### owen by the way

composition of linear fractional transformations

compared to two-by-two matrix multiplications.

consider

in other words,

let f(x) = (Ax+B)/(Cx+D) and

let g(x) =(ax+b)/(cx+d) and

consider the function (“f\circ g”,

i.e. f-composed-with-g). recall

(or trust me on this) that

[f\circ g](x) = f(g(x)); i.e.,

functions compose right-to-left

(“first do gee to ex; then plug in

the answer and do eff *to* gee-of-ex”…

first g, then f… alas. but there it is).

so we have

thus

whereas one also has

so the matrix-multiplication equation

can be obtained from the function-composition equation

merely by applying an eraser here and there.

(my lecture-note-blogging of winter 09 include some

remarks on \mapsto notation and much more

about linear fractional (“mobius”) transformations.)

May 28, 2011 at 3:47 pm

JD’s matrix multiplication thread of 9/’10.

May 30, 2011 at 5:47 pm

Thanks. Useful.