Inverse of Möbius transform - w(z)?

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    Inverse Transform
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Homework Help Overview

The discussion revolves around the inverse of a Möbius transform, specifically the function w(z) = (az + b) / (cz + d). The original poster seeks to understand how to derive the inverse transformation in matrix form.

Discussion Character

  • Conceptual clarification, Problem interpretation

Approaches and Questions Raised

  • Participants explore the nature of the Möbius transform and its relationship to linear transformations. Questions arise about the feasibility of expressing the transform in matrix form and the method for finding its inverse.

Discussion Status

Some participants have provided insights into the limitations of expressing the Möbius transform as a matrix and suggested algebraic methods for finding the inverse. There is an acknowledgment of the complexity involved in inverting the function.

Contextual Notes

There is an indication of fatigue from the original poster, which may affect the depth of engagement in the discussion. The conversation reflects a mix of confusion and attempts to clarify the mathematical concepts involved.

Jaynte
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Homework Statement



I want to know how I could extract T from Tz in matrix form so I can
get [tex]T^{-1}[/tex] to get the inverse of w(z).

w(z)=Tz=[tex]\frac{az+b}{cz+d}[/tex]

Homework Equations





The Attempt at a Solution

 
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You can't. The moebius transform is not linear and only linear transforms can be written as matrices.
 
Thanks.

So how how do I solve it if I don't want to remember the answer [tex]z(w)=\frac{-dw+b}{cw-a}?[/tex]
 
You invert it like you invert any other function. If w=(az+b)/(cz+d), solve that equation for z using algebra.
 
Of course, I am to tired now I should go to sleep :) Thank you.
 

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