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:
 * $F \left({n}\right) = \dfrac {\phi^n - \hat \phi^n} {\sqrt 5}$

First we will prove the following

Lemma
Let $A$ be a square matrix which is neither nilpotent nor idempotent. Let $\lambda$ be an eigenvalue of $A$ and $v$ be the corresponding eigenvector. Then for each positive integer $n$ the following equation holds:

Proof of lemma
The proof proceeds by induction. Clearly, the statement holds for $n=1$. Induction hypothesis: Suppose that $A^n v =\lambda^n v$ holds for some positive integer $n$. Then:

Proof of the theorem
Let $A=\begin{smallmatrix} 1 & 1 \\ 1 & 0 \end{smallmatrix}$.

For each positive integer $n$ we have:
 * $A^n= \begin{pmatrix} F_\left({n+1}\right) & F_n \\ F_n & F_\left({n-1}\right) \end{pmatrix}$.

(See Cassini's Identity for a proof.)

$A$ has the eigenvalues $\phi$ and $\hat \phi$. Now we have that

By the previous lemma we get for a positive integer $n$:

Putting
 * $B = \begin{pmatrix} \phi & {\hat \phi} \\ 1 & 1 \end{pmatrix}$

we get:

Let $B^*$ be the adjugate of $B$. Then:

It follows that

where $I$ is the $2 \times 2$-identity matrix. Hence

Thus:

{{eqn|l= \begin{pmatrix} \phi^n & 0 \\ 0 & \left.{\hat \phi}\right.^n \end{pmatrix} B^{-1} |r= \dfrac{1}{ {\sqrt 5} } \begin{pmatrix} \phi^n & -\phi^n \cdot {\hat phi} \\ -\left.{\hat \phi}\right.^n & \left.{\hat \phi}\right.^n \cdot \phi \end{bmatrix} |r= \dfrac{1}{ {\sqrt 5} } \begin{pmatrix} \phi^n & \phi^{n-1} \\ -\left.{\hat \phi}\right.^n & -\left.{\hat \phi}\right.^{n-1} \end{pmatrix}

So we get:

{{eqn|l= A^n |r= \dfrac{1}{ {\sqrt 5} } B \begin{pmatrix} \phi^n & \phi^{n-1} \\ -\left.{\hat \phi}\right.^n & -\left.{\hat \phi}\right.^{n-1} \end{pmatrix} |r= \dfrac{1}{ {\sqrt 5} } \begin{pmatrix} \phi^{n+1} -\left.{\hat \phi}\right.^{n+1} & \phi^n -\left.{\hat \phi}\right.^n \\ \phi^n -\left.{\hat \phi}\right.^n & \phi^{n-1} -\left.{\hat \phi}\right)^{n-1} \end{pmatrix} }}

Hence the result. {{Qed}} {{namedfor|Jacques Philippe Marie Binet|name2=Leonhard Paul Euler|cat=Binet|cat2=Euler}} It is also known as Binet's Formula.