Note Set 17a: Why we love arbitrary powers of square matrices

We take time to enjoy three reasons to love arbitrary powers of square matrices. With the Big Picture sorted out (mostly), we now travel to new places of matrix magnificence. Square matrices, the monks tell us, can be gadgets: rotators, flippers, system evolvers, and much more. We figure out (1) where on a network a randomly wandering texter, engrossed in their magic rectangle, might be after much time; (2) how solving coupled linear differential equations is all about exponentials of matrices; and (3) how to find the $n$th Fibonacci number.
Best consumed by 2016/10/26