Find mobius transformations that send the upper halfplane to itself Task: Find all $A\in GL(2,\mathbb{C})$, such that the corresponding mobius transformation maps $\mathbb{H}=\{z\in\mathbb{C}\,|\,\Im(z)>0\}$ to itself.
I know that this statement has been asked many times and I know that $A$ has to have real entries with determinant greater than zero, but I have problems deriving on my own, that the entries have to be real. I tried to calculate the imaginary part $\Im\left(\frac{az+b}{cz+d}\right)$ with corresponding matrix $A=\begin{pmatrix} a&b\\ c&d\end{pmatrix}\in GL(2,\mathbb{C})$, but I could not conclude the result. Can anyone give me a hint?
 A: Initially, you might want to show that the real axis must be mapped to the real axis under T.
ie if you let $a$ be real, and suppose that $T(a)$ is not real, then by assumption $Im(T(a))>0. $ (Imaginary part cannot be below the real axis, otherwise $T(\mathbb{H}) \subset  \mathbb{H}$ doesn't hold. )
if you let $b$ be some other point which is very close to $a$ with $Im(b)<0$, then by continuity, $T(b)=c$ is also very close to $T(a)$. As T is a bijection of the Riemann sphere $\to$ Riemann sphere, then $b$ is the unique point for which $c=T(b)$. So if $Im(c)>0$, T cannot map $\mathbb{H}$ onto $\mathbb{H}$. Therefore, $Im(T(a))=0$, so maps any real point to a real point, and therefore the real axis to the real axis. 
In other words, $T$ maps the boundary of $\mathbb{H}$ to its boundary. 
Now, we go on to show that the coefficients $\textit{can be taken to be real}$. Here it need not be the case, that $a,b,c,d$ are all real, but rather that you can find $a,b,c,d$ which are all real. Recall that the mobius transformation corresponding to $(a,b,c,d)$ is the same as the mobius transformation corresponding to $(ta,tb,tc,td)$, with $t$ not necessarily real. 
You now want to show that any mobius transformation that maps the real axis to the real axis needs to have the property that $a,b,c,d$ can be taken to be real.
This gives that $T(z) =\frac{az+b}{cz+d}$, with $a,b,c,d$ chosen to be in$ \mathbb{R}$.
Finally, This transformation sends the real axis to the real axis, but it need not be the case that it sends the upper half plane to itself, but rather to the lower half plane. So to ensure T sends $\mathbb{H}$ to itself, test some complex number in $\mathbb{H}$, ie $T(i) = \frac{(ac+bd)+(ad-bc)i}{c^2+d^2}$,. Then $Im(T(i))>0$, it follows that $ad-bc>0$
