# Math Help - Bilinear transformations

1. ## Bilinear transformations

Prove that a loxodromic transformation can be expressed as a resultant of elliptic and hyperbolic transformations.

Thanks and Regards,
Kalyan.

2. Hi,

Here is a proof I came up with any bilinear transformation $L(\omega, z)$ with two fixed points say $\alpha, \beta$ will be of the form $\frac{\omega - \alpha}{\omega - \beta} = k.\frac{z - \alpha}{z - \beta}$ when $k$ is neither unimodular nor $k \in R$ the transformation is neither elliptical nor hyperbolic in the respective cases, hence loxodromic.

Now for all $k \in C$ we know that $k = r.(\cos\theta + i\sin\theta) , r \in R$ and $|\cos\theta + i\sin\theta| = 1$

$\therefore$ we can express the transformation $L(\omega, z)$ can be expressed as $E(H(\frac{z - \alpha}{z - \beta}))$ where $E(t) = (cis\theta)t$ and $H(t) = r.t , r \in R$
where $E(t), H(t)$ are elliptic and hyperbolic transformations respectively.

The notation used for $E(t), H(t)$ may be confusing but what I mean is its just a matrix multiplication.

Let me know if this proof is ok.

Kalyan.