \mathcal{H} under this action, with identifications z \sim z \pm 1 on vertical boundaries and z \sim -1/z on the arc |z|=1. This quotient \mathrm{SL}(2,\mathbb{Z}) \backslash \mathcal{H}^* (adjoining cusps) forms the modular curve X(1), a compact Riemann surface of genus zero, essential for studying cusp forms—holomorphic modular forms vanishing at cusps like i\infty. Cusp forms of weight k for \mathrm{SL}(2,\mathbb{Z}) have Fourier expansions \sum_{n=1}^\infty a_n q^n with q = e^{2\pi i z}, and for even k \geq 12, their spaces have dimension \dim S_k(\mathrm{SL}(2,\mathbb{Z})) = \left\lfloor \frac{k}{12} \right\rfloor (0 for $2 \leq k < 12 even).^[44] In the context of Galois representations, linear fractional transformations of \mathrm{SL}(2,\mathbb{Z}) and its congruence subgroups define modular curves like X_0(N) = \Gamma_0(N) \backslash \mathcal{H}^*, which parameterize elliptic curves over \mathbb{Q} with level N structure. These curves are defined over \mathbb{Q}, and their étale fundamental groups yield Galois representations \rho: \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \mathrm{GL}_2(\mathbb{F}_\ell) attached to modular forms, realizing the modularity theorem that every such representation arises from a newform. The action identifies points corresponding to isomorphic elliptic curves, linking the geometry of the upper half-plane to the arithmetic of Galois groups over \mathbb{Q}.^[45]^[44] Class field theory connects ideal class groups of imaginary quadratic fields to the j-invariant under the \mathrm{SL}(2,\mathbb{Z})-action. The j-invariant j(\tau) is a modular function invariant under \mathrm{SL}(2,\mathbb{Z}), mapping the fundamental domain to \mathbb{C}, and for \tau with complex multiplication by an order \mathcal{O}_K in K = \mathbb{Q}(\sqrt{-d}), the values j(\tau) for \tau \in \mathcal{H} generate the ring class field, the maximal unramified abelian extension of the Hilbert class field. The Galois group \mathrm{Gal}(H_K/K) is isomorphic to the ideal class group \mathrm{Cl}(K), and the action of units via linear fractional transformations corresponds to the Artin map, with the degree of the extension equaling the class number h_K.^[44] Post-2000 developments in the Langlands program have deepened these connections, with linear fractional transformations parameterizing automorphic forms through geometric realizations on moduli stacks. The geometric Langlands correspondence, advanced by Beilinson-Drinfeld (2004-2005), equates \mathrm{GL}_2(\mathbb{C})-local systems on curves to Hecke eigensheaves on \mathrm{Bun}_{\mathrm{GL}_2}, using the \mathrm{SL}(2,\mathbb{R})-action on the upper half-plane to construct opers and projective connections that encode automorphic representations. Lafforgue's proof of the Langlands correspondence for \mathrm{GL}_n over function fields (2002) and subsequent work on number fields tie cusp forms to Galois representations, with Hecke operators arising from correspondences induced by the group action.^[46]^[47]

References

[1]
Linear Fractional Transformation -- from Wolfram MathWorld
A linear fractional transformation is a composition of translations, rotations, magnifications, and inversions.
[2]
11.7: Fractional Linear Transformations - Mathematics LibreTexts
May 2, 2023 · A linear fractional transformation maps lines and circles to lines and circles. Before proving this, ...Definition: Fractional Linear... · Example 11 . 7 . 1 : Scale and... · Lines and circles
[3]
None
### Summary of Key Points on Möbius Transformations from the Paper
[4]
[PDF] Fractional Linear Transformations - Trinity University
az + b. cz + d. with a,b,c,d ∈ C and ad − bc 6= 0 is called a fractional linear transformation (FLT) (or Möbius transformation). Notice. f ′(z) =a(cz + d) − c( ...
[5]
[PDF] Lesson 33. Linear fractional transformations - Purdue Math
Any linear fractional transformation takes circles and lines to either circles or lines. Note that a line in C becomes a circle passing through. ∞ on the ...
[6]
[PDF] 31. - Linear Fractional Transformations (LFTs).
LFTs Remarks: are also called Möbius transformations. a, b, c, d are complex numbers.Missing: mathematical | Show results with:mathematical
[7]
[PDF] Möbius transformations, also called linear fractional - OSU Math
Recap: Möbius transformations, also called linear fractional are transformations (LFT) conformal automorphisms of. Aut (Ĉ) = {MA A E GL₂ (C)}.Missing: mathematical | Show results with:mathematical
[8]
[PDF] DYNAMICS FOR DISCRETE SUBGROUPS OF SL 2(C) - Yale Math
The group G := PSL2(C) acts on C by Möbius transformations: Åa b c d ã z = az + b cz + d with a, b, c, d ∈ C such that ad − bc = 1.
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[PDF] Geodesics of Hyperbolic Space
Möbius transformations consist of compositions of the following types of maps: • Scalings- maps of the form z 7→ kz for some k ∈ C. • Translations - maps of the ...
[10]
[PDF] Hyperbolic Transformations
Let T be a Möbius transformation. Then T is the composition of translations, dilations, and inversion (g(z) = 1 z. ). Proof: If c = 0, then f(z) = a d z + b.
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None
Summary of each segment:
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[PDF] Linear Fractional Transformations
Feb 10, 2005 · If AT and AS are the matrices associated to the LFT's T and. S, respectively, then AT◦S = AT AS. That is, the matrix associated to T ◦ S is the ...
[13]
[PDF] Groups ofLinear Fractional Transformations
Linear fraction transformations (LFT) are quotients of linear functions of the form. T(z) = Аz+b cz+d. These are analytic for z φ= - dР and are conformal ...Missing: PSL( | Show results with:PSL(
[14]
[PDF] 18.04 S18 Topic 10: Conformal transformations
A linear fractional transformation maps lines and circles to lines and circles. ... Fractional linear transformations preserve symmetry. That is, if 1 and ...
[15]
[PDF] History of Riemann surfaces
Oct 11, 2005 · Poincaré (1882) discovered that the linear fractional transformations give a natural interpretation of non-euclidean geometry. 10 Fitting the ...
[16]
https://math.libretexts.org/Bookshelves/Geometry/Geometry_with_an_Introduction_to_Cosmic_Topology_(Hitchman)/03%3A_Transformations/3.04%3A_Mobius_Transformations
[17]
[PDF] Möbius transformations • Elementary functions of complex variable
Dec 1, 2013 · (d − a)2 + 4bc = (a + d)2 − 4(ad − bc) = tr(A)2 − 4 det(A). If the discriminant is zero, then there is only one fixed point (possi- bly ...
[18]
[PDF] An Introduction to Classical Modular Forms with Kohnen's Proof of ...
If t < 2, we say M is elliptic. If t = 2, M is parabolic. If t > 2,. M is hyperbolic. Proposition 38 If M is a real linear fractional transformation, M 6= I:.
[19]
[PDF] m 597 lecture notes topics in mathematics complex dynamics
Möbius transformations. Möbius transformations, or fractional linear transformations, are maps of the form f(z) = az+b cz+dwith a, b, c, d ∈ C, ad − bc 6 ...<|control11|><|separator|>
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[PDF] Compactification: Riemann sphere, projective space
Nov 22, 2014 · Linear fractional (Möbius) transformations. The general linear group ... action of GL2(C) stabilizes the equivalence classes C× · v used ...
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[PDF] Review of Möbius Transformations
Let C∞ = C ∪ {∞} be the Riemann sphere. Given A = (a b. c d. ) ∈ GL2(C), we define a map ϕA : C∞ → C∞ by ϕ(z) = az + b cz + d . The map ϕA is called a Möbius.
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[PDF] Lecture 8 - The extended complex plane ˆ C, rational functions ...
As Aut (C) can be viewed as the “conformal motions” on the sphere, one may view the action of an LFT on the Riemann sphere dynamically via stereographic ...
[23]
[PDF] Chapter 10 - Poincaré Upper Half Plane Model - Mathematics
PSL2(R) is obtained from GL2(R) by identifying γ with kγ for any k 6= 0. The group PSL2(C) is isomorphic to the group of fractional linear transformations.
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[PDF] 3.1 The hyperbolic plane: half-plane and disc models, isometries
PSL(2, R), acting as fractional linear maps. ← 2 az + b cz + d in the upper half-plane model. We use their fixed points to classify them into types ...
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[PDF] classification of the isometries of the upper half-plane - UChicago Math
Aug 29, 2011 · The group of isometries of the upper half-plane, Isom(H), is gener- ated by the fractional linear transformations in PSL(2, R) together with the ...
[26]
[PDF] Hyperbolic geometry MA 448 - University of Warwick
Mar 1, 2013 · A linear fractional transformation or Möbius map is a map f : ˆC = C ... Note that these elliptic (or parabolic) hyperbolic surfaces correspond to.
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[PDF] The hyperbolic metric and geometric function theory
Also, in any disk or half-plane hyperbolic geodesics are arcs of circles orthogonal to the boundary; in the case of a half- planes we allow half-lines ...
[28]
[PDF] The Modular Group - UCSD Math
This domain tesselates the entire half-plane with congruent hyperbolic triangles – although the two vertical geodesics do become very close at they approach.
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https://www.math.osu.edu/sites/math.osu.edu/files/hyperbolicGeometry.pdf
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[PDF] sl2(z) - keith conrad
The fundamental domain and its translates are called ideal triangles since they are each bounded by three sides and have two endpoints in h but one “endpoint” ...<|control11|><|separator|>
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[PDF] Introductory Lectures on SL(2,Z) and modular forms.
In this way, we encounter the classic prototype for a discrete group action, as first considered by Klein and by Poincaré, the modular group Γ(1) ∼= PSL(2, Z).
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[PDF] Math 113 (2024) Yum-Tong Siu 1 Linear Fractional Transformations ...
The applications of conformal mappings to fluid flow, temperature distri- bution, and electrostatic potential will be formulated in terms of harmonic.
[33]
[PDF] The fundamental theorem of projective geometry
Let K be a field. The fundamental theorem of projective geometry says that an abstract automorphism of the set of lines in Kn which preserves “incidence ...
[34]
[PDF] Basics of Projective Geometry - UPenn CIS
A projective subspace of dimension 0 is a called a (projective) point. A projec- tive subspace of dimension 1 is called a (projective) line, and a projective ...
[35]
Jean-Victor Poncelet | Projective Geometry, Mechanics & Invariant ...
Jean-Victor Poncelet was a French mathematician and engineer who was one of the founders of modern projective geometry. As a lieutenant of engineers in 1812 ...
[36]
Karl von Staudt - Biography - MacTutor - University of St Andrews
By purely projective methods, von Staudt established a complete method for calculating the anharmonic ratios of the most general imaginary elements. ... the ...
[37]
[PDF] Projective Geometry - TU Berlin
The homogeneous coordinates depend on a choice of basis and affine coor- dinates depend on a choice of a basis and an affine hyperplane given by a linear form ...<|separator|>
[38]
Bilinear transformation method for analog-to-digital filter conversion
Bilinear transformation is a method for converting analog filters to discrete equivalents by mapping the s-plane to the z-plane.
[39]
New Stability Criteria for Discrete Linear Systems Based on ... - MDPI
A new criterion for Schur stability is derived by using basic results of the theory of orthogonal polynomials.
[40]
The complex structured singular value - ScienceDirect.com
Uncertain systems are represented using Linear Fractional Transformations (LFTs) which naturally unify the frequency-domain and state space methods. Previous ...Missing: Packwood | Show results with:Packwood
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[PDF] Continued Fractions and Pell's Equation - UChicago Math
Abstract. In this REU paper, I will use some important characteristics of continued fractions to give the complete set of solutions to Pell's equation. I.
[42]
[PDF] Continued Fractions and Linear Fractional Transformations
Abstract. Rational approximations to a square root √k can be produced by iterating the trans- formation f(x) = (dx + k)/(x + d) starting from ∞ for any d ...
[43]
[PDF] pell's equation, ii - keith conrad
In this section, for those who know about continued fractions, we will explain how Pell and generalized Pell equations can be solved with continued fractions.
[44]
[PDF] Modular Functions and Modular Forms
H = {z ∈ C | (z) > 0}. The upper-half plane as a quotient of SL2(R). We saw in the Introduction that there is an action of SL2(R) on H as follows: SL2(R) ...
[45]
[PDF] Modular Curves and Galois Representations
Jun 1, 2025 · c d ∈ SL2(Z) : c ≡ 0 (mod n), a ≡ d ≡ 1 (mod n) . The group Γ1(n) acts on H via linear fractional transformations. γ = a b. c d :H → H τ ...
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[PDF] Lectures on the Langlands Program and Conformal Field Theory
Based on the lectures given by the author at the Les Houches School “Number. Theory and Physics” in March of 2003 and at the DARPA Workshop “Langlands Program ...Missing: developments | Show results with:developments
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https://arxiv.org/abs/math/0501398