Deriving Binet's Formula
From before, we have that . From this we can derive that is equal to the bottom left element of . In fact, in general we have
(Proof left as an exercise to the reader!)
We can use our diagonalization to find a closed form for this value: let the number crunching commence! (I'm writing it out here for pedagogical purposes, but in the real world I am 100% using a computer to do all of this math for me.)
To simplify further, we can use a special property of the golden ratio to rewrite in terms of :
This lets us simplify further:
Remember, is the bottom-left element.1 So, at long last, we have a closed form for :
🥳
That was quite the derivation!2 Our result is known as Binet's formula.
- You can instead multiply this by a vector like we did above: this lets you use any set of initial conditions, not just .↩
- There are other, faster derivations, but I think many of them suffer from a sense of pulling rabbits out of hats. Diagonalization is a very commmon real-world technique, especially for exponentiation of matrices, so it's only natural to try it here.↩
Deriving Binet's Formula
From before, we have that . From this we can derive that is equal to the bottom left element of . In fact, in general we have
(Proof left as an exercise to the reader!)
We can use our diagonalization to find a closed form for this value: let the number crunching commence! (I'm writing it out here for pedagogical purposes, but in the real world I am 100% using a computer to do all of this math for me.)
To simplify further, we can use a special property of the golden ratio to rewrite in terms of :
This lets us simplify further:
Remember, is the bottom-left element.1 So, at long last, we have a closed form for :
🥳
That was quite the derivation!2 Our result is known as Binet's formula.
- You can instead multiply this by a vector like we did above: this lets you use any set of initial conditions, not just .↩
- There are other, faster derivations, but I think many of them suffer from a sense of pulling rabbits out of hats. Diagonalization is a very commmon real-world technique, especially for exponentiation of matrices, so it's only natural to try it here.↩