Yeah, amazing fact, isn't it? This is the action of 2x2 matrices on the set of lines in the plane through the origin, where you identify a line with its intersection with the line y=1. That is, let your matrix act on the point (x,1), taking it to (ax+b, cx+d), and then project back onto the line y=1 to get (ax+b/cx+d, 1). As Mike says, this way lies modular functions on the upper half plane. --Michael On Thu, Feb 11, 2016 at 10:31 AM, Mike Stay <metaweta@gmail.com> wrote:
I agree, it's very cool. It features prominently in the study of the modular group (and therefore all things continued fraction).
On Thu, Feb 11, 2016 at 5:42 AM, David Wilson <davidwwilson@comcast.net> wrote:
For 2 x 2 matrix M define
F(M) = f : R->R : x => (M11 x + M12) / (M21 x + M22).
Then
F(AB) = F(A) o F(B)
So the composition of unreduced order 1 rational functions is isomorphic to the product of 2x2 matrices.
I assume this is well known, but I thought it was pretty cool.
This means that finding, say, all order 1 rational functions f with
f(f(x)) = x
would reduce to finding all 2x2 matrices M with
M^2 = I.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Forewarned is worth an octopus in the bush.