Graduate Theses and Dissertations

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1 University of South Florida Scholar Commons Graduate Theses and Dissertations Graduate School 2005 Fibonacci vectors Ena Salter University of South Florida Follow this and additional works at: Part of the American Studies Commons Scholar Commons Citation Salter, Ena, "Fibonacci vectors" (2005 Graduate Theses and Dissertations This Thesis is brought to you for free and open access by the Graduate School at Scholar Commons It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Scholar Commons For more information, please contact

2 Fibonacci Vectors by Ena Salter A thesis submitted in partial fulfillment of the requirements for the degree of Master of Arts Department of Mathematics College of Arts and Sciences University of South Florida Major Professor: Brian Curtin, PhD Thomas Bieske, PhD David Stone, PhD Date of Approval: July 20, 2005 Keywords: linear algebra, Lucas numbers, product identites, asymptotics, golden ratio c Copyright 2005, Salter

3 Dedication For the three major role models in my life My mom whose courage and strength make me so proud My dad whose unconditional love is my guiding light And for the best teacher in the world, Dr Stone

4 Acknowledgements There are so many people to whom I owe a great deal of appreciation, for without them I would not I have made it thus far First off, I would like to thank the many math professors I had at Georgia Southern for the enormous amount of time they devoted to teaching me that math was more than just numbers and letters Secondly, I would like to thank Dr Brian Curtin for spending countless hours helping me not only understand mathematics but also learn how to program in TEX, his patience will never be forgotten I would also like to thank Dr Thomas Bieske for agreeing to be on my defense committee Of course, I would also like to thank Dr David Stone for seeing in me my desire to learn and inspiring me to become a math teacher Also, thanks to my mother for teaching me the most valuable lesson in lifeto never give up no matter how rough the road may be And a big thank you to my daddy for his endless amount of support both finaincially and emotionally Without the love of my parents and my brother James I would have never been able to accomplish so much in my life Finally, thanks to all of my friends Mindy, Meagan, Erin, Charlotte, Daina, Joey, John, and Michael for listening to me rant and putting up with me when I must have been impossible to be around You guys are truly a blessing in my life and your support shows throughout my thesis I love all of you and I am so thankful to call you guys my friends Through topology, the bad boyfriends, then random allergic reactions, the minor and major illnesses I could always count on you Megas We did it!

5 Table of Contents Abstract iii Introduction 2 Fibonacci and Lucas numbers 3 2 Fundamental Fibonacci and Lucas facts 3 22 A linear algebraic perspective 4 23 The Binet formulas 5 3 Fibonacci and Lucas vectors 0 3 Linear algebraic set up 0 32 Fibonacci and Lucas vectors 3 33 Dot products 4 34 Fibonacci and Lucas identities 7 4 Angles in even dimension 22 4 Dot products with a and b Cosines of angles with a and b Angles between vectors Angles with the axes Dimension two 27 5 Comments on angles in odd dimension 30 5 The angle between a and b Three-dimensional Fibonacci Vectors 3 i

6 6 Asymptotics of angles 33 6 Angles with a Dominant eigenvalues Further directions 36 References 39 About the Author End Page ii

7 Fibonacci Vectors Ena Salter ABSTRACT By the n-th Fibonacci (respectively Lucas vector of length m, we mean the vector whose components are the n-th through (n + m -st Fibonacci (respectively Lucas numbers For arbitrary m, we express the dot product of any two Fibonacci vectors, any two Lucas vectors, and any Fibonacci vector and any Lucas vector in terms of the Fibonacci and Lucas numbers We use these formulas to deduce a number of identities involving the Fibonacci and Lucas numbers iii

8 Introduction Seldom, in the study of mathematics, does one come across a topic so fascinating that it captivates the minds of mathematicians and non-mathematicians alike The Fibonacci sequence, though over 700 years old, still contains many secrets yet to be discovered The famous sequence has intrigued so many people that The Fibonacci Quarterly, a journal devoted solely to the study of anything Fibonacci was created in 963, shortly after the formation of the Fibonacci Society in 962 Very famous during his life, Fibonacci is considered the greatest European mathematician of the middle ages Fibonacci, short for Filius Bonacci son of Bonacci was born around 70, to a Pisan Merchant who freely traveled the expanse of the Byzantine Empire Due to the extensive traveling, Leonardo of Pisa was frequently exposed to Islamic scholars and the mathematics of the Islamic world After his return to Pisa, Fibonacci spent the next 25 years writing books that included much of what he had learned in his travels Though many works were lost, three main works were preserved They are: Liber abaci (202, 228, the Practica geometriae (220, and the Liber quadratorum (225 Fibonacci is credited with being one of the first people to introduce the Hindu-Arabic number system into Europe the system we now use today based of ten digits with its decimal point and a symbol for zero:, 2, 3, 4, 5, 6, 7, 8, 9 and 0 It is in Liber abaci that Fibonacci poses his famous rabbit question, A certain man put a pair of rabbits in a place surrounded on all sides by a wall How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from

9 the second month on becomes productive? The answer to this question involves the famous Fibonacci sequence:,, 2, 3, 5, 8, 3, where the n-th Fibonacci number, denoted F n, is defined for all integers n by the recurrence relation F n+2 F n+ + F n, with starting values F F 2 What seems like such a simple concept appears in almost every scientific field of study from botany to architecture, and biology to painting We are but beings swimming in the sea of Fibonacci where all we must perform is a single stroke to meet a Fibonacci number It is this sequence of numbers that inspires all of the ideas in this paper and to no suprise we find that whenever one works with Fibonacci numbers we obtain beautiful results We will look at vectors of the form f n F n, F n+, F n+2,, F n+m, and call them Fibonacci vectors We will study in depth the Fibonacci vectors in arbitrary dimensions Simultaneously, we shall consider analagous results for Lucas vectors Edouard Lucas was born in France in 842 Following service as an artillery officer during the Franco-Prussian War (870-87, Lucas became professor of mathematics at the Lycée Saint Louis in Paris He later became professor of mathematics at the Lycée Charlemagne, also in Paris Lucas is best known for his studies in number theory We will study the Lucas sequence:, 3, 4, 7,, 8, 29, where the n-th Lucas number, denoted L n, is defined for all integers n by the recurrence relation L n+2 L n+ + L n, with starting values L, L 2 3 We look at analogous Lucas vectors of the form l n L n, L n+, L n+2,, L n+m We will begin our study by recalling some well-known facts concerning the Fibonacci and Lucas vectors in two dimensions We then generalize these facts to arbitrary dimension From very elementary linear algebraic considerations we shall derive a number of nontrivial relations involving the Fibonacci and Lucas numbers It turns out that vectors in a fixed dimension lie in a single plane We consider angles between the vectors of interest 2

10 2 Fibonacci and Lucas numbers 2 Fundamental Fibonacci and Lucas facts The Fibonacci numbers F n are defined for all integers n by the second order recurrence relation F n+2 F n+ + F n (2 and initial conditions F F 2 (22 The Lucas numbers L n are defined for all integers n by the same second order recurrence relation as the Fibonacci numbers L n+2 L n+ + L n (23 but initial conditions L, L 2 3 (24 We recall some relations involving the Fibonacci and Lucas numbers These facts are well-known and can be found in most basic references, eg [3, 4] Theorem 2 ([3] For all integers n, F 2n F 2 n+ F 2 n, (25 3

11 F 2n F 2n+ F2n 2 +, (26 F n F k F n k+ + F k F n k, (27 F 2n+ Fn 2 + Fn+, 2 (28 F n L n + L n+, (29 F n ( n+ F n, (20 L n ( n L n (2 22 A linear algebraic perspective We recall a well-known linear algebraic approach to the Fibonacci numbers This can be found in [4] and many elementary linear algebra books, eg [7, 9] This approach appears to be due to Binet Set T 0 (222 Theorem 22 ([4] Let T be as in Eqn (222 For all integers n, let fn [F n, F n+ ] t Then f n+ T f n In particular, for all integers k f n+ T n k+ fk (223 Theorem 22 can be used to prove a number of Fibonacci identities Observe that the rows and columns of T are simply f 0 and f, so that T n+ has rows and columns f n and f n Thus, the simple observation that T n T n k T k gives the identity (27 See [4] Recently, Askey [, 2] and Huang [5] have given matrix theoretic proofs in this spirit of other Fibonacci identities as an interesting application of matrix multiplication 4

12 Our work applies this sort of argument in arbitrary dimension Rather than use matrix multiplication, we consider the dot product of vectors with components consisting of consecutive Fibonacci numbers This is a natural generalization since the entries in a matrix product can be viewed as a dot product of a row with a column We shall treat the Lucas vectors in the same manner 23 The Binet formulas We recall closed-form formulas for the Fibonacci and Lucas numbers known as the Binet Formulas These formulas are often found as an application of Theorem 22 in elementary linear algebra textbooks We recall these formulas and the common linear algebraic derivation now Lemma 23 The matrix T of Eqn (222 has characteristic polynomial x 2 x Thus it has eigenvalues α and β 5 2 The associated eigenvectors are respectively α, β Lemma 232 For all integers n, f n ln α n α n α β α α + β n β β n β, Proof Observe that f 0 0 l0 2 α 0 α 0 α β α α + β 0 β β 0 β, 5

13 Now, f n T n f0 T n α β α α β α n α T n β β n β by Eqn (223 by Lem 23 Theorem 233 For all integers n, F n αn β n α β, (234 L n α n + β n (235 Proof Equate the first components on each side of these equations of Lemma 232 Equations (234 and (235 are generally known as the Binet formulas for F n and L n (although previously known to Euler and Daniel Bernoulli [4] We shall make extensive use of the Binet formulas, so we present some formulas involving α and β and some alternate versions of the Binet formulas Lemma 234 αβ, (236 α + β, (237 α β 5, (238 α 2 + 5α, (239 β 2 + 5β, (2320 α 2 β 2, (232 β 2 α 2 (2322 6

14 Proof Immediate since α and β are the roots of x 2 x Lemma 235 [4] For all integers n, α n F n α + F n, (2323 β n F n β + F n (2324 Lemma 236 For all integers n and n 2, α n β n 2 + α n 2 β n ( n L n2 n, (2325 α n β n 2 α n 2 β n ( n + (α βf n2 n (2326 Proof Observe that α n β n 2 2 (αn β n 2 + α n β n 2 ( ( n β n 2 n + ( n 2 α n n 2 2 α n 2 β n 2 (αn 2 β n + α n 2 β n ( ( n 2 β n n 2 + ( n α n 2 n 2 by Eqn (236, by Eqn (236 Thus α n β n 2 + α n 2 β n ( ( n (α n 2 n + β n 2 n + ( n 2 (α n n 2 + β n n 2 2 ( n L n2 n + ( n 2 L n n 2 2 by Eqn (235 ( n L n2 n by Eqn (2 ( n 2 L n n 2 by Eqn (2 Similarly, α n β n 2 α n 2 β n 2 ( ( n ( α n 2 n + β n 2 n + ( n 2 (α n n 2 β n n 2 7

15 (α β ( ( n + F n2 n 2 + ( n 2 F n n 2 ( n+ (α βf n2 n by Eqn (20 by Eqn (234 ( n 2 (α βf n n 2 by Eqn (20 Corollary 237 For all integers n, ( n ( n α β + ( n L 2n, β α ( n ( n α β ( n (α βf 2n β α Proof Take n n and n 2 n in (2325 and (2326 Lemma 238 For all integers n, F n α n β n ( α 2n ( n α 2 ( β 2n ( n β 2 (2327, (2328 L n α2n + ( n α n (2329 β2n + ( n β n (2330 Proof Compute F n αn β n by Eqn (234 α β α2n α n β n α α 2 αβ α ( n α 2n ( n by Eqn (236, α n α 2 L n α n + β n 8

16 α2n + α n β n α n α2n + ( n α n Equations (2329 and (2330 are derived similarly 9

17 3 Fibonacci and Lucas vectors In this chapter we study Fibonacci and Lucas vectors in arbitrary dimension from a linear algebraic perspective 3 Linear algebraic set up Throughout this section m shall be a fixed positive integer Define an m m matrix T by T ( Lemma 3 The matrix T of Eqn (3 has characteristic polynomial x (m (x 2 x Thus the non-zero eigenvalues of T are α ( + 5/2 and β ( 5/2, each with geometric and algebraic multiplicity The eigenspaces associated with α 0

18 and β are spanned respectively by a α α 2 α m and b β β 2 β m Proof Elementary linear algebra Observe that the matrix T and vectors a and b depend upon the dimension m Since m will always be clear from context, we shall suppress this dependence in notation We shall frequently use the formula for the sum of a finite geometric series Lemma 32 For all real numbers c and all positive integers m, m j0 c j cm c (32 Lemma 33 { Fm (α βα m if m is even, a a L m α m if m is odd, { Fm (α ββ m if m is even, b b L m β m if m is odd, { 0 if m is even, a b if m is odd (33 (34 (35 Proof By definition of dot product a a m j0 m j0 α 2j ( j α by Eqn (236 β

19 ( α/βm ( α/β by Eqn (32 ( m α m β m β m α β α m (α m ( m β m by Lem 234 Suppose m is even Then ( α a a α m m β m (α β by Eqn (236 α β α m (α βf m by Eqn (234 Now suppose m is odd Then a a α m (α m + β m by Eqn (236 α m L m by Eqn (235 The computation of b b is similar, so we omit it By definition of dot product a b m j0 α j β j m ( j by Eqn (236 j0 { 0 if m is even, if m is odd Recall that for any vector v, the square of the length of v is v 2 v v 2

20 Corollary 34 a b a b { Fm (α βα m if m is even, Lm α m if m is odd, { Fm (α ββ m if m is even, Lm β m if m is odd, { Fm (α β if m is even, L m if m is odd, (36 (37 (38 32 Fibonacci and Lucas vectors Definition 32 For all positive integers m and for all integers n, define f n m F n F n+, l m n L n L n+ F n+m L n+m We refer to f n m and l m n as the n-th Fibonacci and Lucas vectors of length m, respectively The vectors f n and l n of Section 22 are just f n 2 and l 2 n in the present notation We shall carry the dimension m in the notation as we shall have occasion below to use more than one value of m in the same equation A number of other relations among the Fibonacci and Lucas numbers generalize to the corresponding vectors Observe that the f n m and l m n satisfy the vector recurrence relation x n+2 x n+ + x n The analog of Theorem 22 is the following Lemma 322 Let T be as in Eqn (3 Then for all integers n, f m n+ T f m n, 3

21 l m n+ T l m n Thus for all integers k n +, f m n+ T n k+ f m k, l m n+ T n k+ l m k Vector versions of the Binet formulas hold here Theorem 323 For all integers n, f n m (α n a β n b, α β (329 l m n α n a + β n b (320 In particular, f m n and l m n lie in the plane spanned by a and b Proof The j-th entry of f n m is F n+j (α n+j β n+j /(α β The j-th entry of (α n a β n b /(α β is (α n α j β n β j /(α β The first equation follows since both sides have the same entries A similar argument gives the second equation The analog of Lemma 235 is the following Lemma 324 For all integers n, α n a α f m n + f m n, β n b β f m n + f m n Proof Compute corresponding entries on each side of the equations and note that they are equal by Lemma Dot products We compute the dot products of Fibonacci and Lucas vectors 4

22 Theorem 33 For all positive integers m and for all integers n and n 2, f m n f m n 2 { Fm F n +n 2 +m if m is even, (L 5 ml n +n 2 +m ( n L n2 n if m is odd (33 Proof f n m f n m 2 (α n a β n b (α n 2 a β n 2 b by Eqn (329 α β α β ( α n +n 2 a a + β n +n 2 b b (α n β n 2 + α n 2 β n a b 5 Suppose m is even Then f n m f n m 2 F m ( α n +n 2 +m β n +n 2 +m by Lem 33 α β F m F n +n 2 +m by Eqn (234 Now suppose m is odd Then f n m f n m 2 ( Lm (α n +n 2 +m + β n +n 2 +m (α n β n 2 + α n 2 β n 5 5 (L ml n +n 2 +m ( n L n2 n by Eqns (234, (2325 Corollary 332 For all positive integers m and for all integers n and n 2, f m n { 2 Fm F 2n+m if m is even, (L 5 ml 2n+m 2( n if m is odd (332 Proof Observe that f m n 2 f m n f m n 5

23 Theorem 333 For all positive integers m and for all integers n and n 2, { l m n l 5Fm F m n +n 2 +m if m is even, n 2 L m L n +n 2 +m + ( n L n2 n if m is odd (333 Proof l m n l m n 2 (α n a + β n b (α n 2 a + β n 2 b by Eqn (329 ( α n +n 2 a a + β n +n 2 b b + (α n β n 2 + α n 2 β n a b Suppose m is even Then l m n l m n 2 F m (α β ( α n +n 2 +m β n +n 2 +m by Lem 33 5F m ( α n +n 2 +m β n +n 2 +m /(α β by Eqn (238 5F m F n +n 2 +m by Eqn (234 Now suppose m is odd Then l m n l m n 2 L m (α n +n 2 +m + β n +n 2 +m + (α n β n 2 + α n 2 β n by Lem 33 L m L n +n 2 +m + ( n L n2 n by Eqn (235 Corollary 334 For all positive integers m and for all integers n, { l m n 2 5Fm F 2n+m if m is even, L m L 2n+m + 2( n if m is odd (334 Corollary 335 For all positive integers m and for all integers n and n 2, l m n l m n 2 5 f m n f m n 2 (335 6

24 Proof Clear from (33 and (333 Theorem 336 For all positive integers m and for all integers n and n 2, { f n m l 5Fm L m n +n 2 +m if m is even, n 2 L m F n +n 2 +m + ( n+ F n2 n if m is odd (336 Proof f m n l m n 2 (α n a β n b (α n 2 a + β n 2 b by Thm 323 α β ( α n +n 2 a a β n +n 2 b b + (α n β n 2 α n 2 β n a b α β Suppose m is even Then f m n l m n 2 F m (α β ( α n +n 2 +m + β n +n 2 +m by Lem 33 5F m L n +n 2 +m by Eqns (235 and (238 Now suppose m is odd Then f n m l m n 2 α β L ( m α n +n 2 +m β n +n 2 +m + (α n β n 2 α n 2 β n by Thm 323 L m F n +n 2 +m + ( n+ F n2 n by Eqn ( Fibonacci and Lucas identities In this section we state a number of identities involving the Fibonacci and Lucas numbers which follow from the dot product formulas of the previous section Theorem 34 For all positive integers m and for all integers n and n 2, m j0 F n +jf n2 +j { Fm F n +n 2 +m if m is even, (L 5 ml n +n 2 +m ( n L n2 n if m is odd (347 7

25 Proof Observe that both sides are equal to f m n f m n 2 by Theorem 33 and the definition of dot product From this basic identity a number of other identities can be derived We state two of the simplest consequences now Corollary 342 For all positive integers m and for all integers n, { m Fm F 2n+m if m is even, Fn+j 2 (L 5 ml 2n+m 2( n if m is odd j0 Proof Take n n 2 in Eqn (347 Corollary 343 For all positive integers m and for all integers k and n, { Fm F 2n k+m 2 if m is even, F n+m 2 F n k+m + F n F n k ( Lm L 5 2n k+m 2( n k L k if m is odd Proof By the definition of dot product, the left-hand side is f n m f n k m f n m 2 m 2 f n k+ For m even, Theorem 33 gives f m n f m n k f m 2 n f m 2 n k+ F m F 2n k+m 2 F m 2 F 2n k+m 2 (F m F m 2 F 2n k+m 2 F m F 2n k+m 2 For m odd, Theorem 33 gives that f m n f m n k f m 2 n f m 2 n k+ equals 5 ( Lm L n +n k+m ( n L n k n+ 5 5 (L m 2L n+n k++m 2 ( n L n k+ n ( Lm L 2n k+m 2 ( n L k+ 5 (L m 2L 2n k+m 2 ( n L k+ 5 L ml 2n k+m 2 5 L m 2L 2n k+m 2 5 ( n L k+ + 5 ( n L k+ 8

26 5 L 2n k+m 2 (L m L m 2 5 L k+2( n 5 L m L 2n k+m 2 2( n L k+ The idea of the previous corollary can be used to derive a number of other more complicated identities involving more summands of the same sort Theorem 344 For all integers n and n 2 and for all positive integers m, { m 5Fm F n +n 2 +m if m is even, L n +jl n2 +j L m L n +n 2 +m + ( n L n2 n if m is odd j0 (348 Proof Observe that both sides are equal to f m n f m n 2 by Theorem 333 and the definition of dot product Corollary 345 For all positive integers m and for all integers n, { m 5Fm F 2n+m if m is even, L 2 n+j L m L 2n+m + 2( n if m is odd j0 Proof Take n n 2 in Eqn (348 Corollary 346 For all positive integers m and for all integers k and n, { 5Lm L 2n k+m 2 if m is even, L n+m 2 L n k+m + L n L n k L m L 2n k+m 2 + 2( n L k+ if m is odd Proof By the definition of dot product, the left-hand side is l m n l m n k l m 2 n l m 2 n k+ For m is even, Theorem 333 gives l m n l m n k l m 2 n l m 2 n k+ 5F m F 2n k+m 2 5F m 2 F 2n k+m 2 9

27 5(F m F m 2 F 2n k+m 2 5F m F 2n k+m 2 For m odd, Theorem 333 gives that l m n l m n k l m 2 n ( Lm L n +n k+m + ( n L n k n+ l m 2 n k+ equals (L m 2 L n+n k++m 2 + ( n L n k+ n ( L m L 2n k+m 2 + ( n L k+ (Lm 2 L 2n k+m 2 + ( n L k+ L m L 2n k+m 2 L m 2 L 2n k+m 2 + ( n L k+ ( n L k+ L 2n k+m 2 (L m L m 2 + 2( n L k+ L m L 2n k+m 2 + 2( n L k+ Theorem 347 For all integers n and n 2 and for all positive integers m, { m 5Fm L n +n 2 +m if m is even, F n +jl n2 +j L m F n +n 2 +m + ( n+ F n2 n if m is odd j0 (349 Proof Observe that both sides are equal to f m n l m n 2 by Theorem 336 and the definition of dot product Corollary 348 For all positive integers m and for all integers n, { m 5Fm L 2n+m if m is even, F n+j L n+j L m F 2n+m + ( n+ if m is odd j0 Proof Take n n 2 in Eqn (349 20

28 Corollary 349 For all integers k, n and for all postive integers m, { 5Fm L 2n k+m 2 if m is even F n+m 2 L n k+m + F n L n k L m F 2n k+m 2 + 2( n F k+ if m is odd Proof By the definition of dot product, the left-hand side is f n l m m n k f n m 2 l m 2 n k+ For m even, Theorem 336 gives f m n l m n k f m 2 n l m 2 n k+ 5F m L 2n k+m 2 5F m 2 L 2n k+m 2 5(F m F m 2 L 2n k+m 2 5F m L 2n k+m 2 For m odd, Theorem 336 gives that f m n l m n k f m 2 n ( Lm F n +n k+m + ( n + F n k n+ ( L m 2 F n+n k++m 2 + ( n+ F n k+ n l m 2 n k+ equals (L m F 2n k+m 2 + ( n F k+ ( L m 2 F 2n k+m 2 + ( n+ F k+ L m F 2n k+m 2 L m 2 F 2n k+m 2 + ( n F k+ ( n+ F k+ F 2n k+m 2 (L m L m 2 + 2( n F k+ L m F 2n k+m 2 + 2( n F k+ 2

29 4 Angles in even dimension In this chapter we study the angles between the various vectors which we studied so far We shall restrict our attention to the case of even dimension as the formulas become much more involved in odd dimension We shall comment on the odd dimensional case in Chapter 5 Recall that the angle θ between arbitrary vectors v and v 2 satisfies cos θ v v 2 v v 2 4 Dot products with a and b In this section, we compute the dot products of the vectors a and b with the Fibonacci and Lucas vectors Lemma 4 For all integers n and for all positive even integers m, f m n a F m α n+m, (4 f m n b F m β n+m (42 Proof f n m a (α n a β n b a by Thm 323 α β (α n a a β n b a α β α β αn (F m (α βα m by Lem 33 F m α n+m 22

30 Lemma 42 For all integers n and for all positive even integers m, l m n a F m (α βα n+m, (43 l m n b F m (α ββ n+m (44 Proof Compute l m n a (α n a + β n b a by Thm 323 α n a a + β n b a α n F m (α βα m by Lem 33 F m (α βα n+m Equation (44 is proved similarly 42 Cosines of angles with a and b In light of Theorem 323, it is interesting to consider the angles of the Fibonacci and Lucas vectors with a and b Recall that all the vectors of interest lie in the same plane Lemma 42 For all integers k and all positive integers m, f m 2k and l m 2k+ opposite side of a as b, and f m 2k+ and l m 2k are on the same side of a as b are on the Proof Observe that β < 0, so β 2k is positive and β 2k is negative The result follows from Theorem 323 Lemma 422 Let φ n,m and χ n,m denote the respective angles between f m n and a and 23

31 between f m n and b Then cos φ n,m (α β /2 α 2n+m, (425 F 2n+m cos χ n,m (α β /2 β 2n+m (426 F 2n+m Proof Compute cos φ n,m Also cos χ n,m f m n a f m n a F m α n+m Fm F 2n+m Fm (α βα m by Lem 4, Cors 34, 332 (α β /2 α 2n+m F 2n+m f m n b f m n b F m α n+m Fm F 2n+m Fm (α ββ m by Lem 4, Cors 34, 332 (α β /2 β 2n+m F 2n+m Lemma 423 Let λ n,m and µ n,m denote the respective angles between l m n between l m n and b Then and a and cos λ n,m (α β /2 α 2n+m, F 2n+m (427 cos µ n,m (α β /2 β 2n+m F 2n+m (428 24

32 Proof Compute cos λ n,m l n m a l n m a F m (α βα n+m 5Fm F 2n+m Fm (α βα m by Lem 4, Cors 34, 332 (α β /2 α 2n+m F 2n+m Equation (428 is proved similarly Corollary 424 For all integers n and for all even integers m, φ n,m > φ n+,m Proof Observe that φ n,m is accute since the coefficient of a in the expression of fn m in (329 is positive To show that φ n,m > φ n+,m, it suffices to show that cos φ n,2k < cos φ n+,m Note that α j /F j > α j+ /F j+, so the result follows from Equation (425 Corollary 425 For all integers n and for all positive integers t, φ n,2t φ n+t,2 λ n+t λ n,2t In particular, the sequences of angles {φ j,2t } j, { λ j,2t } j, and {φ t+j,2 } j are equal Proof Immediate from (425 and (427 Since the sequences of angles are simply tails of those in dimension two, we shall take another look at the Fibonacci vectors of length 2 in the Section Angles between vectors In this section we study the angles between Fibonacci and Lucas vectors We fix m to be a positive even integer 25

33 Lemma 43 For all integers n and n 2, let ζ n,n 2,m denote the angle between f m n and f m n 2 Then cos ζ n,n 2,m F n +n 2 +m F2n +m F 2n2 +m Proof Follows directly from Theorem 33 and Corollary 332 Lemma 432 For all integers n and n 2, let η n,n 2,m denote the angle between l m n and l m n 2 Then cos η n,n 2,m F n +n 2 +m F2n +m F 2n2 +m Proof Follows directly from Theorem 333 and Corollary 334 Lemma 433 For all integers n and n 2, let θ n,n 2,m denote the angle between f m n and l m n 2 Then cos θ n,n 2,m 5L n +n 2+m F2n +m F 2n2 +m Proof Follows directly from Theorem 336 and Corollaries 332 and 334 The various angles which we have studied are not independent Observe that the angle between fn m and fn m 2 can be computed by either adding or subtracting their respective angles with a If n and n 2 differ by an even number they are on the same side of a so we subtract, and if n and n 2 differ by an odd number they are on opposite sides of a so we add Thus for n > n 2, ζ n,n 2,m φ n,m + ( n n 2 φ n2,m Similarly, η n,n 2,m λ n,m + ( n n 2 + λ n2,m, θ n,n 2,m φ n,m + ( n n 2 λ n2,m 26

34 44 Angles with the axes Let x i denote the unit vector in the direction of the i-th coordinate axis and let m be a positive even integer Observe that f n m x i F n+i and l m n x i L n+i Lemma 44 For all integers n, let ω m,n,i be the angle between f n m and the i-th standard unit vector then cos ω m,n,i F n+i Fm F 2n+m Lemma 442 For all integers n, let ι m,n,i be the angle between l m n and the i-th standard unit vector then cos ι m,n,i L n+i 5Fm F 2n+m 45 Dimension two In this section we discuss the Fibonacci and Lucas vectors of length two This is directly applicable to all even dimensional cases as discussed in Section 42 We begin with a result of Lucas which gives a nice geometric condition on the Fibonacci vectors of length 2 Theorem 45 (Lucas The endpoints of f n 2 lie on two hyperbolas given by the equation y 2 yx x 2 ± In Chapter 6 we will consider the limit of the sequence of angles of the Fibonacci and Lucas vectors with a In dimension 2 we can also easily consider the angles with the axes and then deduce the angles with a Lemma 452 [4] For all integers k, F n+k lim α k (459 n F n 27

35 Lemma 453 For all integers n, lim cos ω 2,n,x n lim cos ω 2,n,y n lim cos ι 2,n,x n lim cos ι 2,n,y n + α, 2 α + α, 2 + α, 2 α + α 2 Proof By Theorem 44 cos ω 2,n,x F n F 2 n + Fn+ 2 + (Fn+ /F n 2 Thus by Lemma 452 lim cos ω 2,n,x /( + α 2 n The remaining limits are proven similarly It follows from the previous lemma that the limiting unit vectors for the directions of f 2 n and l 2 n are both + α 2 α + α 2 a In otherwords the unit directions approach that of a We shall see that this is the case in all dimensions We note that in dimension two a and b form an orthogonal basis for the whole vector space From Theorem 323, we see that the transformation matrix from a, b to f 2 n and f 2 n 2 is αn β n α n 2 β n 2 28

36 Since this matrix is essentially Vandermonde it is invertible In other words, any two disinct Fibonacci vectors form a basis for R 2 Similarly for Lucas vectors 29

37 5 Comments on angles in odd dimension Although we do not carry out the computations of the cosines of angles in odd dimension because the results are not so nice, some computations do turn out nicely We present them in this Chapter 5 The angle between a and b Recall that a and b form a basis for the plane containing all of the Fibonacci and Lucas vectors When m is even, a and b are orthogonal When m is odd we have the following Theorem 5 Assume that m is odd Then the cosine of the angle between a and b is /Lm Proof Let τ denote the angle between a and b Then cos τ a b a b L m by Cor 34 Lemma 52 Suppose m is odd Then the vector a + (L m α m b is orthogonal to a and on the same side of a as b 30

38 Proof Compute a ( a + (L m α m b a a + (L m α m a b L m α m + L m α m by Lem Three-dimensional Fibonacci Vectors In this section we discuss the geometric interpretations of these results in 3 dimensions This case is nicer than the general odd dimensional case Example 52 With the notation of Lemma 44, lim cos ω 3,n,x /(2α, n lim cos ω 3,n,y /2, n lim cos ω 3,n,z α/2 n So, the limiting unit vector is,, α 2α 2 2, α, 2α α2 Thus, geometrically the limiting line is given by r(t t(, α, α 2 or x t, y αt, and z α 2 t Now consider the determinant of the matrix f n F n F n+ F n+2 f m F m F m+ F m+2 fr F r F r+ F r+2 The latter determinant equals zero since the last column is the sum of the first two columns Therefore, the vectors are dependent (coplanar We assume without loss of generality n < m < r, then f r ( m+n ( F r m F m n f n + ( F r n F m n f m Taking f n f and f m f 2 we write f r F r 2 f + F r f2 Thus, all vectors f n fall in the plane determined by f and f 2 This plane has normal vector n,, and equation 3

39 z x + y Each vector f n / f n lies in the intersection of this plane with the unit sphere x 2 + y 2 + z 2 this intersection is a great circle of this sphere 32

40 6 Asymptotics of angles We consider the ultimate behavior of the angles of the Fibonacci and Lucas vectors with a and b 6 Angles with a Lemma 6 With reference to Lemma 422, for all even integers m, lim cos φ n,m n That is to say the direction of the Fibonacci vectors approaches that of a Proof By Lemma 425, cos φ n,m (α β /2 α 2n+m /F 2n+m Thus we consider F k /α k By Equation (234, F k /α k (α β (α k β k /α k (α β ( (β/α k Observe that β/α < and α β Thus lim F k/α k k α β It follows that lim cos φ n,m n Lemma 62 With reference to Lemma 423, for all even integers m, lim cos λ n,m n 33

41 That is to say the direction of the Lucas vectors approaches that of a Proof Similar to that of Lemma 6 62 Dominant eigenvalues In the last section we saw that in even dimension, the sequences of the Fibonacci and Lucas vectors approach in direction that of the vector a We now give another proof of this fact which does not depend upon the parity of the dimension We use a variation of the well-known power method to do so Definition 62 Suppose λ is an eigenvalue of a square matrix A that is larger in absolute value than any other eigenvalue of A Then λ is called the dominant eigenvalue of A An eigenvector corresponding to λ is called a dominant eigenvector of A Dominant eigenvalues play an important role in the study of matrices The power method provides a means of finding the dominant eigenvalue and eigenvector We state one of the more general forms of the power method Theorem 622 Assume that the m m complex matrix A has a dominant eigenvalue λ and a unique dominant eigenvector v up to scalar multiples Then the sequence A x/ A x, A 2 x/ A 2 x, A 3 x/ A 3 x,, converges to v/ v for any intitial vector x except for a set of measure zero In the literature one finds various criteria on the initial vector for the convergence of the above sequence to a dominant eigenvector Clearly, any vector in the sum of eigenspaces other than that of the dominant eigenvalue cannot converge a dominant eigenvalue For a diagonalizable matrix, this is almost a sufficient condition Theorem 623 Assume that the m m complex matrix A is diagonalizable and has a dominant eigenvalue λ and a unique dominant eigenvector v up to scalar multiples Let v v, v 2,, v m denote a basis of eigenvectors for the complex vector space of 34

42 columns vectors with complex entries and length m Let x c v + c 2 v c m v m with c 0 Then the sequence A x/ A x, A 2 x/ A 2 x, A 3 x/ A 3 x, converges to v/ v Proof Compute A x A(c v + c 2 v c m v m c A v + c 2 A v c m A v m c λ v + c 2 λ 2 v c m λ m v m, where λ i is the eigenvalue associated with v i for all i By repeated multiplication by A, A k x c λ k v + c 2 λ k 2 v c m λ k m v m Thus A k x λ k [ c v + c 2 ( λ2 λ k ( ] k λr v c r v r λ But λ is a dominant eigenvalue, so it is larger in absolute value than all other eigenvalues, so that λ i /λ < for i 2, 3,, r Thus each fraction (λ i /λ k approaches 0 as k goes to infinity Let us modify the previous theorem slightly to make it more applicable to our situation Theorem 624 Assume that the m m complex matrix A as a dominant eigenvalue λ and dominant eigenvector v Let v v, v 2,, v r denote a maximal set of linearly independent nonzero eigenvectors and let v r+, v r+2,, v s denote a basis of generalized eigenvectors for the generalized eigenspace associated with zero Let x c v + c 2 v c r v r with c 0 Then the sequence A x/ A x, A 2 x/ A 2 x, A 3 x/ A 3 x, converges to v/ v 35

43 Proof Observe that for some j, A j v i is an eigenvector of A for all i ( i s Now apply Theorem 623 with A restricted to the eigenspaces and y A j x in place of x Now the result follows since A k y A k+j x Observe that the matrix T of Eqn (3 has eigenvalues 0, α, β with respective algebraic multiplicites m 2,, and respective geometric multiplicites,, Thus the complex vector space of complex vectors of length m has a basis consisting of nonzero eigenvectors and generalized eigenvectors associated with 0 A basis of the generalized eigenspace associated with 0 has a basis of generalized eigenvectors consisting of e, e 0 2,, e m Thus the column vector space has a basis a, b, e, e 2,, e m 2 This gives us the following Corollary 625 Let T be as in Eqn (3 Let x c a a + c v b + c e + c m 2 e m 2 with c α 0 Then the sequence T x/ T x, T 2 x/ T 2 x, T 3 x/ T 3 x, converges to a/ a Corollary 626 The sequences of directions of the Fibonacci and Lucas vectors approach that of the vector a Proof Immediate from Lemma 322 and Corollary Further directions We have begun to consider a number of variations on the work presented in this thesis 36

44 Consider the reverse Fibonacci and Lucas vectors r n m F n+m F n+m 2, t m n L n+m L n+m 2 F n L n Identical results hold for these vectors as did for f m n, l m n However, a number of very interesting identities arise when we consider f m n r m n, f m n t m n, l m n r m n, and l m n t m n Consider the Fibonacci and Lucas vectors with a step of p: f m,p n F n F n+p F n+(m p, l m,p n L n L n+p L n+(m p Our computations still work out since this constant step gives rise to geometric series in α and β as before Again, interesting identities arise from the computations of dot products Different steps can be mixed, as well as reverse vectors considered To some extent our computations can be carried out for any second order recurrence x n+2 cx n + dx n+ However, when c (so αβ many nice properties vanish A number of other problems suggest themselves Consider the Kronecker products of the Fibonacci vectors We expect that such considerations will give rise to identities involving sums of longer products Reduce Fibonacci and Lucas vectors mod k for any modulus k These vectors cycle through a finite collection rather than approach a given line There are 37

45 many deep open problems concerning the Fibonacci numbers modulo k Perhaps we may gain some insight from a consideration of these vectors 38

46 References [] RA Askey, Fibonacci and related sequences, Mathematics Teacher 97 (2004, 6 9 [2] RA Askey, Fibonacci and Lucas numbers, Mathematics Teacher 98 (2005, [3] VE Hoggatt, Fibonacci and Lucas Numbers, Houghton Mifflin Company, New York, NY, 969 [4] R Honsberger, Mathematical Gems III, Math Assoc America, Washington, DC, 985 [5] D Huang, Fibonacci identities matrices, and graphs, Mathematics Teacher 98 (2005, [6] VJ Katz, A History Of Mathematics, An Introduction, 2nd ed, Addison- Wesley, Reading, MA, 998 [7] RE Larson and BH Edwards, Elementary Linear Algebra, DC Heath and Company, Lexington, MA, 969 [8] L Sadum, Applied Linear Algebra: the decoupling principle, Prentice Hall, Upper Saddle River, NJ, 200 [9] G Strang, Introduction to Linear Algebra, 2nd ed, Wellesley-Cambridge Press, Wellesley, MA,

47 About the Author Growing up I would have never dreamed that I would write a thesis in mathematics After all, the one subject I feared the most was mathematics Then one day it all changed while I was in college at Georgia Southern University It was there that I met people who were willing to take the time out to explain the many things that confused me and it was there that I decided I wanted to become a math professor For many years my fear of mathematics made me shy away from learning about several other subjects It is my goal to make sure that no student of mine will ever let the subject of mathematics get in the way of the dreams they wish to pursue I hope that in my lifetime I can be half the teacher that my professors were to me