Matrix Representation of the Fibonacci Numbers
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General formula for the Fibonacci sequence
The problem is to obtain a general formula for the Fibonacci numbers; that is, a formula of the type [Equation goes here - download the original to see it.] where f is a function. To find this formula we use a matrix representation of the Fibonacci numbers. This is given by [Equation goes here - download the original to see it.] To verify this, we start with the matrix [Equation goes here - download the original to see it.] , then[Equation goes here - download the original to see it.] Continuing [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] So in terms of the original problem about rabbit populations, the first term in the matrix column is the number of reproductive pairs, and the second term is the number of non-reproductive pairs. So the matrix does represent the Fibonacci sequence. Let [Equation goes here - download the original to see it.] , then let us use the idea of the decomposition of this matrix into its canonical diagonal form to solve the problem. That is, we will write T in the form [Equation goes here - download the original to see it.] where D is a diagonal matrix containing eigenvalues along its diagonal, and Q and Q-1 are inverses of each other. To find the diagonal matrix and the eigenvalues, we use the formul [Equation goes here - download the original to see it.] where is an eigenvalue That is [Equation goes here - download the original to see it.] Hence [Equation goes here - download the original to see it.] Applying the quadratic formula to this, we get [Equation goes here - download the original to see it.] That is [Equation goes here - download the original to see it.] This gives the diagonal matrix as [Equation goes here - download the original to see it.] or [Equation goes here - download the original to see it.] Now to find the eigenvectors [Equation goes here - download the original to see it.] To find the inverse of Q we use the formula [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] Now [Diagram goes here - download the original to see it.] hence [Equation goes here - download the original to see it.] and [Equation goes here - download the original to see it.] To find the nth term of the series we have [Equation goes here - download the original to see it.] That is [Equation goes here - download the original to see it.] Hence [Equation goes here - download the original to see it.] or [Equation goes here - download the original to see it.] Since [Equation goes here - download the original to see it.] this means that as n becomes very large [Equation goes here - download the original to see it.] Also the ratio of two successive numbers in the sequence is [Equation goes here - download the original to see it.] This number is called the golden ratio. Other second order recurrence relations can be solved in the same way by a matrix representation.
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Contents of
Matrix Representation of the Fibonacci Numbers
1 Matrix representation of the Fibonacci numbers
2 General formula for the Fibonacci sequence
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