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8/7/2019 MIT Slides on Diagonalisatiojn
1/124
Introduction
The Three Proofs
Unifying the Proofs
Conclusion
The Fibonacci numbersProofs of Binets formula and other mysteries
Qiaochu Yuan
Department of MathematicsMassachusetts Institute of Technology
February 21, 2009 / HMMT
Qiaochu Yuan The Fibonacci numbers
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Introduction
The Three Proofs
Unifying the Proofs
Conclusion
Outline
1 Introduction
Whats Going On?
Background
2 The Three ProofsProof 1: Linear Independence
Proof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
3 Unifying the ProofsThe shift operator
4 Conclusion
Qiaochu Yuan The Fibonacci numbers
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Introduction
The Three Proofs
Unifying the Proofs
Conclusion
Whats Going On?
Background
What Were Going To Do
Definition: The Fibonacci numbers satisfy
F0 = 0, F1 = 1, Fn+2 = Fn+1 + Fn
Theorem (Binets formula): Fn =1
5
1+5
2
n
152
n
Were going to run through three proofs,
Were going to learn why theyre really the same,
We might even do other cool stuff!
Qiaochu Yuan The Fibonacci numbers
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Introduction
The Three Proofs
Unifying the Proofs
Conclusion
Whats Going On?
Background
What Were Going To Do
Definition: The Fibonacci numbers satisfy
F0 = 0, F1 = 1, Fn+2 = Fn+1 + Fn
Theorem (Binets formula): Fn =1
5
1+5
2
n
152
n
Were going to run through three proofs,
Were going to learn why theyre really the same,
We might even do other cool stuff!
Qiaochu Yuan The Fibonacci numbers
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5/124
Introduction
The Three Proofs
Unifying the Proofs
Conclusion
Whats Going On?
Background
What Were Going To Do
Definition: The Fibonacci numbers satisfy
F0 = 0, F1 = 1, Fn+2 = Fn+1 + Fn
Theorem (Binets formula): Fn =1
5
1+5
2
n
152
n
Were going to run through three proofs,
Were going to learn why theyre really the same,
We might even do other cool stuff!
Qiaochu Yuan The Fibonacci numbers
I d i
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Introduction
The Three Proofs
Unifying the Proofs
Conclusion
Whats Going On?
Background
What Were Going To Do
Definition: The Fibonacci numbers satisfy
F0 = 0, F1 = 1, Fn+2 = Fn+1 + Fn
Theorem (Binets formula): Fn =1
5
1+5
2
n
152
n
Were going to run through three proofs,
Were going to learn why theyre really the same,
We might even do other cool stuff!
Qiaochu Yuan The Fibonacci numbers
I t d ti
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Introduction
The Three Proofs
Unifying the Proofs
Conclusion
Whats Going On?
Background
Linear Algebra Is Important!
Definition: A (complex) vector space V is a set with:
Vector addition v + u,Scalar multiplication c v, c C,
A zero vector v + 0 = v.
Examples: Cn,C[x], the solutions to y + y = 0
Qiaochu Yuan The Fibonacci numbers
Introduction
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Introduction
The Three Proofs
Unifying the Proofs
Conclusion
Whats Going On?
Background
Linear Algebra Is Important!
Definition: A (complex) vector space V is a set with:
Vector addition v + u,Scalar multiplication c v, c C,
A zero vector v + 0 = v.
Examples: Cn,C[x], the solutions to y + y = 0
Qiaochu Yuan The Fibonacci numbers
Introduction
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Introduction
The Three Proofs
Unifying the Proofs
Conclusion
Whats Going On?
Background
Linear Algebra Is Important!
Definition: A (complex) vector space V is a set with:
Vector addition v + u,Scalar multiplication c v, c C,
A zero vector v + 0 = v.
Examples: Cn,C[x], the solutions to y + y = 0
Qiaochu Yuan The Fibonacci numbers
Introduction
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Introduction
The Three Proofs
Unifying the Proofs
Conclusion
Whats Going On?
Background
Linear Algebra Is Important!
Definition: A (complex) vector space V is a set with:
Vector addition v + u,Scalar multiplication c v, c C,
A zero vector v + 0 = v.
Examples: Cn,C[x], the solutions to y + y = 0
Qiaochu Yuan The Fibonacci numbers
Introduction
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Introduction
The Three Proofs
Unifying the Proofs
Conclusion
Whats Going On?
Background
Linear Algebra Is Important!
Definition: A (complex) vector space V is a set with:
Vector addition v + u,Scalar multiplication c v, c C,
A zero vector v + 0 = v.
Examples: Cn,C[x], the solutions to y + y = 0
Qiaochu Yuan The Fibonacci numbers
Introduction
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Introduction
The Three Proofs
Unifying the Proofs
Conclusion
Whats Going On?
Background
Linear Algebra Is Important!
Definition: The dimension d of V is the smallest number of
vectors v1,...vd such that:
Every v is of the form c1v1 + ... + cdvd,
This representation is unique,
c1v1 + ... + cdvd = 0 if and only if c1 = ... = cd = 0 (linearindependence).
We call (vi) a basisof V and say that it spans V.
Examples:
dimCn = n, dimC[x] = , dim{y|y + y = 0} = 2
Qiaochu Yuan The Fibonacci numbers
Introduction
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The Three Proofs
Unifying the Proofs
Conclusion
Whats Going On?
Background
Linear Algebra Is Important!
Definition: The dimension d of V is the smallest number of
vectors v1,...vd such that:
Every v is of the form c1v1 + ... + cdvd,
This representation is unique,
c1v1 + ... + cdvd = 0 if and only if c1 = ... = cd = 0 (linearindependence).
We call (vi) a basisof V and say that it spans V.
Examples:
dimCn = n, dimC[x] = , dim{y|y + y = 0} = 2
Qiaochu Yuan The Fibonacci numbers
Introduction
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The Three Proofs
Unifying the Proofs
Conclusion
Whats Going On?
Background
Linear Algebra Is Important!
Definition: The dimension d of V is the smallest number of
vectors v1,...vd such that:
Every v is of the form c1v1 + ... + cdvd,
This representation is unique,
c1v1 + ... + cdvd = 0 if and only if c1 = ... = cd = 0 (linearindependence).
We call (vi) a basisof V and say that it spans V.
Examples:
dimCn = n, dimC[x] = , dim{y|y + y = 0} = 2
Qiaochu Yuan The Fibonacci numbers
Introduction
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The Three Proofs
Unifying the Proofs
Conclusion
Whats Going On?
Background
Linear Algebra Is Important!
Definition: The dimension d of V is the smallest number of
vectors v1,...vd such that:
Every v is of the form c1v1 + ... + cdvd,
This representation is unique,
c1v1 + ... + cdvd = 0 if and only if c1 = ... = cd = 0 (linearindependence).
We call (vi) a basisof V and say that it spans V.
Examples:
dimCn = n, dimC[x] = , dim{y|y + y = 0} = 2
Qiaochu Yuan The Fibonacci numbers
Introduction
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The Three Proofs
Unifying the Proofs
Conclusion
Whats Going On?
Background
Linear Algebra Is Important!
Definition: The dimension d of V is the smallest number of
vectors v1,...vd such that:
Every v is of the form c1v1 + ... + cdvd,
This representation is unique,
c1v1 + ... + cdvd = 0 if and only if c1 = ... = cd = 0 (linearindependence).
We call (vi) a basisof V and say that it spans V.
Examples:
dimCn = n, dimC[x] = , dim{y|y + y = 0} = 2
Qiaochu Yuan The Fibonacci numbers
Introduction
Th Th P f Wh G i O ?
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The Three Proofs
Unifying the Proofs
Conclusion
Whats Going On?
Background
Linear Algebra Is Important!
Definition: The dimension d of V is the smallest number of
vectors v1,...vd such that:
Every v is of the form c1v1 + ... + cdvd,
This representation is unique,
c1v1 + ... + cdvd = 0 if and only if c1 = ... = cd = 0 (linearindependence).
We call (vi) a basisof V and say that it spans V.
Examples:
dimCn = n, dimC[x] = , dim{y|y + y = 0} = 2
Qiaochu Yuan The Fibonacci numbers
Introduction
Th Th P f Wh t G i O ?
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The Three Proofs
Unifying the Proofs
Conclusion
Whats Going On?
Background
Linear Algebra Is Important!
Definition: A linear transformation T : V1 V2 is a function
such that:T(v1 + v2) = T(v1) + T(v2),
T(cv1) = cT(v1).
Examples: n n matrices, T(P(x)) = xP(x), ddx
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three Proofs Whats Going On?
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The Three Proofs
Unifying the Proofs
Conclusion
What s Going On?
Background
Linear Algebra Is Important!
Definition: A linear transformation T : V1 V2 is a function
such that:T(v1 + v2) = T(v1) + T(v2),
T(cv1) = cT(v1).
Examples: n n matrices, T(P(x)) = xP(x), ddx
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three Proofs Whats Going On?
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The Three Proofs
Unifying the Proofs
Conclusion
What s Going On?
Background
Linear Algebra Is Important!
Definition: A linear transformation T : V1 V2 is a function
such that:T(v1 + v2) = T(v1) + T(v2),
T(cv1) = cT(v1).
Examples: n n matrices, T(P(x)) = xP(x), ddx
Qiaochu Yuan The Fibonacci numbers
IntroductionThe Three Proofs Whats Going On?
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The Three Proofs
Unifying the Proofs
Conclusion
What s Going On?
Background
Linear Algebra Is Important!
Definition: A linear transformation T : V1 V2 is a function
such that:T(v1 + v2) = T(v1) + T(v2),
T(cv1) = cT(v1).
Examples: n n matrices, T(P(x)) = xP(x), ddx
Qiaochu Yuan The Fibonacci numbers
IntroductionThe Three Proofs Whats Going On?
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The Three Proofs
Unifying the Proofs
Conclusion
What s Going On?
Background
Linear Algebra Is Important!
Definition: An eigenvectorof T : V V is a v such thatT(v) = v.
is called an eigenvalue. If V has dimension n, T has at
most n eigenvalues.
Most T are characterized by their eigenvectors and
eigenvalues.
Qiaochu Yuan The Fibonacci numbers
IntroductionThe Three Proofs Whats Going On?
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The Three Proofs
Unifying the Proofs
Conclusion
What s Going On?
Background
Linear Algebra Is Important!
Definition: An eigenvectorof T : V V is a v such thatT(v) = v.
is called an eigenvalue. If V has dimension n, T has at
most n eigenvalues.
Most T are characterized by their eigenvectors and
eigenvalues.
Qiaochu Yuan The Fibonacci numbers
IntroductionThe Three Proofs Whats Going On?
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Unifying the Proofs
Conclusion
g
Background
Linear Algebra Is Important!
Definition: An eigenvectorof T : V V is a v such thatT(v) = v.
is called an eigenvalue. If V has dimension n, T has at
most n eigenvalues.
Most T are characterized by their eigenvectors and
eigenvalues.
Qiaochu Yuan The Fibonacci numbers
IntroductionThe Three Proofs
Proof 1: Linear Independence
Proof 2: Partial Fraction Decomposition
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Unifying the Proofs
Conclusion
Proof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
Motivation: Geometric Series
What are the solutions to sn+1 = asn?
sn = ans0! (Induction)
"First-order" version of what we want to look at.
Identity to keep in mind: an+1s0 = a(an)s0
Qiaochu Yuan The Fibonacci numbers
IntroductionThe Three Proofs
Proof 1: Linear Independence
Proof 2: Partial Fraction Decomposition
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Unifying the Proofs
Conclusion
Proof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
Motivation: Geometric Series
What are the solutions to sn+1 = asn?
sn = ans0! (Induction)
"First-order" version of what we want to look at.
Identity to keep in mind: an+1s0 = a(an)s0
Qiaochu Yuan The Fibonacci numbers
IntroductionThe Three Proofs
Proof 1: Linear Independence
Proof 2: Partial Fraction Decomposition
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Unifying the Proofs
Conclusion
Proof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
Motivation: Geometric Series
What are the solutions to sn+1 = asn?
sn = ans0! (Induction)
"First-order" version of what we want to look at.
Identity to keep in mind: an+1s0 = a(an)s0
Qiaochu Yuan The Fibonacci numbers
IntroductionThe Three Proofs
U if i h P f
Proof 1: Linear Independence
Proof 2: Partial Fraction Decomposition
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Unifying the Proofs
Conclusion
Proof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
Motivation: Geometric Series
What are the solutions to sn+1 = asn?
sn = ans0! (Induction)
"First-order" version of what we want to look at.
Identity to keep in mind: an+1s0 = a(an)s0
Qiaochu Yuan The Fibonacci numbers
IntroductionThe Three Proofs
U if i th P f
Proof 1: Linear Independence
Proof 2: Partial Fraction Decomposition
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Unifying the Proofs
Conclusion
oo a t a act o eco pos t o
Proof 3: Diagonalization
Fibonacci-type Sequences
Definition: A sequence sn is of Fibonacci typeif
sn+2 = sn+1 + sn.Proposition: The set S of Fibonacci-type sequences forms a
vector space.
Proof.
(sn+2 + tn+2) = (sn+1 + tn+1) + (sn + tn),
csn+2 = csn+1 + csn,
0 = 0 + 0 (zero vectors are important!)
Qiaochu Yuan The Fibonacci numbers
IntroductionThe Three Proofs
Unifying the Proofs
Proof 1: Linear Independence
Proof 2: Partial Fraction Decomposition
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Unifying the Proofs
Conclusion
p
Proof 3: Diagonalization
Fibonacci-type Sequences
Definition: A sequence sn is of Fibonacci typeif
sn+2 = sn+1 + sn.Proposition: The set S of Fibonacci-type sequences forms a
vector space.
Proof.
(sn+2 + tn+2) = (sn+1 + tn+1) + (sn + tn),
csn+2 = csn+1 + csn,
0 = 0 + 0 (zero vectors are important!)
Qiaochu Yuan The Fibonacci numbers
IntroductionThe Three Proofs
Unifying the Proofs
Proof 1: Linear Independence
Proof 2: Partial Fraction Decomposition
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Unifying the Proofs
ConclusionProof 3: Diagonalization
Fibonacci-type Sequences
Definition: A sequence sn is of Fibonacci typeif
sn+2 = sn+1 + sn.Proposition: The set S of Fibonacci-type sequences forms a
vector space.
Proof.
(sn+2 + tn+2) = (sn+1 + tn+1) + (sn + tn),
csn+2 = csn+1 + csn,
0 = 0 + 0 (zero vectors are important!)
Qiaochu Yuan The Fibonacci numbers
IntroductionThe Three Proofs
Unifying the Proofs
Proof 1: Linear Independence
Proof 2: Partial Fraction Decomposition
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Unifying the Proofs
ConclusionProof 3: Diagonalization
Fibonacci-type Sequences
Definition: A sequence sn is of Fibonacci typeif
sn+2 = sn+1 + sn.Proposition: The set S of Fibonacci-type sequences forms a
vector space.
Proof.
(sn+2 + tn+2) = (sn+1 + tn+1) + (sn + tn),
csn+2 = csn+1 + csn,
0 = 0 + 0 (zero vectors are important!)
Qiaochu Yuan The Fibonacci numbers
IntroductionThe Three Proofs
Unifying the Proofs
Proof 1: Linear Independence
Proof 2: Partial Fraction Decomposition
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Unifying the Proofs
ConclusionProof 3: Diagonalization
Finding a Basis
Proposition: S has dimension 2; it has basis Fn and Fn1.Proof.
Suppose s0 = a, s1 = b. Then,s2 = a+ b, s3 = a+ 2b, s4 = 2a+ 3b, s5 = 3a+ 5b, s6 =5a+ 8b...
In fact sn = aFn1 + bFn, n > 1. (Induction)
The first two values determine the whole sequence, sothere arent any others.
Qiaochu Yuan The Fibonacci numbers
IntroductionThe Three Proofs
Unifying the Proofs
Proof 1: Linear Independence
Proof 2: Partial Fraction Decomposition
P f 3 Di li i
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Unifying the Proofs
ConclusionProof 3: Diagonalization
Finding a Basis
Proposition: S has dimension 2; it has basis Fn and Fn1.Proof.
Suppose s0 = a, s1 = b. Then,s2 = a+ b, s3 = a+ 2b, s4 = 2a+ 3b, s5 = 3a+ 5b, s6 =5a+ 8b...
In fact sn = aFn1 + bFn, n > 1. (Induction)
The first two values determine the whole sequence, sothere arent any others.
Qiaochu Yuan The Fibonacci numbers
IntroductionThe Three Proofs
Unifying the Proofs
Proof 1: Linear Independence
Proof 2: Partial Fraction Decomposition
P f 3 Di li ti
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y g
ConclusionProof 3: Diagonalization
Finding a Basis
Proposition: S has dimension 2; it has basis Fn and Fn1.Proof.
Suppose s0 = a, s1 = b. Then,s2 = a+ b, s3 = a+ 2b, s4 = 2a+ 3b, s5 = 3a+ 5b, s6 =5a+ 8b...
In fact sn = aFn1 + bFn, n > 1. (Induction)The first two values determine the whole sequence, so
there arent any others.
Qiaochu Yuan The Fibonacci numbers
IntroductionThe Three Proofs
Unifying the Proofs
Proof 1: Linear Independence
Proof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
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y g
ConclusionProof 3: Diagonalization
Finding a Basis
Proposition: S has dimension 2; it has basis Fn and Fn1.Proof.
Suppose s0 = a, s1 = b. Then,s2 = a+ b, s3 = a+ 2b, s4 = 2a+ 3b, s5 = 3a+ 5b, s6 =5a+ 8b...
In fact sn = aFn1 + bFn, n > 1. (Induction)The first two values determine the whole sequence, so
there arent any others.
Qiaochu Yuan The Fibonacci numbers
IntroductionThe Three Proofs
Unifying the Proofs
Proof 1: Linear Independence
Proof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
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ConclusionProof 3: Diagonalization
Finding a Basis
Proposition: S has dimension 2; it has basis Fn and Fn1.Proof.
Suppose s0 = a, s1 = b. Then,s2 = a+ b, s3 = a+ 2b, s4 = 2a+ 3b, s5 = 3a+ 5b, s6 =5a+ 8b...
In fact sn = aFn1 + bFn, n > 1. (Induction)The first two values determine the whole sequence, so
there arent any others.
Qiaochu Yuan The Fibonacci numbers
IntroductionThe Three Proofs
Unifying the Proofs
Proof 1: Linear Independence
Proof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
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ConclusionProof 3: Diagonalization
A Fancy Word For Guessing
Now we use the ansatz sn = an:
sn+2 = sn+1 + sn if and only if an+2 = an+1 + an,
If and only if a2 = a+ 1,
If and only if a= 1
52 .
= 1+
52 is the golden ratio, =
152 is the othergolden
ratio.
Qiaochu Yuan The Fibonacci numbers
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IntroductionThe Three Proofs
Unifying the Proofs
C l i
Proof 1: Linear Independence
Proof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
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Conclusionoo 3 ago a at o
A Fancy Word For Guessing
Now we use the ansatz sn = an:
sn+2 = sn+1 + sn if and only if an+2 = an+1 + an,
If and only if a2 = a+ 1,
If and only if a= 1
52 .
= 1+
52 is the golden ratio, =
152 is the othergolden
ratio.
Qiaochu Yuan The Fibonacci numbers
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41/124
IntroductionThe Three Proofs
Unifying the Proofs
Conclusion
Proof 1: Linear Independence
Proof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
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Conclusion
A Fancy Word For Guessing
Now we use the ansatz sn = an:
sn+2 = sn+1 + sn if and only if an+2 = an+1 + an,
If and only if a2 = a+ 1,
If and only if a= 1
52 .
= 1+
52 is the golden ratio, =
152 is the othergolden
ratio.
Qiaochu Yuan The Fibonacci numbers
IntroductionThe Three Proofs
Unifying the Proofs
Conclusion
Proof 1: Linear Independence
Proof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
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Conclusion
Guessing Works!
Proposition: n, n form a basis for S.
Proof.
Suppose s0 = a, s1 = b and sn = c1n + c2
n.
Then s0 = c1 + c2 and s1 = c1 + c2.
Can we solve this system? Yes!
c1 =ba , c2 =
ba
So every Fibonacci-type sequencehas this form.
Corollary: Fn =nn
Qiaochu Yuan The Fibonacci numbers
IntroductionThe Three Proofs
Unifying the Proofs
Conclusion
Proof 1: Linear Independence
Proof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
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Conclusion
Guessing Works!
Proposition: n, n form a basis for S.
Proof.
Suppose s0 = a, s1 = b and sn = c1n + c2
n.
Then s0 = c1 + c2 and s1 = c1 + c2.
Can we solve this system? Yes!
c1 =ba , c2 =
ba
So every Fibonacci-type sequencehas this form.
Corollary: Fn =nn
Qiaochu Yuan The Fibonacci numbers
IntroductionThe Three Proofs
Unifying the Proofs
Conclusion
Proof 1: Linear IndependenceProof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
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Conclusion
Guessing Works!
Proposition: n, n form a basis for S.
Proof.
Suppose s0 = a, s1 = b and sn = c1n + c2
n.
Then s0 = c1 + c2 and s1 = c1 + c2.
Can we solve this system? Yes!
c1 =ba , c2 =
ba
So every Fibonacci-type sequencehas this form.
Corollary: Fn =nn
Qiaochu Yuan The Fibonacci numbers
IntroductionThe Three Proofs
Unifying the Proofs
Conclusion
Proof 1: Linear IndependenceProof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
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Guessing Works!
Proposition: n, n form a basis for S.
Proof.
Suppose s0 = a, s1 = b and sn = c1n + c2
n.
Then s0 = c1 + c2 and s1 = c1 + c2.
Can we solve this system? Yes!
c1 =ba , c2 =
ba
So every Fibonacci-type sequencehas this form.
Corollary: Fn =nn
Qiaochu Yuan The Fibonacci numbers
IntroductionThe Three Proofs
Unifying the Proofs
Conclusion
Proof 1: Linear IndependenceProof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
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Guessing Works!
Proposition: n, n form a basis for S.
Proof.
Suppose s0 = a, s1 = b and sn = c1n + c2
n.
Then s0 = c1 + c2 and s1 = c1 + c2.
Can we solve this system? Yes!
c1 =ba , c2 =
ba
So every Fibonacci-type sequencehas this form.
Corollary: Fn =nn
Qiaochu Yuan The Fibonacci numbers
IntroductionThe Three Proofs
Unifying the Proofs
Conclusion
Proof 1: Linear IndependenceProof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
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Guessing Works!
Proposition: n, n form a basis for S.
Proof.
Suppose s0 = a, s1 = b and sn = c1n + c2
n.
Then s0 = c1 + c2 and s1 = c1 + c2.
Can we solve this system? Yes!
c1 =ba , c2 =
ba
So every Fibonacci-type sequencehas this form.
Corollary: Fn =nn
Qiaochu Yuan The Fibonacci numbers
IntroductionThe Three Proofs
Unifying the Proofs
Conclusion
Proof 1: Linear IndependenceProof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
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Guessing Works!
Proposition: n, n form a basis for S.
Proof.
Suppose s0 = a, s1 = b and sn = c1n + c2
n.
Then s0 = c1 + c2 and s1 = c1 + c2.
Can we solve this system? Yes!
c1 =ba , c2 =
ba
So every Fibonacci-type sequencehas this form.
Corollary: Fn =nn
Qiaochu Yuan The Fibonacci numbers
IntroductionThe Three Proofs
Unifying the Proofs
Conclusion
Proof 1: Linear IndependenceProof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
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Generating Functions
Definition: The (ordinary) generating functionof sn is
s0 + s1x + s2x2 + ....
Examples:
1 + x + x2 + ... = 11xs0 + as0x + a
2s0x2 + ... = a01sx
1 + 2x + 3x2 + ... = 1(1x)2
n0
+
n1
x +
n2
x2 + ... = (1 + x)n
Qiaochu Yuan The Fibonacci numbers
IntroductionThe Three Proofs
Unifying the Proofs
Conclusion
Proof 1: Linear IndependenceProof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
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Generating Functions
Definition: The (ordinary) generating functionof sn is
s0 + s1x + s2x2 + ....
Examples:
1 + x + x2 + ... = 11xs0 + as0x + a
2s0x2 + ... = a01sx
1 + 2x + 3x2 + ... = 1(1x)2
n0
+
n1
x +
n2
x2 + ... = (1 + x)n
Qiaochu Yuan The Fibonacci numbers
IntroductionThe Three Proofs
Unifying the Proofs
Conclusion
Proof 1: Linear IndependenceProof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
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52/124
Generating Functions
Definition: The (ordinary) generating functionof sn is
s0 + s1x + s2x2 + ....
Examples:
1 + x + x2 + ... = 11xs0 + as0x + a
2s0x2 + ... = a01sx
1 + 2x + 3x2 + ... = 1(1x)2
n0
+
n1
x +
n2
x2 + ... = (1 + x)n
Qiaochu Yuan The Fibonacci numbers
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Introduction
The Three Proofs
Unifying the Proofs
Conclusion
Proof 1: Linear IndependenceProof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
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Generating Functions
Definition: The (ordinary) generating functionof sn is
s0 + s1x + s2x2 + ....
Examples:
1 + x + x2 + ... = 11xs0 + as0x + a
2s0x2 + ... = a01sx
1 + 2x + 3x2 + ... = 1(1x)2
n0
+
n1
x +
n2
x2 + ... = (1 + x)n
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three Proofs
Unifying the Proofs
Conclusion
Proof 1: Linear IndependenceProof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
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55/124
A Very Special Function
So let F(x) = F0 + F1x + F2x2 + F3x
3 + ....
Then xF(x) = F0x + F1x2 + F2x
3 + ....
Then x2
F(x) = F0x2
+ F1x3
+ ....Then xF(x) + x2F(x) = F0x + F2x
2 + F3x3 + ....
But this is also F(x) F0 F1x + F0x = F(x) x!
It follows that (1 x x2)F(x) = x, so F(x) = x1xx2 .
Corollary: F
110
= 1089 = 0.0112358...
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three Proofs
Unifying the Proofs
Conclusion
Proof 1: Linear IndependenceProof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
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A Very Special Function
So let F(x) = F0 + F1x + F2x2 + F3x
3 + ....
Then xF(x) = F0x + F1x2 + F2x
3 + ....
Then x2
F(x) = F0x2
+ F1x3
+ ....Then xF(x) + x2F(x) = F0x + F2x
2 + F3x3 + ....
But this is also F(x) F0 F1x + F0x = F(x) x!
It follows that (1 x x2)F(x) = x, so F(x) = x1xx2 .
Corollary: F
110
= 1089 = 0.0112358...
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three Proofs
Unifying the Proofs
Conclusion
Proof 1: Linear IndependenceProof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
A V S i l F i
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A Very Special Function
So let F(x) = F0 + F1x + F2x2 + F3x
3 + ....
Then xF(x) = F0x + F1x2 + F2x
3 + ....
Then x
2
F(x) = F0x2
+ F1x3
+ ....Then xF(x) + x2F(x) = F0x + F2x
2 + F3x3 + ....
But this is also F(x) F0 F1x + F0x = F(x) x!
It follows that (1 x x2)F(x) = x, so F(x) = x1xx2 .
Corollary: F
110
= 1089 = 0.0112358...
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three Proofs
Unifying the Proofs
Conclusion
Proof 1: Linear IndependenceProof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
A V S i l F ti
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58/124
A Very Special Function
So let F(x) = F0 + F1x + F2x2 + F3x
3 + ....
Then xF(x) = F0x + F1x2 + F2x
3 + ....
Then x
2
F(x) = F0x2
+ F1x3
+ ....Then xF(x) + x2F(x) = F0x + F2x
2 + F3x3 + ....
But this is also F(x) F0 F1x + F0x = F(x) x!
It follows that (1 x x2)F(x) = x, so F(x) = x1xx2 .
Corollary: F
110
= 1089 = 0.0112358...
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three Proofs
Unifying the Proofs
Conclusion
Proof 1: Linear IndependenceProof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
A V S i l F ti
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59/124
A Very Special Function
So let F(x) = F0 + F1x + F2x2 + F3x
3 + ....
Then xF(x) = F0x + F1x2 + F2x
3 + ....
Then x
2
F(x) = F0x2
+ F1x3
+ ....Then xF(x) + x2F(x) = F0x + F2x
2 + F3x3 + ....
But this is also F(x) F0 F1x + F0x = F(x) x!
It follows that (1 x x2)F(x) = x, so F(x) = x1xx2 .
Corollary: F
110
= 1089 = 0.0112358...
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three Proofs
Unifying the Proofs
Conclusion
Proof 1: Linear IndependenceProof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
A V S i l F ti
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60/124
A Very Special Function
So let F(x) = F0 + F1x + F2x2 + F3x
3 + ....
Then xF(x) = F0x + F1x2 + F2x
3 + ....
Then x
2
F(x) = F0x
2
+ F1x
3
+ ....Then xF(x) + x2F(x) = F0x + F2x
2 + F3x3 + ....
But this is also F(x) F0 F1x + F0x = F(x) x!
It follows that (1 x x2)F(x) = x, so F(x) = x1xx2 .
Corollary: F
110
= 1089 = 0.0112358...
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three Proofs
Unifying the Proofs
Conclusion
Proof 1: Linear IndependenceProof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
A Very Special Function
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61/124
A Very Special Function
So let F(x) = F0 + F1x + F2x2 + F3x
3 + ....
Then xF(x) = F0x + F1x2 + F2x
3 + ....
Then x
2
F(x) = F0x
2
+ F1x
3
+ ....Then xF(x) + x2F(x) = F0x + F2x
2 + F3x3 + ....
But this is also F(x) F0 F1x + F0x = F(x) x!
It follows that (1 x x2)F(x) = x, so F(x) = x1xx2 .
Corollary: F
110
= 1089 = 0.0112358...
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three Proofs
Unifying the Proofs
Conclusion
Proof 1: Linear IndependenceProof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
Remember Those Partial Fractions?
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Remember Those Partial Fractions?
1 = x + x2 1x2
= 1x + 1, so:
1 x x2 = (1 x)(1 x). Moreover,x
1xx2 =
A
1x+ B
1xfor some A, B,
Hence x = A(1 x) + B(1 x).
This gives A = 1 , B =
1 , so...
F(x) = 1
11x
11x
!
F(x) = 1
(1 1) + ( )x + (2 2)x2 + ...
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three Proofs
Unifying the Proofs
Conclusion
Proof 1: Linear IndependenceProof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
Remember Those Partial Fractions?
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63/124
Remember Those Partial Fractions?
1 = x + x2 1x2
= 1x + 1, so:
1 x x2 = (1 x)(1 x). Moreover,x
1xx2 =
A
1x+ B
1xfor some A, B,
Hence x = A(1 x) + B(1 x).
This gives A = 1 , B =
1 , so...
F(x) = 1
11x
11x
!
F(x) = 1
(1 1) + ( )x + (2 2)x2 + ...
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three Proofs
Unifying the Proofs
Conclusion
Proof 1: Linear IndependenceProof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
Remember Those Partial Fractions?
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64/124
Remember Those Partial Fractions?
1 = x + x2 1x2
= 1x + 1, so:
1 x x2 = (1 x)(1 x). Moreover,x
1xx2 =
A
1x+ B
1xfor some A, B,
Hence x = A(1 x) + B(1 x).
This gives A = 1 , B =
1 , so...
F(x) = 1
11x
11x
!
F(x) = 1
(1 1) + ( )x + (2 2)x2 + ...
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three Proofs
Unifying the Proofs
Conclusion
Proof 1: Linear IndependenceProof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
Remember Those Partial Fractions?
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Remember Those Partial Fractions?
1 = x + x2 1x2
= 1x + 1, so:
1 x x2 = (1 x)(1 x). Moreover,x
1xx2 =
A
1x+ B
1xfor some A, B,
Hence x = A(1 x) + B(1 x).
This gives A = 1 , B =
1 , so...
F(x) = 1
1
1x 1
1x
!
F(x) = 1
(1 1) + ( )x + (2 2)x2 + ...
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three Proofs
Unifying the Proofs
Conclusion
Proof 1: Linear IndependenceProof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
Remember Those Partial Fractions?
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66/124
Remember Those Partial Fractions?
1 = x + x2 1x2
= 1x + 1, so:
1 x x2 = (1 x)(1 x). Moreover,x
1xx2 =
A
1x+ B
1xfor some A, B,
Hence x = A(1 x) + B(1 x).
This gives A = 1 , B =
1 , so...
F(x) = 1
1
1x 1
1x
!
F(x) = 1
(1 1) + ( )x + (2 2)x2 + ...
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three Proofs
Unifying the Proofs
Conclusion
Proof 1: Linear IndependenceProof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
Remember Those Partial Fractions?
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67/124
Remember Those Partial Fractions?
1 = x + x2 1x2
= 1x + 1, so:
1 x x2 = (1 x)(1 x). Moreover,x
1xx2 =
A
1x+ B
1xfor some A, B,
Hence x = A(1 x) + B(1 x).
This gives A = 1 , B =
1 , so...
F(x) = 1
1
1x 1
1x
!
F(x) = 1
(1 1) + ( )x + (2 2)x2 + ...
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three ProofsUnifying the Proofs
Conclusion
Proof 1: Linear IndependenceProof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
Remember Those Partial Fractions?
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68/124
Remember Those Partial Fractions?
1 = x + x2 1x2
= 1x + 1, so:
1 x x2 = (1 x)(1 x). Moreover,x
1xx2 =
A
1x+ B
1xfor some A, B,
Hence x = A(1 x) + B(1 x).
This gives A = 1 , B =
1 , so...
F(x) = 1
1
1x 1
1x
!
F(x) = 1
(1 1) + ( )x + (2 2)x2 + ...
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three ProofsUnifying the Proofs
Conclusion
Proof 1: Linear IndependenceProof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
Recurrences as Linear Transformations
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Recurrences as Linear Transformations
(a, b) (b, a+ b) is a linear transformation on C2
Matrix form:
1 1
1 0
b
a
=
b+ a
b
(a, b) (b, a+b) (a+b, a+2b) (a+2b, 2a+3b) ...
In fact Fn =
1 1
1 0
n=
Fn+1 Fn
Fn Fn1
. (Induction)
Corollary: Fn+1Fn
1 F2n = (1)
n. (Determinant)
How do we quickly compute the powers of a matrix?
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three ProofsUnifying the Proofs
Conclusion
Proof 1: Linear IndependenceProof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
Recurrences as Linear Transformations
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70/124
Recurrences as Linear Transformations
(a, b) (b, a+ b) is a linear transformation on C2
Matrix form:
1 1
1 0
b
a
=
b+ a
b
(a, b) (b, a+b) (a+b, a+2b) (a+2b, 2a+3b) ...
In fact Fn =
1 1
1 0
n=
Fn+1 Fn
Fn Fn1
. (Induction)
Corollary: Fn+1Fn
1 F2n = (1)
n. (Determinant)
How do we quickly compute the powers of a matrix?
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three ProofsUnifying the Proofs
Conclusion
Proof 1: Linear IndependenceProof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
Recurrences as Linear Transformations
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71/124
Recurrences as Linear Transformations
(a, b) (b, a+ b) is a linear transformation on C2
Matrix form:
1 1
1 0
b
a
=
b+ a
b
(a, b) (b, a+b) (a+b, a+2b) (a+2b, 2a+3b) ...
In fact Fn =
1 1
1 0
n=
Fn+1 Fn
Fn Fn1
. (Induction)
Corollary: Fn+1Fn
1 F2n = (1)
n. (Determinant)
How do we quickly compute the powers of a matrix?
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three ProofsUnifying the Proofs
Conclusion
Proof 1: Linear IndependenceProof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
Recurrences as Linear Transformations
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72/124
ecu e ces as ea a s o at o s
(a, b) (b, a+ b) is a linear transformation on C2
Matrix form:
1 1
1 0
b
a
=
b+ a
b
(a, b) (b, a+b) (a+b, a+2b) (a+2b, 2a+3b) ...
In fact Fn =
1 1
1 0
n=
Fn+1 Fn
Fn Fn1
. (Induction)
Corollary: Fn+1Fn
1 F2n = (1)
n. (Determinant)
How do we quickly compute the powers of a matrix?
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three ProofsUnifying the Proofs
Conclusion
Proof 1: Linear IndependenceProof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
Recurrences as Linear Transformations
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73/124
(a, b) (b, a+ b) is a linear transformation on C2
Matrix form:
1 1
1 0
b
a
=
b+ a
b
(a, b) (b, a+b) (a+b, a+2b) (a+2b, 2a+3b) ...
In fact Fn =
1 1
1 0
n=
Fn+1 Fn
Fn Fn1
. (Induction)
Corollary: Fn+1Fn
1 F2n = (1)
n. (Determinant)
How do we quickly compute the powers of a matrix?
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three ProofsUnifying the Proofs
Conclusion
Proof 1: Linear IndependenceProof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
Recurrences as Linear Transformations
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74/124
(a, b) (b, a+ b) is a linear transformation on C2
Matrix form:
1 1
1 0
b
a
=
b+ a
b
(a, b) (b, a+b) (a+b, a+2b) (a+2b, 2a+3b) ...
In fact Fn =
1 1
1 0
n=
Fn+1 Fn
Fn Fn1
. (Induction)
Corollary: Fn+1Fn
1 F2n = (1)
n. (Determinant)
How do we quickly compute the powers of a matrix?
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three ProofsUnifying the Proofs
Conclusion
Proof 1: Linear IndependenceProof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
Changing Basis
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g g
x, y = xe1 + ye2; ei is the standard basis
x, y = zv1 + wv2; how do we compute z, w?
yx
=
v2 v1 w
z
v2 v1
1 yx
=
w
z
We can also write linear transformations in a different
basis!
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three ProofsUnifying the Proofs
Conclusion
Proof 1: Linear IndependenceProof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
Changing Basis
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76/124
g g
x, y = xe1 + ye2; ei is the standard basis
x, y = zv1 + wv2; how do we compute z, w?
yx
=
v2 v1 w
z
v2 v1
1 yx
=
w
z
We can also write linear transformations in a different
basis!
Qiaochu Yuan The Fibonacci numbers
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77/124
Introduction
The Three ProofsUnifying the Proofs
Conclusion
Proof 1: Linear Independence
Proof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
Changing Basis
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x, y = xe1 + ye2; ei is the standard basis
x, y = zv1 + wv2; how do we compute z, w?
yx
=
v2 v1 w
z
v2 v1
1 yx
=
w
z
We can also write linear transformations in a different
basis!
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three ProofsUnifying the Proofs
Conclusion
Proof 1: Linear Independence
Proof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
Changing Basis
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79/124
x, y = xe1 + ye2; ei is the standard basis
x, y = zv1 + wv2; how do we compute z, w?
yx
=
v2 v1 w
z
v2 v1
1 yx
=
w
z
We can also write linear transformations in a different
basis!
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three ProofsUnifying the Proofs
Conclusion
Proof 1: Linear Independence
Proof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
Linear Transformations in a Different Basis
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80/124
Suppose Av = u but we want to change to a basis P. Then:
If new coordinates are v, u then v = Pv, u = Pu
So APv = Pu, or P1APv = u
New matrix in new basis is P1AP, a conjugate of AGoal: find basis such that new matrix is easier to work with
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three ProofsUnifying the Proofs
Conclusion
Proof 1: Linear Independence
Proof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
Linear Transformations in a Different Basis
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81/124
Suppose Av = u but we want to change to a basis P. Then:
If new coordinates are v, u then v = Pv, u = Pu
So APv = Pu, or P1APv = u
New matrix in new basis is P1AP, a conjugate of AGoal: find basis such that new matrix is easier to work with
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three ProofsUnifying the Proofs
Conclusion
Proof 1: Linear Independence
Proof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
Linear Transformations in a Different Basis
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82/124
Suppose Av = u but we want to change to a basis P. Then:
If new coordinates are v, u then v = Pv, u = Pu
So APv = Pu, or P1APv = u
New matrix in new basis is P1AP, a conjugate of AGoal: find basis such that new matrix is easier to work with
Qiaochu Yuan The Fibonacci numbers Introduction
The Three ProofsUnifying the Proofs
Conclusion
Proof 1: Linear Independence
Proof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
Linear Transformations in a Different Basis
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83/124
Suppose Av = u but we want to change to a basis P. Then:
If new coordinates are v, u then v = Pv, u = Pu
So APv = Pu, or P1APv = u
New matrix in new basis is P1AP, a conjugate of AGoal: find basis such that new matrix is easier to work with
Qiaochu Yuan The Fibonacci numbers Introduction
The Three ProofsUnifying the Proofs
Conclusion
Proof 1: Linear Independence
Proof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
Linear Transformations in a Different Basis
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Suppose Av = u but we want to change to a basis P. Then:
If new coordinates are v, u then v = Pv, u = Pu
So APv = Pu, or P1APv = u
New matrix in new basis is P1AP, a conjugate of AGoal: find basis such that new matrix is easier to work with
Qiaochu Yuan The Fibonacci numbers Introduction
The Three ProofsUnifying the Proofs
Conclusion
Proof 1: Linear Independence
Proof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
The Power of Eigenvectors
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Proposition: , are the eigenvalues of F. The eigenvectors
are
1
,
1
Proof.
1 1
1 0
1 =
+ 1
2 = + 1 by definition (and same for )
Dimension 2, so there are no other eigenvalues or
eigenvectors.
Computation can be done in general: we can find acharacteristic polynomial.
With P =
1 1
, P1FP =
0
0
, a diagonal matrix
Qiaochu Yuan The Fibonacci numbers Introduction
The Three ProofsUnifying the Proofs
Conclusion
Proof 1: Linear Independence
Proof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
The Power of Eigenvectors
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86/124
Proposition: , are the eigenvalues of F. The eigenvectors
are
1
,
1
Proof.
1 1
1 0
1 =
+ 1
2 = + 1 by definition (and same for )
Dimension 2, so there are no other eigenvalues or
eigenvectors.
Computation can be done in general: we can find acharacteristic polynomial.
With P =
1 1
, P1FP =
0
0
, a diagonal matrix
Qiaochu Yuan The Fibonacci numbers
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87/124
Introduction
The Three ProofsUnifying the Proofs
Conclusion
Proof 1: Linear Independence
Proof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
The Power of Eigenvectors
8/7/2019 MIT Slides on Diagonalisatiojn
88/124
Proposition: , are the eigenvalues of F. The eigenvectors
are
1
,
1
Proof.
1 1
1 0
1 =
+ 1
2 = + 1 by definition (and same for )
Dimension 2, so there are no other eigenvalues or
eigenvectors.
Computation can be done in general: we can find acharacteristic polynomial.
With P =
1 1
, P1FP =
0
0
, a diagonal matrix
Qiaochu Yuan The Fibonacci numbers Introduction
The Three ProofsUnifying the Proofs
Conclusion
Proof 1: Linear Independence
Proof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
The Power of Eigenvectors
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89/124
Proposition: , are the eigenvalues of F. The eigenvectors
are
1
,
1
Proof.
1 1
1 0
1 =
+ 1
2 = + 1 by definition (and same for )
Dimension 2, so there are no other eigenvalues or
eigenvectors.
Computation can be done in general: we can find acharacteristic polynomial.
With P =
1 1
, P1FP =
0
0
, a diagonal matrix
Qiaochu Yuan The Fibonacci numbers Introduction
The Three ProofsUnifying the Proofs
Conclusion
Proof 1: Linear Independence
Proof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
The Power of Eigenvectors
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90/124
Proposition: , are the eigenvalues of F. The eigenvectors
are
1
,
1
Proof.
1 1
1 0
1 =
+ 1
2 = + 1 by definition (and same for )
Dimension 2, so there are no other eigenvalues or
eigenvectors.
Computation can be done in general: we can find acharacteristic polynomial.
With P =
1 1
, P1FP =
0
0
, a diagonal matrix
Qiaochu Yuan The Fibonacci numbers Introduction
The Three ProofsUnifying the Proofs
Conclusion
Proof 1: Linear Independence
Proof 2: Partial Fraction Decomposition
Proof 3: Diagonalization
The Power of Eigenvectors
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91/124
Conjugation respects multiplication: P1FnP =
n 0
0 n
Back to old basis: Fn
= P n 0
0 n
P1
Computations give
Fn = 1
n+1 n+1 n n
n n n1 n1
as desired
Qiaochu Yuan The Fibonacci numbers Introduction
The Three ProofsUnifying the Proofs
Conclusion
Proof 1: Linear Independence
Proof 2: Partial Fraction DecompositionProof 3: Diagonalization
The Power of Eigenvectors
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92/124
Conjugation respects multiplication: P1FnP =
n 0
0 n
Back to old basis: Fn
= P n 0
0 n
P1
Computations give
Fn = 1
n+1 n+1 n n
n n n1 n1
as desired
Qiaochu Yuan The Fibonacci numbers Introduction
The Three ProofsUnifying the Proofs
Conclusion
Proof 1: Linear Independence
Proof 2: Partial Fraction DecompositionProof 3: Diagonalization
The Power of Eigenvectors
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93/124
Conjugation respects multiplication: P1FnP =
n 0
0 n
Back to old basis: Fn
= P n 0
0 n
P1
Computations give
Fn = 1
n+1 n+1 n n
n n n1 n1
as desired
Qiaochu Yuan The Fibonacci numbers Introduction
The Three ProofsUnifying the Proofs
Conclusion
The shift operator
Eigendecomposition as a General Principle
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94/124
What do the three proofs have in common?
Geometric series n, n appeared naturally
General idea of decomposing into a sum of simpler partsappeared naturally
"Simpler parts" = geometric series = eigenvectors?
Identify linear transformations acting in the first two proofs
Qiaochu Yuan The Fibonacci numbers Introduction
The Three ProofsUnifying the Proofs
Conclusion
The shift operator
Eigendecomposition as a General Principle
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95/124
What do the three proofs have in common?
Geometric series n, n appeared naturally
General idea of decomposing into a sum of simpler partsappeared naturally
"Simpler parts" = geometric series = eigenvectors?
Identify linear transformations acting in the first two proofs
Qiaochu Yuan The Fibonacci numbers Introduction
The Three ProofsUnifying the Proofs
Conclusion
The shift operator
Eigendecomposition as a General Principle
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96/124
What do the three proofs have in common?
Geometric series n, n appeared naturally
General idea of decomposing into a sum of simpler partsappeared naturally
"Simpler parts" = geometric series = eigenvectors?
Identify linear transformations acting in the first two proofs
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three ProofsUnifying the Proofs
Conclusion
The shift operator
Eigendecomposition as a General Principle
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97/124
What do the three proofs have in common?
Geometric series n, n appeared naturally
General idea of decomposing into a sum of simpler partsappeared naturally
"Simpler parts" = geometric series = eigenvectors?
Identify linear transformations acting in the first two proofs
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three ProofsUnifying the Proofs
Conclusion
The shift operator
Eigendecomposition as a General Principle
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98/124
What do the three proofs have in common?
Geometric series n, n appeared naturally
General idea of decomposing into a sum of simpler partsappeared naturally
"Simpler parts" = geometric series = eigenvectors?
Identify linear transformations acting in the first two proofs
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three ProofsUnifying the Proofs
Conclusion
The shift operator
Interesting Linear Transformations
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99/124
Vector spaces: spaces of sequences (proof 1), spaces of
functions (proof 2)
We want an be to associated with an eigenvalue of a
Sequences: take T(sn) = sn+1; then an is an eigenvector
with eigenvalue a
Functions: take T(f(x)) = f(x)f(0)x ; then1
1ax is aneigenvector with eigenvalue a
Shift operators:
T : a0 + a1x + a2x2 + ... a1 + a2x + a3x
2 + ...
Eigenvectors are precisely the geometric series,eigenvalues are their common ratios
Caution: spaces are infinite-dimensional
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three ProofsUnifying the Proofs
Conclusion
The shift operator
Interesting Linear Transformations
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100/124
Vector spaces: spaces of sequences (proof 1), spaces of
functions (proof 2)
We want an be to associated with an eigenvalue of a
Sequences: take T(sn) = sn+1; then an is an eigenvector
with eigenvalue a
Functions: take T(f(x)) = f(x)f(0)x ; then1
1ax is aneigenvector with eigenvalue a
Shift operators:
T : a0 + a1x + a2x2 + ... a1 + a2x + a3x
2 + ...
Eigenvectors are precisely the geometric series,eigenvalues are their common ratios
Caution: spaces are infinite-dimensional
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three Proofs
Unifying the Proofs
Conclusion
The shift operator
Interesting Linear Transformations
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101/124
Vector spaces: spaces of sequences (proof 1), spaces of
functions (proof 2)
We want an be to associated with an eigenvalue of a
Sequences: take T(sn) = sn+1; then an is an eigenvector
with eigenvalue a
Functions: take T(f(x)) = f(x)f(0)x ; then1
1ax is aneigenvector with eigenvalue a
Shift operators:
T : a0 + a1x + a2x2 + ... a1 + a2x + a3x
2 + ...
Eigenvectors are precisely the geometric series,eigenvalues are their common ratios
Caution: spaces are infinite-dimensional
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three Proofs
Unifying the Proofs
Conclusion
The shift operator
Interesting Linear Transformations
V f ( f 1) f
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102/124
Vector spaces: spaces of sequences (proof 1), spaces of
functions (proof 2)
We want an be to associated with an eigenvalue of a
Sequences: take T(sn) = sn+1; then an is an eigenvector
with eigenvalue a
Functions: take T(f(x)) = f(x)f(0)x ; then1
1ax is aneigenvector with eigenvalue a
Shift operators:
T : a0 + a1x + a2x2 + ... a1 + a2x + a3x
2 + ...
Eigenvectors are precisely the geometric series,eigenvalues are their common ratios
Caution: spaces are infinite-dimensional
Qiaochu Yuan The Fibonacci numbers
http://find/http://goback/8/7/2019 MIT Slides on Diagonalisatiojn
103/124
Introduction
The Three Proofs
Unifying the Proofs
Conclusion
The shift operator
Interesting Linear Transformations
V t f ( f 1) f
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104/124
Vector spaces: spaces of sequences (proof 1), spaces of
functions (proof 2)
We want an be to associated with an eigenvalue of a
Sequences: take T(sn) = sn+1; then an is an eigenvector
with eigenvalue a
Functions: take T(f(x)) = f(x)f(0)x ; then1
1ax is aneigenvector with eigenvalue a
Shift operators:
T : a0 + a1x + a2x2 + ... a1 + a2x + a3x
2 + ...
Eigenvectors are precisely the geometric series,eigenvalues are their common ratios
Caution: spaces are infinite-dimensional
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three Proofs
Unifying the Proofs
Conclusion
The shift operator
Interesting Linear Transformations
Vector spaces: spaces of sequences (proof 1) spaces of
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105/124
Vector spaces: spaces of sequences (proof 1), spaces of
functions (proof 2)
We want an be to associated with an eigenvalue of a
Sequences: take T(sn) = sn+1; then an is an eigenvector
with eigenvalue a
Functions: take T(f(x)) = f(x)f(0)x ; then1
1ax is aneigenvector with eigenvalue a
Shift operators:
T : a0 + a1x + a2x2 + ... a1 + a2x + a3x
2 + ...
Eigenvectors are precisely the geometric series,eigenvalues are their common ratios
Caution: spaces are infinite-dimensional
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three Proofs
Unifying the Proofs
Conclusion
The shift operator
A Generic Description
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106/124
A sequence s is Fibonacci-type if
T2s = Ts+ s (T2 T 1)(s) = 0
When T is the shift on sequences, this is the usual
definitionWhen T is division by x, this is the definition of the
generating function
When T = F, this is the identity F2 F I = 0
(characteristic polynomial)
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three Proofs
Unifying the Proofs
Conclusion
The shift operator
A Generic Description
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107/124
A sequence s is Fibonacci-type if
T2s = Ts+ s (T2 T 1)(s) = 0
When T is the shift on sequences, this is the usual
definitionWhen T is division by x, this is the definition of the
generating function
When T = F, this is the identity F2 F I = 0
(characteristic polynomial)
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three Proofs
Unifying the Proofs
Conclusion
The shift operator
A Generic Description
http://find/http://goback/8/7/2019 MIT Slides on Diagonalisatiojn
108/124
A sequence s is Fibonacci-type if
T2s = Ts+ s (T2 T 1)(s) = 0
When T is the shift on sequences, this is the usual
definitionWhen T is division by x, this is the definition of the
generating function
When T = F, this is the identity F2 F I = 0(characteristic polynomial)
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three Proofs
Unifying the Proofs
Conclusion
The shift operator
A Generic Description
http://find/http://goback/8/7/2019 MIT Slides on Diagonalisatiojn
109/124
A sequence s is Fibonacci-type if
T2s = Ts+ s (T2 T 1)(s) = 0
When T is the shift on sequences, this is the usual
definitionWhen T is division by x, this is the definition of the
generating function
When T = F, this is the identity F2 F I = 0(characteristic polynomial)
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three Proofs
Unifying the Proofs
Conclusion
The shift operator
A Generic Proof
http://find/http://goback/8/7/2019 MIT Slides on Diagonalisatiojn
110/124
Operators like T can be added and multiplied:
T2 T 1 = (T )(T )
T a : a0 + a1x + a2x2 + ...
(a1 a0a) + (a2 a1a)x + (a3 a2a)x2 + ...
(T a)s = 0 a is a geometric series with common ratioa
Operators commute, so we just take everything that is
annihilated by both operators
So we expect solutions to be span of{s|(T )s = 0}, {s|(T )s = 0}
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three Proofs
Unifying the Proofs
Conclusion
The shift operator
A Generic Proof
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111/124
Operators like T can be added and multiplied:
T2 T 1 = (T )(T )
T a : a0 + a1x + a2x2 + ...
(a1 a0a) + (a2 a1a)x + (a3 a2a)x2 + ...
(T a)s = 0 a is a geometric series with common ratioa
Operators commute, so we just take everything that is
annihilated by both operators
So we expect solutions to be span of{s|(T )s = 0}, {s|(T )s = 0}
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three Proofs
Unifying the Proofs
Conclusion
The shift operator
A Generic Proof
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112/124
Operators like T can be added and multiplied:
T2 T 1 = (T )(T )
T a : a0 + a1x + a2x2 + ...
(a1 a0a) + (a2 a1a)x + (a3 a2a)x2 + ...
(T a)s = 0 a is a geometric series with common ratioa
Operators commute, so we just take everything that is
annihilated by both operators
So we expect solutions to be span of{s|(T )s = 0}, {s|(T )s = 0}
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three Proofs
Unifying the Proofs
Conclusion
The shift operator
A Generic Proof
http://find/http://goback/8/7/2019 MIT Slides on Diagonalisatiojn
113/124
Operators like T can be added and multiplied:
T2 T 1 = (T )(T )
T a : a0 + a1x + a2x2 + ...
(a1 a0a) + (a2 a1a)x + (a3 a2a)x2 + ...
(T a)s = 0 a is a geometric series with common ratioa
Operators commute, so we just take everything that is
annihilated by both operators
So we expect solutions to be span of{s|(T )s = 0}, {s|(T )s = 0}
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three Proofs
Unifying the Proofs
Conclusion
The shift operator
A Generic Proof
http://find/http://goback/8/7/2019 MIT Slides on Diagonalisatiojn
114/124
Operators like T can be added and multiplied:
T2 T 1 = (T )(T )
T a : a0 + a1x + a2x2 + ...
(a1 a0a) + (a2 a1a)x + (a3 a2a)x2 + ...
(T a)s = 0 a is a geometric series with common ratioa
Operators commute, so we just take everything that is
annihilated by both operators
So we expect solutions to be span of{s|(T )s = 0}, {s|(T )s = 0}
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three Proofs
Unifying the Proofs
Conclusion
The shift operator
Generalization
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115/124
Theorem: Let (sn) satisfy sn+k = ak1sn+k1 + ... + a0sn andlet P(x) = xk ak1xk1 ... a0 = (x 1)m1 ...(x r)mr.Then:
There exist polynomials p1,...pr with deg pi < mi suchthat...
sn = p1(n)n1 + ... + pr(n)
nr
Example: sn+2 = 4sn+1 4sn sn = (c1n+ c0)2n
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three Proofs
Unifying the ProofsConclusion
The shift operator
Generalization
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116/124
Theorem: Let (sn) satisfy sn+k = ak1sn+k1 + ... + a0sn andlet P(x) = xk ak1xk1 ... a0 = (x 1)m1 ...(x r)mr.Then:
There exist polynomials p1,...pr with deg pi < mi suchthat...
sn = p1(n)n1 + ... + pr(n)
nr
Example: sn+2 = 4sn+1 4sn sn = (c1n+ c0)2n
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three Proofs
Unifying the ProofsConclusion
The shift operator
Generalization
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117/124
Theorem: Let (sn) satisfy sn+k = ak1sn+k1 + ... + a0sn andlet P(x) = xk ak1xk1 ... a0 = (x 1)m1 ...(x r)mr.Then:
There exist polynomials p1,...pr with deg pi < mi suchthat...
sn = p1(n)n1 + ... + pr(n)
nr
Example: sn+2 = 4sn+1 4sn sn = (c1n+ c0)2n
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three Proofs
Unifying the ProofsConclusion
The shift operator
Generalization
http://find/http://goback/8/7/2019 MIT Slides on Diagonalisatiojn
118/124
Theorem: Let (sn) satisfy sn+k = ak1sn+k1 + ... + a0sn andlet P(x) = xk ak1xk1 ... a0 = (x 1)m1 ...(x r)mr.Then:
There exist polynomials p1,...pr with deg pi < mi suchthat...
sn = p1(n)n1 + ... + pr(n)
nr
Example: sn+2 = 4sn+1 4sn sn = (c1n+ c0)2n
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three Proofs
Unifying the ProofsConclusion
Conclusion
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119/124
Binets formula is best understood as eigendecomposition.
Eigendecomposition can appear in many situations.
Further reading: artofproblemsolv-
ing.com/Forum/weblog_entry.php?t=175257, 212490,215833, 217361
Even further reading:
www.math.upenn.edu/ wilf/DownldGF.html
GOOD LUCK ON GUTS!
Qiaochu Yuan The Fibonacci numbers
http://find/http://goback/8/7/2019 MIT Slides on Diagonalisatiojn
120/124
Introduction
The Three Proofs
Unifying the ProofsConclusion
Conclusion
8/7/2019 MIT Slides on Diagonalisatiojn
121/124
Binets formula is best understood as eigendecomposition.
Eigendecomposition can appear in many situations.
Further reading: artofproblemsolv-
ing.com/Forum/weblog_entry.php?t=175257, 212490,215833, 217361
Even further reading:
www.math.upenn.edu/ wilf/DownldGF.html
GOOD LUCK ON GUTS!
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three Proofs
Unifying the ProofsConclusion
Conclusion
http://find/http://goback/8/7/2019 MIT Slides on Diagonalisatiojn
122/124
Binets formula is best understood as eigendecomposition.
Eigendecomposition can appear in many situations.
Further reading: artofproblemsolv-
ing.com/Forum/weblog_entry.php?t=175257, 212490,215833, 217361
Even further reading:
www.math.upenn.edu/ wilf/DownldGF.html
GOOD LUCK ON GUTS!
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three Proofs
Unifying the ProofsConclusion
Conclusion
http://find/http://goback/8/7/2019 MIT Slides on Diagonalisatiojn
123/124
Binets formula is best understood as eigendecomposition.
Eigendecomposition can appear in many situations.
Further reading: artofproblemsolv-
ing.com/Forum/weblog_entry.php?t=175257, 212490,215833, 217361
Even further reading:
www.math.upenn.edu/ wilf/DownldGF.html
GOOD LUCK ON GUTS!
Qiaochu Yuan The Fibonacci numbers
Introduction
The Three Proofs
Unifying the ProofsConclusion
Conclusion
http://find/http://goback/8/7/2019 MIT Slides on Diagonalisatiojn
124/124
Binets formula is best understood as eigendecomposition.
Eigendecomposition can appear in many situations.
Further reading: artofproblemsolv-
ing.com/Forum/weblog_entry.php?t=175257, 212490,215833, 217361
Even further reading:
www.math.upenn.edu/ wilf/DownldGF.html
GOOD LUCK ON GUTS!
Qiaochu Yuan The Fibonacci numbers
http://find/http://goback/