Cool idea possibly extendable to other sequences; Getting the golden ratio from the Fibonnacci Sequence via eigenvector decomposition.

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!

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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.

M =

[[1,1],

[1,0]]

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.

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I wonder what the eigenvectors mean, or what else this can be used to understand

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Nevin Manimala

Nevin Manimala is interested in blogging and finding new blogs https://nevinmanimala.com

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