From 115cc22af2ba00e76f123f2a382afc44ff9710d4 Mon Sep 17 00:00:00 2001
From: jxu <7989982+jxu@users.noreply.github.com>
Date: Fri, 18 Apr 2025 23:11:59 -0400
Subject: [PATCH 1/2] Fibonacci: restore matrix power form

Maybe a dotted line would show the matrix [[F2, F1],[F1,F0]] can be viewed as two column vectors.
Using only the matrix power saves one matrix-vector multiply.
---
 src/algebra/fibonacci-numbers.md | 14 +++++++++++++-
 1 file changed, 13 insertions(+), 1 deletion(-)

diff --git a/src/algebra/fibonacci-numbers.md b/src/algebra/fibonacci-numbers.md
index e63e52081..e1acfdcb0 100644
--- a/src/algebra/fibonacci-numbers.md
+++ b/src/algebra/fibonacci-numbers.md
@@ -157,7 +157,19 @@ F_{n}
 \end{pmatrix}
 $$
 
-where $F_1 = 1, F_0 = 0$.
+where $F_1 = 1, F_0 = 0$. 
+In fact, since 
+$$
+\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}
+= \begin{pmatrix} F_2 & F_1 \\ F_1 & F_0 \end{pmatrix}
+$$
+
+we can use the matrix directly:
+
+$$
+\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}^n
+= \begin{pmatrix} F_{n+1} & F_n \\ F_n & F_{n-1} \end{pmatrix}
+$$
 
 Thus, in order to find $F_n$ in $O(\log  n)$ time, we must raise the matrix to n. (See [Binary exponentiation](binary-exp.md))
 

From 59c31747c57735cfe83639f9a1dac64bb800dd78 Mon Sep 17 00:00:00 2001
From: Oleksandr Kulkov <adamant.pwn@gmail.com>
Date: Sat, 19 Apr 2025 09:56:55 +0200
Subject: [PATCH 2/2] Update src/algebra/fibonacci-numbers.md

---
 src/algebra/fibonacci-numbers.md | 1 +
 1 file changed, 1 insertion(+)

diff --git a/src/algebra/fibonacci-numbers.md b/src/algebra/fibonacci-numbers.md
index e1acfdcb0..dd8e1bd1a 100644
--- a/src/algebra/fibonacci-numbers.md
+++ b/src/algebra/fibonacci-numbers.md
@@ -159,6 +159,7 @@ $$
 
 where $F_1 = 1, F_0 = 0$. 
 In fact, since 
+
 $$
 \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}
 = \begin{pmatrix} F_2 & F_1 \\ F_1 & F_0 \end{pmatrix}