Fibonacci Numbers - Matrix Form

Matrix Form

A 2-dimensional system of linear difference equations that describes the Fibonacci sequence is

\begin{align} {F_{k+2} \choose F_{k+1}} &= \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix} {F_{k+1} \choose F_{k}} \\ \vec F_{k+1} &= A \vec F_{k} \end{align}

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:

\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}^n = \begin{pmatrix} F_{n+1} & F_n \\ F_n & F_{n-1} \end{pmatrix}.

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:

\begin{align} {F_m}{F_n} + {F_{m-1}}{F_{n-1}} &= F_{m+n-1}\\ F_{n+1}F_{m} + F_n F_{m-1} &= F_{m+n} \end{align}

In particular, with ,

\begin{align} F_{2n-1} &= F_n^2 + F_{n-1}^2\\ F_{2n} &= (F_{n-1}+F_{n+1})F_n\\ &= (2F_{n-1}+F_n)F_n \end{align}

Read more about this topic:  Fibonacci Numbers

Famous quotes containing the words matrix and/or form:

    “In all cultures, the family imprints its members with selfhood. Human experience of identity has two elements; a sense of belonging and a sense of being separate. The laboratory in which these ingredients are mixed and dispensed is the family, the matrix of identity.”
    —Salvador Minuchin (20th century)

    “Quintilian [educational writer in Rome around A.D. 100] thought that the earliest years of the child’s life were crucial. Education should start earlier than age seven, within the family. It should not be so hard as to give the child an aversion to learning. Rather, these early lessons would take the form of play—that embryonic notion of kindergarten.”
    —C. John Sommerville (20th century)