フィボナッチ数列の行列表示

by nomura · 2024年12月4日

フィボナッチ数列の行列表示
フィボナッチ数列は行列で表すと次のようになる。

(1)漸化式

\[ \left(\begin{array}{c} F_{n+2}\\ F_{n+1} \end{array}\right)=\left(\begin{array}{cc} 1 & 1\\ 1 & 0 \end{array}\right)\left(\begin{array}{c} F_{n+1}\\ F_{n} \end{array}\right) \]

(2)一般項

\[ \left(\begin{array}{cc} F_{n+1} & F_{n}\\ F_{n} & F_{n-1} \end{array}\right)=\left(\begin{array}{cc} 1 & 1\\ 1 & 0 \end{array}\right)^{n} \]

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\(F_{n}\)はフィボナッチ数列

(1)

\begin{align*} \left(\begin{array}{cc} 1 & 1\\ 1 & 0 \end{array}\right)\left(\begin{array}{c} F_{n+1}\\ F_{n} \end{array}\right) & =\left(\begin{array}{c} F_{n+1}+F_{n}\\ F_{n+1} \end{array}\right)\\ & =\left(\begin{array}{c} F_{n+2}\\ F_{n+1} \end{array}\right) \end{align*}

(2)

漸化式は\(F_{n+2}=F_{n+1}+F_{n}\)なので、\(n=-1\)を代入して、\(F_{1}=F_{0}+F_{-1}\)より、\(F_{-1}=F_{1}-F_{0}=1-0=1\)とする。
\begin{align*} \left(\begin{array}{c} F_{n+1}\\ F_{n} \end{array}\right) & =\left(\begin{array}{cc} 1 & 1\\ 1 & 0 \end{array}\right)\left(\begin{array}{c} F_{n}\\ F_{n-1} \end{array}\right)\\ & =\left(\begin{array}{cc} 1 & 1\\ 1 & 0 \end{array}\right)^{n}\left(\begin{array}{c} F_{1}\\ F_{0} \end{array}\right) \end{align*} \begin{align*} \left(\begin{array}{c} F_{n}\\ F_{n-1} \end{array}\right) & =\left(\begin{array}{cc} 1 & 1\\ 1 & 0 \end{array}\right)\left(\begin{array}{c} F_{n-1}\\ F_{n-2} \end{array}\right)\\ & =\left(\begin{array}{cc} 1 & 1\\ 1 & 0 \end{array}\right)^{n}\left(\begin{array}{c} F_{0}\\ F_{-1} \end{array}\right) \end{align*} これより、
\begin{align*} \left(\begin{array}{cc} F_{n+1} & F_{n}\\ F_{n} & F_{n-1} \end{array}\right) & =\left(\begin{array}{cc} 1 & 1\\ 1 & 0 \end{array}\right)^{n}\left(\begin{array}{cc} F_{1} & F_{0}\\ F_{0} & F_{-1} \end{array}\right)\\ & =\left(\begin{array}{cc} 1 & 1\\ 1 & 0 \end{array}\right)^{n}\left(\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right)\\ & =\left(\begin{array}{cc} 1 & 1\\ 1 & 0 \end{array}\right)^{n} \end{align*} となり題意は成り立つ。