It wouldn’t be a blog if I didn’t link to cool material I found on other blogs, right? So here is a beautiful video made by Doug Arnold and Jonathan Rogness demonstrating a geometric way to think of Mobius transformations of the plane. I learned about this from Good Math, Bad Math. The pictures are very nice, but it took me a while to figure out why they worked and what was bothering me.
The group of Mobius transformations is PSL(2,C), where the P means that we are modding out by the central subgroup generated by the identity. Of course, SL(2,C) contains SU(2), so PSL(2,C) contains PSU(2), which is just SO(3,R). So this is the subgroup that corresponds to the rotations of the sphere in the video.
What is the subgroup that corresponds to the dilations and translations? That’s simple enough : it is the complex maps which are of the form with a a positive real. In the representation of Mobius transformations by elements of PSL(2,C), this is . In other words, it is also a subgroup — the subgroup of upper triangular elements of PSL(2,C) with positive real entries on the diagonal.
So the hidden theorem in this video is that every element of PSL(2,C) can be uniquely represented as the product of a unitary matrix and an upper triangual matrix with positive real diagonal entries (each with determinant one). This is the complex form of the QR-decomposition — the matrix factorization theorem which falls out of the Gram-Schmidt procedure. Specifically, given a Mobius transformation in PSL(2,C), write it in the form QR with Q unitary and R upper triangular. Then this video displays Q as the rotation of the sphere and R as the dilation/rotation.
Now that I see this, I know why the video works and what puzzled me about it. The video gave me a good way to think about the elements of PSL(2,C) but it didn’t give me a good way to think about the group structure. Every time I tried to see the multiplication operation in terms of that rolling sphere I got very confused. And now I see why! Multiplying matrices in QR form is really hard. So this gives a challenge. Is there a way to use this nice picture of the rolling sphere to see multiplication, or even just to see geodesics in PSL(2,C)?