eigenvalues and eigenvectors of a Möbius transformation

The fixed points of a Möbius transformation satisfy the requirement

$$\frac{az+b}{cz+d}$$ z = (34)

which translates into the quadratic equation

cz² + (d-a) z - b = 0 (35)

with roots

$$\frac{1}{c} \left\{\frac{a-d}{2}\pm\surd \left[\left(\frac{a+d}{2}\right)^2-1\right]\right\}.$$

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 $\infty$, 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