Cassini's Identity/Lemma

Lemma for Cassini's Identity

 * $\forall n \in \Z_{>1}: \begin{bmatrix}

F_{n + 1} & F_n \\ F_n      & F_{n - 1} \end{bmatrix} = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}^n$

Basis for the Induction

 * $\begin{bmatrix}

F_2 & F_1    \\ F_1 & F_0 \end{bmatrix} = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix} = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}^1$

Induction Hypothesis
For $k \in \Z_{>1}$, it is assumed that:


 * $\begin{bmatrix}

F_{k + 1} & F_k \\ F_k      & F_{k - 1} \end{bmatrix} = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}^k$

It remains to be shown that:


 * $\begin{bmatrix}

F_{k + 2} & F_{k + 1} \\ F_{k + 1} & F_k \end{bmatrix} = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}^{k + 1}$

Induction Step
The induction step follows from conventional matrix multiplication:

So by induction:
 * $\begin{bmatrix}

F_{n + 1} & F_n \\ F_n      & F_{n - 1} \end{bmatrix} = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}^n$