Möbius transformation/Proofs

Fixed Points

The article claims that for $c\; e\; 0$, the two roots are :$gamma\; =\; frac\{(a\; -\; d)\; pm\; sqrt\{(a\; -\; d)^2\; +\; 4\; c\; b\{2\; c\}$

of the quadratic equation:$c\; gamma^2\; -\; (a\; -\; d)\; gamma\; -\; b\; =\; 0\; ,$which follows from the fixed point equation : $gamma=$agamma +b}over {cgamma +d by multiplying both sides with the denominator $cgamma\; +d$ and collecting equal powers of $gamma$. Note that the quadratic equation degenerates into a linear equation if $c=0$, this corresponds to the situation that one of the fixed points is the point at infinity. In this case the second fixed point is finite if $a-d\; e\; 0$ otherwise the point at infinity is a fixed point "with multiplicity two" (the case of a pure translation).

Note that

:$(a\; -\; d)^2\; +\; 4\; c\; b\; =(a\; -\; d)^2\; +\; 4ad\; -4\; =\; (a+d)^2-4\; =\; mbox\{tr\}^2mathfrak\{H\}\; -\; 4$