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## eigenvalues and eigenvectors of a Möbius transformation

The fixed points of a Möbius transformation satisfy the requirement

 z = (34)

which translates into the quadratic equation
 cz2 + (d-a) z - b = 0 (35)

with roots In case c=0 the quadratic term is absent and the single fixed point would be b/(d-a) unless (d-a) were zero. That alternative would leave z without constraint and require b to vanish as well as c. But then all points would be fixed and the transformation would be the identity. And if b did not vanish, the equation would read z=z+b which could not be satisfied unless z were , which might as well be considered admissible.

Another case of a single root occurs when the radicand is zero leaving (a-d)/2c as the only fixed point. Otherewise there are always two distinct fixed points    Next: hyperbolic, parabolic, elliptic transformations Up: Functions of a complex Previous: Möbius transformations representable as
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2001-04-05