The purpose of this post is to acquire a deeper appreciation of Mobius transformations. Typically, one simply encounters these as maps \(\mathcal M:\textbf C\cup\{\infty\}\to\textbf C\cup\{\infty\}\) on the Riemann sphere \(\textbf C\cup\{\infty\}\) of the form \(\mathcal M(z):=\frac{az+b}{cz+d}\) for \(ad-bc\neq 0\) without enough emphasis being placed on just how incredibly general and versatile this form is. The following problems (followed by their solutions) are meant to expose the full scope of possibilities that come with Mobius transformations.
Problem #\(1\): Solve the equation \(\frac{5}{8}=\frac{11-3x}{4+7x}\) for \(x\in\textbf R\).
Solution #\(1\): Exploit the invariance under Mobius transformations of a suitable cross-ratio:
\[\frac{(x-0)(\infty-(-4/7))}{(x-\infty)(0-(-4/7))}=\frac{(5/8-11/4)(-3/7-\infty)}{(5/8-(-3/7))(11/4-\infty)}\]
\[-\frac{7x}{4}=\frac{-17/8}{59/56}\]
\[x=\frac{68}{59}\]
Problem #\(2\): Solve the equation \(\frac{5}{8}=\frac{11-3x}{3+7x}\) for \(x\in\textbf R\).
Solution #\(2\): The trick is that any traceless Mobius transformation with \(a=-d\) is an involution. So the answer is just:
\[x=\frac{11-3\times(5/8)}{3+7\times(5/8)}=\frac{73}{59}\]
Problem #\(3\): Redo Problem #\(1\) but using the trick in Problem #\(2\).
Solution #\(3\): Since \(\text{lcm}(3,4)=12\), one has:
\[\frac{5}{8}=\frac{11-3x}{4+7x}\Rightarrow\frac{4}{3}\times\frac{5}{8}=\frac{4}{3}\times\frac{11-3x}{4+7x}\]
So:
\[x=\frac{4}{3}\times\frac{11-3\left(\frac{4}{3}\times\frac{5}{8}\right)}{4+7\left(\frac{4}{3}\times\frac{5}{8}\right)}=\frac{68}{59}\]
In general, the Mobius group is isomorphic to \(\cong\text{PGL}_2(\textbf C)\). This is incredibly powerful because it provides an explicit bridge between complex analysis and linear algebra. In this language, the question of which Mobius transformations are involutions \(\mathcal M^2=1\) boils down to which \(2\times 2\) complex invertible matrices \(M=\begin{pmatrix}a&b\\c&d\end{pmatrix}\) satisfy \(M^2=1\). By the Cayley-Hamilton theorem, any matrix with eigenvalues \(\lambda=\pm 1\) will satisfy such an equation…
What’s the connection with unitary quantum logic gates on qubits? Give examples with 1D scattering (which is similar to Fresnel equations). Even for certain classes of intuitive \(2\)D matrices like rotations, what do the corresponding Mobius maps say? And also is there any connection between evaluating a Mobius map on some \(z\) and multiplying the corresponding matrix by a C^2 vector?