Matrix Form
A 2-dimensional system of linear difference equations that describes the Fibonacci sequence is
The eigenvalues of the matrix A are and, and the elements of the eigenvectors of A, and, are in the ratios and Using these facts, and the properties of eigenvalues, we can derive a direct formula for the nth element in the Fibonacci series as an analytic function of n:
The matrix has a determinant of −1, and thus it is a 2×2 unimodular matrix. This property can be understood in terms of the continued fraction representation for the golden ratio:
The Fibonacci numbers occur as the ratio of successive convergents of the continued fraction for, and the matrix formed from successive convergents of any continued fraction has a determinant of +1 or −1.
The matrix representation gives the following closed expression for the Fibonacci numbers:
Taking the determinant of both sides of this equation yields Cassini's identity
Additionally, since for any square matrix A, the following identities can be derived:
In particular, with ,
Read more about this topic: Fibonacci Number
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