The Fibonacci Matrix
This was the content of the final lecture of a linear algebra class I took. It was magical to learn about the explicit formula for Fibonacci numbers found via eigendecomposition.<p>One funny trick that brought some realism to the lecture: If a and b are the golden ratio and its conjugate, then f_n = a^n + b^n. But since |b| < 1, you can just do f_n = nearest_integer(a^n).