- #1

murshid_islam

- 458

- 19

- TL;DR 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:

[tex]\begin{pmatrix}

1 & 1 \\

1 & 0\\

\end{pmatrix} ^ n =

\begin{pmatrix}

F_{n+1} & F_n \\

F_n & F_{n-1}\\

\end{pmatrix} [/tex]

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

[tex]\begin{pmatrix}

1 & 1 \\

1 & 0\\

\end{pmatrix} ^ n =

\begin{pmatrix}

F_{n+1} & F_n \\

F_n & F_{n-1}\\

\end{pmatrix} [/tex]

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