Tried several times in the past to write this with detailed step by step derivation but kept failing but just wanted to share this cool concept that I learned in a stochastic processes class.
We know that eventually Fibonacci numbers in the sequence reduces from a sum of the previous two terms to being a multiple of the previous term. The ratio is the golden ratio phi.
Idk if others felt this way, but I wondered like while I was in K-12 if it was possible to get a closed form formula for a general term in the sequence. Then decided it was too hard but not really knowing if it was possible or not.
Well heres a way you can!
Consider a matrix M and a vector v of terms (n-1,n-2) in the sequence such that M*v, we get (n,n-1)
Then M is a matrix that is used to generate Fibonacci numbers.
You can diagonalize this and lo and behold one of the eigenvalues is the golden ratio
The other one is less than 1 so you can see how repeated applications of the Fibonacci matrix diminishes its effect. That’s why the sequence converges to the golden ratio geometric series.
I wonder what the eigenvectors mean, or what else this can be used to understand