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:

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

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:

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

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
$$\begin{pmatrix}<br /> a+b & b \\<br /> b & a\\<br /> \end{pmatrix} =<br /> <br /> \begin{pmatrix}<br /> F_2 & F_1 \\<br /> F_1 & F_0\\<br /> \end{pmatrix}$$

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

 

[anuttarasammyak]

Though same as post #2,