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College of the Redwoods

Fun With Fibonacci

Angie Tapia, Cameron Trujillo

December 7, 2012

1 Introduction

1.1 In the Beginning

...there was Pythagoras. This nearly mythical mathematician put geometry on the mapand supposedly coined the term "Mathematics" (or "that which is learned"). He was aholy man, a prophet, and is purported to have been a seer, but he taught his ferventdisciples that all is number. The Pythagoreans are credited with the discovery of the

so-called "Golden Ratio". Figure 1.1

Figure 1.1: The Frequently Manifested "Golden Ratio"

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1.2 And After That...

...there was Leonardo of Pisa. He was a major dude, and a son of Italy proper. Leonardosmain mission in life was to spread the good word of the Hindu-Arabic number system tothe western world at large. His book, the Liber abaci (or "book of abacus"), really had

nothing to do with the abacus at all, but was among the first of western books aboutnumbers as we know them. Within its hallowed pages can be found the inklings ofgreatness in the form of a lagamorphic confundity. Leonardo, also known as Fibonacci,posed the question:

How many pairs of rabbits will be produced in a year, beginning witha single pair, if in every month each pair bears a new pair which becomesproductive from the second month on?

Fibonaccis quandry led mathematicians for generations to come on a wild ride throughthe annals of numerical history. The so called "Fibonacci Sequence" was born. A closerlook at this ever-expanding rabbit family reveals the series

1, 1, 2, 3, 5, 13, 21, 34,...,FnWith Fn =Fn1 + Fn2

Or, in other words, a sequence of numbers wherein each entry is the sum of the twopreceeding entries.

1.3 The End of the Beginning

As it turns out, a rather simple application of the tenets of the most holy of mathemat-ics, linear algebra, reveals an interesting relationship between the Golden Ratio of thePythagoreans and Fibonaccis copiously copulating rabbits.

2 What Well Need To Do...

To fully comprehend how the Fibonacci Sequence applies in Linear Algebra, a few termsmust first be defined and explained.

Eigenvector: An eigenvector of an n x n matrix A is a nonzero vector x such thatAx=xfor some scalar .

Eigenvalue: A scalar is called an eigenvalue ofA if there is a nontrivial solution x

ofAx=x; such an xis called an eigenvector corresponding to .

Eigenspace(of A corresponding to ): The set of all solutions of Ax=x, where is an eigenvalue of A. Consists of the zero vector and all eigenvectors corresponding to.

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Theorem 1: The eigenvalues of a triangular matrix are the entries on its main di-agonal.

Theorem 2: Ifv1,...,vr are eigenvectors that correspond to distinct eigenvalues 1,...,rof an nx nmatrix A, then the set {v1,...,vr} is linearly independent.

detA=

1r,product of pivots in U

0, when A is not invertible

Characteristic Equation: The scalar equation det(A - I)=0 is the characteristicequation ofA.

The scalar is an eigenvalue of an nx nmatrix A if and only if satisfies the charac-teristic equation det(A - I)=0 .

Invertible Matrix: A square matrix that possesses an inverse.

The Invertible Matrix Theorem (continued)Let A be an nx nmatrix. Then A is invertible if and only if:s. The number ) is not an eigenalue ofAt. The determinant ofA is not zero.

The Diagonalization Theorem :An nx nmatrix A is diagonizable if and only ifA has nlinearly independent eigenvec-tors.In fact, A=PDP1, with D a diagonal matrix, if and only if the columns of P are nlinearly independent eigenvectors ofA. In this case, the diagonal entries ofDare eigen-

values of A that correspond, respectively, to the eigenvectors in P.

Eigenvector Basis: A is diagonizable if and only if there are enough eigenvectorsto form a basis ofRn. Such a basis is called an eigenvector basis ofRn.

Theorem 6: An nx nmatrix with ndistinct eigenvalues is diagonizable.

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3 Maths, baby. Maths...

As stated prviously, the nthentry in the fibonacci sequence is defined by

Fn =Fn1 + Fn2

Pretty easy, right? Well not so fast, cowboy. This definition is all fine and well, so longas we have Fn1 andFn2 defined.To get where were going, we need to view the systemas a linear transformation. Lets begin by looking at the Fibonacci Sequence as a Systemof linear equations

Fn =Fn1 + Fn2

Fn1 =Fn1 + 0 FnFn1

=

1 1

1 0

Fn1Fn2

(3.1)

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