Cassini's Identity/Lemma

Lemma for Cassini's Identity

$\forall n \in \Z_{>1}: \begin{bmatrix} F_{n + 1} & F_n \\ F_n & F_{n - 1} \end{bmatrix} = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}^n$

Proof

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:

$\ds \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}^{k+1}$

$=$

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

Induction Hypothesis

$\ds $

$=$

$\ds \begin{bmatrix} F_{k + 1} + F_k & F_{k + 1} \\ F_k + F_{k - 1} & F_k \end{bmatrix}$

Definition of Matrix Product

$\ds $

$=$

$\ds \begin{bmatrix} F_{k + 2} & F_{k + 1} \\ F_{k + 1} & F_k \end{bmatrix}$

So by induction:

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

$\blacksquare$

Sources

1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.8$: Fibonacci Numbers: Exercise $6$