Euler-Binet Formula/Proof 2

Theorem
The Fibonacci numbers have a closed-form solution:
 * $F \left({n}\right) = \dfrac {\phi^n - \left({1 - \phi}\right)^n} {\sqrt 5} = \dfrac {\phi^n - \left({-1 / \phi}\right)^n} {\sqrt 5}$

where $\phi$ is the golden mean.

Putting $\hat \phi = 1 - \phi = -\dfrac 1 \phi$ this can be written:
 * $F \left({n}\right) = \dfrac {\phi^n - \hat \phi^n} {\sqrt 5}$

Proof
Let $A = \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}$.

Let $I$ be the $2 \times 2$ identity matrix.

First we will prove:

For all positive integers $n$ we have:

Proof
We will prove the statement by mathematical induction. Since $F_2 =1$, we have:

So the statement is true for $n=1$.

Induction hypothesis: Suppose that the statement is true for $n \geq 1$. Then:

But $n \ge 1$ implies $n+1 \gt 1$ and $n+2 \gt 1$, so
 * $\displaystyle F_\left({n+1}\right) +F_n =F_\left({n+2}\right)$,

and
 * $\displaystyle F_n +F_\left({n-1}\right) =F_\left({n+1}\right)$.

Thus:

So the statement is true for $n+1$.

$A$ has the eigenvalues $\phi$ and $\hat \phi$.

Now we have that

This shows that:
 * $\displaystyle \begin{pmatrix} \phi \\ 1 \end{pmatrix}$

is an [[Definition:Eigenvector|Eigenvector] of $\phi$, and
 * $\displaystyle \begin{pmatrix} \hat \phi \\ 1 \end{pmatrix}$

is an eigenvector of $\hat \phi$.

Thus:
 * $\displaystyle \begin{pmatrix} \frac \phi {\sqrt 5} \\ \frac 1 {\sqrt 5} \end{pmatrix}$

is an [[Definition:Eigenvector|Eigenvector] of $\phi$, and
 * $\displaystyle \begin{pmatrix} \frac \hat \phi {\sqrt 5} \\ \frac 1 {\sqrt 5} \end{pmatrix}$

is an eigenvector of $\hat \phi$.

By Eigenvalue of Matrix Powers we get for a positive integer $n$:

From \ref{nth_pw} we get:
 * $\displaystyle A^n \begin{pmatrix} 1 \\ 0 \end{pmatrix} =\begin{pmatrix} F_\left({n+1}\right) \\ F_\left({n}\right) \end{pmatrix}$.

Substituting

we get:

Hence the result.

It is also known as Binet's Formula.