I Matrix representation for closed-form expression for Fibonacci numbers

[murshid_islam]

TL;DR

Matrix representation for closed-form expression for Fibonacci numbers:

From the wikipedia page for Fibonacci numbers, I got that the matrix representation for closed-form expression for Fibonacci numbers is:

\begin{pmatrix}<br /> 1 &amp; 1 \\<br /> 1 &amp; 0\\<br /> \end{pmatrix} ^ n =<br /> \begin{pmatrix}<br /> F_{n+1} &amp; F_n \\<br /> F_n &amp; F_{n-1}\\<br /> \end{pmatrix}

That only works when F_0 = 0 and F_1 = 1. How can I find the matrix representation for arbitrary starting values, for example, when F_0 = a and F_1 = b?

 

[martinbn]

Summary:: Matrix representation for closed-form expression for Fibonacci numbers:

From the wikipedia page for Fibonacci numbers, I got that the matrix representation for closed-form expression for Fibonacci numbers is:

\begin{pmatrix}<br /> 1 &amp; 1 \\<br /> 1 &amp; 0\\<br /> \end{pmatrix} ^ n =<br /> \begin{pmatrix}<br /> F_{n+1} &amp; F_n \\<br /> F_n &amp; F_{n-1}\\<br /> \end{pmatrix}

That only works when F_0 = 0 and F_1 = 1. How can I find the matrix representation for arbitrary starting values, for example, when F_0 = a and F_1 = b?

Start with
\begin{pmatrix}<br /> a+b &amp; b \\<br /> b &amp; a\\<br /> \end{pmatrix} =<br /> <br /> \begin{pmatrix}<br /> F_2 &amp; F_1 \\<br /> F_1 &amp; F_0\\<br /> \end{pmatrix}<br />

Then multiply repeatedly on the left by
\begin{pmatrix}<br /> 1 &amp; 1 \\<br /> 1 &amp; 0\\<br /> \end{pmatrix} <br />

 

[anuttarasammyak]

Though same as post #2,