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Linear Fractional Transformation

Linear fractional transformation, abbreviated as LFT, is a type of transformation that is represented by a fraction consisting of a linear numerator and a linear denominator. When a linear fractional transformation is performed, symmetry is always maintained. In the complex plane, a linear fractional transformation takes lines or circles onto lines or circles.

In the complex plane, linear fractional transformation is known as a Mobius transformation. Linear fractional transformation is widely used in control theory to analyze systems such as a damped harmonic oscillator. In this article, we will learn more about LFT, how to solve problems based on this concept using complex analysis, and certain applications.

1.	What is Linear Fractional Transformation?
2.	Linear Fractional Transformation Complex Analysis
3.	Linear Fractional Transformation Applications
4.	FAQs on Linear Fractional Transformation

What is Linear Fractional Transformation?

A linear fractional transformation can be defined as a transformation where

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

Here, az + b and cz + d are linear. Also, a, b, c, d, and z are complex numbers.

This transformation can be associated with a matrix given by \(\begin{bmatrix} a & b\\ c & d \end{bmatrix}\)

Further, the inverse of such a transformation is given by f^-1(w) = \(\frac{dw-b}{-cw+a}\). Using matrix notation the inverse of an LFT can be expressed as \(\begin{bmatrix} d & -b\\ -c & a\end{bmatrix}\)

The condition for invertibility is ad – bc ≠ 0. Suppose ad - bc = 0 then f(z) will be a constant function and is not considered an LFT.

A linear fractional transformation is a conformal mapping because this transformation preserves local angles. LFT is a composition of translations, inversions, dilations and rotations. Every linear fractional transformation, with the exception of f(z) = z, will have either one or two fixed points. If a point does not change under the application of a map then it is known as a fixed point.

Some special values of the linear fractional transformation function are given as follows:

f(∞) = a / c, if c ≠ 0

f(-d / c) = ∞, if c ≠ 0

Linear Fractional Transformation

There are some properties of linear fractional transformations that can help to solve questions with increased speed and accuracy. These are given as follows:

w = az, |a| = 1 (Rotation)
w = az + b (Translation)
w = 1/z. This LFT produces an inversion in a unit circle.

Linear Fractional Transformation Complex Analysis

Complex analysis is a technique that is used to evaluate very complicated functions. Such functions have complex outputs and complex arguments thus, they exist in the complex plane. The property that linear fractional transformation maps lines and circles to lines and circles can be proved using complex analysis. Complex analysis usually deals with holomorphic functions. Holomorphic functions can be defined as complex functions that are differentiable everywhere in a complex plane.

Linear Fractional Transformation Formula

Complex analysis can be used to find the linear fractional transformation that takes \(z_{i}\) to \(w_{i}\) if there are two sets of three points \(z_{1}\), \(z_{2}\), \(z_{3}\) and \(w_{1}\), \(w_{2}\), \(w_{3}\). Here, the general form of z = x + iy and w = u + iv. The equation of such a linear fractional transformation is expressed as follows:

\(\frac{(w-w_{1})(w_{2}-w_{3})}{(w-w_{3})(w_{2}-w_{1})}\) = \(\frac{(z-z_{1})(z_{2}-z_{3})}{(z-z_{3})(z_{2}-z_{1})}\)

This is also known as the LFT cross-ratio formula. It implies that the cross-ratio is invariant under linear fractional transformation.

Linear Fractional Transformation Applications

Linear fractional transformation finds use in mathematics and engineering. This concept is vastly used in classical geometry, control theory, group theory, and number theory. Some applications are listed as follows:

Linear fractional transformation is applied to the scattering theory of differential equations to understand how wave and particle scattering occurs.
In electronic and electrical engineering, linear fractional transformation is used for gain scheduling.
In electronic control systems, a complicated feedback system can be simplified using LFT.

Related Articles:

Important Notes on Linear Fractional Transformation

Linear fractional transformation (LFT) is a type of transformation that is a composition of dilation, translations, inversions, and rotations.
It can be expressed as f(z) = \(\frac{az+b}{cz+d}\), where the numerator and the denominator are linear.
A linear fractional transformation is a conformal mapping as well as preserves symmetry.
An LFT maps circles and lines to circles and lines.

Examples on Linear Fractional Transformation

Example 1: Find a Linear fractional transformation with the property f(1) = 1 + i, f(i) = 1 - i, f(-i) = 2
Solution: We know that f(z) = \(\frac{az+b}{cz+d}\)
Choosing an arbitrary value a = 1, f(z) = \(\frac{z+b}{cz+d}\)
Now plugging in values
f(1) = 1 + i;
\(\frac{1+b}{c+d}\) = 1 + i or - b + (1 + i)c + (1 + i)d = 1
f(i) = 1 - i;
\(\frac{i+b}{ci+d}\) = 1 - i or - b + (1 + i) c + (1 - i)d = i.
f(-i) = 2;
\(\frac{-i+b}{-ci+d}\) = 2 or - b - 2ic + 2d = - i
Solving these three simultaneous equations we get,
b = \(\frac{-4}{5} - \frac{3}{5}i\)
c = \(\frac{3}{10} + \frac{i}{10}\)
d = \(\frac{-1}{2} - \frac{i}{2}\)
Answer: \(\frac{z-\frac{4}{5} - \frac{3}{5}i}{(\frac{3}{10} + \frac{i}{10})z-\frac{1}{2} - \frac{i}{2}}\) is the required linear fractional transformation.
Example 2: If f(z) corresponds to a matrix A = \(\begin{bmatrix} a & b\\ c & d \end{bmatrix}\) and g(z) corresponds to B = \(\begin{bmatrix} e & f\\ g & h \end{bmatrix}\), then prove that f( g(z)) = AB
Solution: g(z) can be written as \(\frac{ez+f}{gz+h}\).
f(g(z)) = f(\(\frac{ez+f}{gz+h}\)) = \(\frac{a(\frac{ez+f}{gz+h})+b}{c(\frac{ez+f}{gz+h})+d}\)
= \(\frac{(ae +bg)z+af+bh}{(ce+dg)z+cf+dh}\)
AB = \(\begin{bmatrix} a & b\\ c & d \end{bmatrix}\)\(\begin{bmatrix} e & f\\ g & h \end{bmatrix}\) = \(\begin{bmatrix} ae+bg & af+bh\\ ce + dg & cf +dh \end{bmatrix}\)
Example 3: Find the linear fractional transformation that maps the points \(z_{1}\) = - i, \(z_{2}\) = 0, \(z_{3}\) = i to \(w_{1}\) = -1, \(w_{2}\) = i, \(w_{3}\) = 1.
Solution: Using the linear fractional transformation cross-ratio formula,
\(\frac{(w-w_{1})(w_{2}-w_{3})}{(w-w_{3})(w_{2}-w_{1})}\) = \(\frac{(z-z_{1})(z_{2}-z_{3})}{(z-z_{3})(z_{2}-z_{1})}\)
Substituting the given values in this equation,
\(\frac{(w+1)(i-1)}{(w-1)(i+1)}\) = \(\frac{(z+i)(-i)}{(z-i)(i)}\)
On solving for w we get,
\(\frac{w+1}{w-1}=\frac{-zi + 1}{zi + 1}\times \frac{i+1}{i-1}\)
\(\frac{w+1}{w-1}=\frac{-zi^{2}+i-zi+1}{zi^{2} +i-zi-1}\)
w = - i \(\frac{z-i}{z+1}\)
Answer: The linear fractional transformation is given by w = - i \(\frac{z-i}{z+1}\)

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Practice Questions on Linear Fractional Transformation

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FAQs on Linear Fractional Transformation

What is Linear Fractional Transformation in Math?

Linear fractional transformation in math is a composition of translation, rotation, dilation, and inversion. It is represented by a fraction that contains a linear numerator and a linear denominator.

How Do You Represent Linear Fractional Transformation?

In the fractional form, a linear fractional transformation is given by f(z) = \(\frac{az+b}{cz+d}\) where, a, b, c, d, and z are complex numbers belonging to the complex plane.

Is a Linear Fractional Transformation a Conformal Mapping?

Yes, a linear fractional transformation is a conformal mapping as this transformation preserves angles locally.

What is Linear Fractional Transformation Complex Analysis?

Complex analysis is a method that is used to solve problems on linear fractional transformations. Functions that are complicated and belong to the complex plane can be evaluated using complex analysis.

What is the Cross-Ratio Formula in Linear Fractional Transformation?

The cross-ratio formula for linear fractional tranformation is given by \(\frac{(w-w_{1})(w_{2}-w_{3})}{(w-w_{3})(w_{2}-w_{1})}\) = \(\frac{(z-z_{1})(z_{2}-z_{3})}{(z-z_{3})(z_{2}-z_{1})}\). This implies that the points \(z_{1}\), \(z_{2}\), \(z_{3}\) will get mapped to the points \(w_{1}\), \(w_{2}\), \(w_{3}\).

Can a Matrix be Associated With a Linear Fractional Transformation?

A linear fractional tranformation given by f(z) = \(\frac{az+b}{cz+d}\) can be represented using a matrix as \(\begin{bmatrix} a & b\\ c & d \end{bmatrix}\).

What is the Inverse of a Linear Fractional Transformation?

The inverse of a linear fractional tranformation given by f(z) = \(\frac{az+b}{cz+d}\) is f^-1(w) = \(\frac{dw-b}{-cw+a}\) provided ad – bc ≠ 0. The matrix notation for the inverse is \(\begin{bmatrix} d & -b\\ -c & a\end{bmatrix}\).

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