# Fractional Linear Transformation Question

• joeblow
In summary, the conversation discusses a question about FLTs (Mobius Transformations) and how any FLT can be written as an FLT with determinant 1. There is a discussion about the uniqueness of the constants in the Mobius map and how the determinant is multiplied by a constant. The conversation also mentions a possible typo in the problem and raises a question about the approach to solving it.
joeblow
I am a graduate assistant and was asked a question about FLTs (Mobius Transformations). The student was asked to prove that any FLT can be written as an FLT with determinant 1.

However, I can't make sense of that. If I look at the possible Jordan Canonical forms of 2-by-2's, it would seem that the matrix
[x 0]
[0 y]
where x and y are distinct eigenvalues cannot be represented as an FLT with determinant 1 (since it would require finding a complex number that when multiplied with both x and y gives 1 which violates the uniqueness of multiplicative inverse).

To fix notation: a Mobius map is a map of the form
$$z\mapsto \frac{az+b}{cz+d}$$
for some complex constants $a,b,c,d$ satisfying $ad-bc\neq0$. It is this number $ad-bc$ that is meant by the 'determinant' in the problem (though that name is not justified quite yet, see later).

The constants are not, however, unique for anyone map: if you multiply all of them by a nonzero number, $\lambda$, you get the same map. The 'determinant' is then multiplied by $\lambda^2$, so by choice of $\lambda$ we can pick it to be one.

The name determinant comes from the fact that there is an isomorphism between Mobius maps and the projective linear group $SL(2,\mathbb{C})/\{1,-1\}$ where a,b,c,d become the entries of the matrix (the freedom to still pick $\lambda=\pm1$ is why we have to take the quotient).

joeblow said:
I am a graduate assistant and was asked a question about FLTs (Mobius Transformations). The student was asked to prove that any FLT can be written as an FLT with determinant 1.

However, I can't make sense of that. If I look at the possible Jordan Canonical forms of 2-by-2's, it would seem that the matrix
[x 0]
[0 y]
where x and y are distinct eigenvalues cannot be represented as an FLT with determinant 1 (since it would require finding a complex number that when multiplied with both x and y gives 1 which violates the uniqueness of multiplicative inverse).

## 1. What is a fractional linear transformation?

A fractional linear transformation, also known as a Möbius transformation, is a mathematical function that maps points on a complex plane to other points on the same plane. It is represented by the formula f(z) = (az + b) / (cz + d), where a, b, c, and d are complex numbers and z is the input value.

## 2. What is the significance of a fractional linear transformation in mathematics?

Fractional linear transformations are important in many areas of mathematics, including complex analysis, geometry, and algebra. They have applications in solving differential equations, conformal mapping, and symmetry groups. They are also used in computer graphics and image processing.

## 3. How is a fractional linear transformation different from a linear transformation?

A linear transformation is a function that maps points in a vector space to other points in the same vector space. It is represented by the formula f(x) = ax + b, where a and b are constants and x is the input value. A fractional linear transformation, on the other hand, maps points on a complex plane to other points on the same plane using a more complex formula that involves division and complex numbers.

## 4. Are there any limitations to fractional linear transformations?

Yes, there are certain limitations to fractional linear transformations. For example, the denominator in the formula (cz + d) cannot be equal to zero, as this would result in an undefined value. Additionally, certain values of a, b, c, and d can cause the function to have a singularity, where it is not defined or becomes infinite.

## 5. How are fractional linear transformations used in real-world applications?

Fractional linear transformations have various real-world applications, including in engineering, physics, and economics. They are used to model complex systems and make predictions in fields such as fluid dynamics, electrical circuits, and financial markets. They are also utilized in designing and analyzing control systems and electronic filters.

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