Fibonacci numbers The n th Fibonacci number is generated by...

Fibonacci numbers The $n$ th Fibonacci number is generated by the recursion equation $F_{n}=F_{n-1}+F_{n-2}$ with $F_{0}=F_{1}=1 .$ Define the two new variables $x_{n}=F_{n}$ and $y_{n}=F_{n-1} .$ (a) Show that the vector $\mathbf{Z}_{n}=\left[ \begin{array}{l}{x_{n}} \\ {y_{n}}\end{array}\right]$ obeys the recursion $\mathbf{z}_{n}=A \mathbf{z}_{n-1},$ where $$A=\left[ \begin{array}{cc}{1} & {1} \\ {1} & {0}\end{array}\right]$$ (b) Find the eigenvalues of $A$ (c) Can the Perron-Frobenius Theorem be applied to the matrix $A ?$ If so, what is the long-term behavior of the vector $\mathbf{Z}_{n} ?$ (d) Find a formula for the $n$ th Fibonacci number.

Fibonacci numbers The $n$ th Fibonacci number is generated by the recursion equation $F_{n}=F_{n-1}+F_{n-2}$ with $F_{0}=F_{1}=1 .$ Define the two new variables $x_{n}=F_{n}$ and $y_{n}=F_{n-1} .$
(a) Show that the vector $\mathbf{Z}_{n}=\left[ \begin{array}{l}{x_{n}} \\ {y_{n}}\end{array}\right]$ obeys the recursion $\mathbf{z}_{n}=A \mathbf{z}_{n-1},$ where $$A=\left[ \begin{array}{cc}{1} & {1} \\ {1} & {0}\end{array}\right]$$
(b) Find the eigenvalues of $A$
(c) Can the Perron-Frobenius Theorem be applied to the matrix $A ?$ If so, what is the long-term behavior of the vector $\mathbf{Z}_{n} ?$
(d) Find a formula for the $n$ th Fibonacci number.

Key Concepts

Perron-Frobenius Theorem

The Perron-Frobenius theorem applies to non-negative, often irreducible, square matrices and guarantees the existence of a unique largest real eigenvalue, known as the Perron root, along with a corresponding eigenvector with strictly positive components. This theorem is useful for understanding the asymptotic behavior of processes modeled by such matrices, as it implies that repeated applications of the matrix will cause the system to align with the direction of this dominant eigenvector.

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are fundamental concepts in linear algebra that describe the action of a matrix on vectors in a way that scales them without changing their direction. By finding the eigenvalues of the matrix associated with a recurrence, one can determine the growth rates of the components of the sequence and thus derive properties such as long-term behavior and closed-form solutions.

Closed-Form Formula (Binet's Formula)

A closed-form formula provides an explicit expression for the nth term of a sequence in terms of n, without requiring recursion. For sequences defined by linear recurrences with constant coefficients, solving the characteristic equation leads to such formulas. In the case of Fibonacci numbers, this process yields Binet's formula, which expresses the nth Fibonacci number in terms of powers of the golden ratio and its conjugate.

Recurrence Relations

A recurrence relation is an equation that expresses each term of a sequence as a function of its preceding terms. It provides a way to define sequences recursively, which is particularly useful in many areas of mathematics including computer science and combinatorics. In the context of Fibonacci numbers, the relation F? = F??? + F??? with initial conditions defines the entire sequence.

Matrix Representation of Linear Recurrences

A linear recurrence relation, particularly of second order, can be represented as a first order system by using vectors and matrices. This transformation allows the use of linear algebra techniques, such as eigenvalue analysis, to study and solve the recurrence. In the Fibonacci case, the vector [F?, F???] and a specific 2x2 matrix encapsulate the recursive behavior of the sequence.

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