/sci/ - Science & Math

>>12536712
>geometric intuition
The stackexchange example anon posted works, and it's very nice, considering it's all integers, but it doesn't really have any clear geometric intuition to it, so I'll post a different one.
Recall that the matrix [math]R = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}[/math] rotates coordinates by ninety degrees.
The matrices [math]A' = \begin{pmatrix} 2 & 0 \\ 0 & 1/2 \end{pmatrix}[/math] and [math]B' = \begin{pmatrix} 3 & 0 \\ 0 & 1/3 \end{pmatrix}[/math] both expand the first coordinate and flatten the second by inverse factors.
These don't have finite order, but [math]A = A'R = \begin{pmatrix} 0 & 2 \\ -1/2 & 0 \end{pmatrix}[/math] and [math]B = B'R = \begin{pmatrix} 0 & 3 \\ -1/3 & 0 \end{pmatrix}[/math] do, because they rotate and then expand/flatten.
When you compute [math]AB[/math], what happens is that you rotate and expand/flatten, but these expansions have different factors, so you get [math]AB = \begin{pmatrix} -2/3 & 0 \\ 0 & -3/2 \end{pmatrix}[/math], because the expansions and flattenings don't cancel out since the factors are different.

Formerly >>11100698
Why the hell did you lads actually wait for me edition.