I ran into an interesting page which purported to be about various ways of looking at (or deriving) the trigonometric identity
sin2 + cos2 = 1
but the site has lots of stuff including the trail I'm following here, which involves the Fibonacci numbers. Strang derives the Binet formula for the nth Fibonacci number, Fn, using linear algebra:
Fn = 1/√5 [φ1n - φ2n]
where φ1 is the golden mean or golden ratio, and
φ1 + φ2 = 1
The approach is very nice, because it uses the eigenvalues and eigenvectors of the matrix
[ 1 1 ]
[ 1 0 ]
which involves solving λ2 - λ -1 = 0, whose solutions are φ1 and φ2. That is, φ1 and φ2 are eigenvalues of that matrix.
However, I got lost in the middle of Strang's version (in the book), and cast about for another explanation. I found a great one here.
This video turns out to be one of a large number by MathDoctorBob. I am very impressed with the quality of these videos, based on the first half-dozen or so that I've looked at.
Finally, back at the first site, we use two limits (as n => ∞):
φ1n => ∞
φ2n = (1 - φ1)n => 0
and the Binet formula to get a limit for the ratio with large n:
Fn+1 / Fn = φ