A fixed point of a transformation w=f(z) is a point Zo such that f(Zo)=Zo, Show that every linear transformation, with the exception of the identity transformation w=z, has at most two fixed points in the extended plane.
A fixed point of a transformation w=f(z) is a point Zo such that f(Zo)=Zo, Show that every linear transformation, with the exception of the identity transformation w=z, has at most two fixed points in the extended plane.
If is a linear transformation then . Say that . If then .
- If and then the equation is not solvable and so has no fixed points in . It does however have as a fixed point on the Riemann sphere.
- If and then is the identity.
- If then is a fixed point in the otherfixed point is because if because otherwise is not a fixed point.
The question asks about linear transformations, but the thread is headed "linear fractional transformations", in other words those of the form . In that case, a fixed point must satisfy . This is clearly a quadratic equation for z, except when b=c=0 and a=d, when it becomes identically true.