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hyperbolic, parabolic, elliptic transformations

If the eigenvectors of the matrix representation of a Möbius transformation are its fixed points, there remains the question of interpreting the eigenvalues. A good way to find this out is to use a coordinate system in which the representation is diagonal (or failing that, in the Jordan normal form). It also helps to recall that the representation matrix might just as well be unimodular, with reciprocal eigenvalues. In canonical form, the fixed points are zero and infinity, or else just infinity.

In the former case, represent the complex number z by u/v and consider the eigenvalue equation

$\displaystyle \left[ \begin{array}{cc} \lambda & 0 \\  0 & \lambda^{-1} \end{array} \right]
\left[ \begin{array}{c} u \\  v \end{array} \right]$ = $\displaystyle \left[ \begin{array}{c} \lambda u \\  \lambda^{-1}v \end{array} \right],$ (36)

which transforms u/v into $\lambda u / \lambda^{-1} v$, or in other words z becomes $\lambda^2 z$, and the eigenvalue squared is a multiplier.

There are essentially three cases to consider. If the multiplier is real, the transformation moves points radially - outward if the factor is greater than 1, inward if it is less than 1. In the first case, $\infty$ is a stable attractor and 0 is an unstable repeller; in the second case their roles are reversed. Transformations in this class are usually called hyperbolichyperbolic Möbius transformation mappings.

If the multiplier is complex, but with absolute value 1, both 0 and $\infty$ are neutral fixed points, the general movement of points according to the transformation being a rotation. Such transformations are usually called ellipticelliptic Möbius transformation mappings.

In both these restricted cases it is convenient to think of the circles fixed with respect to the transformation. In the first case, they are radii which are arcs of constant angle passing through zero and $\infty$; in the second case they are concentric circles surrounding the two fixed points.

The third case combines the previous two, contemplating a general complex multplier. It results in a composite of the other types, given the polar representation of a complex number. These are called loxodromicloxodromic Möbius transformation mappings.

The confluent case, where the two fixed points have coalesced, gives the Jordan normal form of the Möbius transformation, which would read

$\displaystyle \left[ \begin{array}{cc} 1 & a \\  0 & 1 \end{array} \right]
\left[ \begin{array}{c} u \\  v \end{array} \right]$ = $\displaystyle \left[ \begin{array}{c} u + a v \\  v \end{array} \right],$ (37)

and transforms z into z+a, which is a translation. The invariant circles in this case are straight lines in the direction of the translation, which are circles through $\infty$. The confluent mappings are called parabolicparabolic Möbius transformations mappings, to complete the analogy with the classification of conic sections.

There is a degenerate case not so far mentioned, in which z maps into 1/z, and it to be seen as inversion in a circle - the unit circle.

When the matrix of the Möbius transformation is not diagonal, the ability to map any three points into any other three points should be used to place the fixed points at 0 and $\infty$, the diagonal form of the transformation executed, and the inverse mapping be used to restore the fixed points. Of course, all other points will have moved in the process but knowing that circles map into circles, the mapping can be followed along their arcs.

Hyperbolic mappings will move points away from the repelling fixed point along circular arcs toward the attracting point. Elliptic transformations will shepherd points towards the line joining the fixed points or draw them away into the vast space remote from the fixed points. Parabolic transformations behave similarly, without there being any space between the coalesced fixed points. The Smith Chart is based on a parabolic transformation.

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