Section 5.1Eigenvectors and Eigenvalues

Eigenvectors and Eigenvalues

•Useful throughout pure and applied mathematics. •Used to study difference equations and continuous dynamical systems. •Provide critical information in engineering design•Arise naturally in such fields as physics and

chemistry.•Used in statistics to analyze multicollinearity

Example:

•Let

•Consider Av for some v

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1

31v

1

12v 3

1

1v

Definition:

•The Eigenvector of Anxn is a nonzero vector x such that Ax=λx for some scalar λ.

•λ is called an Eigenvalue of A

Statistics (multicollinearity)

•Where y is the dependent response vector and the x’s are the independent explanatory vectors•The β’s are least squares regression coefficients•εi are errors

•We desire linear independence between x vectors•Can use Eigen analysis to determine

iiii xxxy .......22110

From Definition:•Ax = λx = λIx•Ax – λIx = 0•(A – λI)x = 0

•Observations:1. λ is an eigenvalue of A iff (A – λI)x= 0 has non-

trivial solutions A – λI is not invertible IMT all false

2. The set {xεRn: (A – λI)x= 0} is the nullspace of (A – λI)x= 0, A a subspace of Rn

3. The set of all solutions is called the eigenspace of A corresponding to λ

Example

Show that 2 is an eigenvalue of

and find the corresponding eigenvectors.

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33A

Comments:

Warning: The method just used (row reduction) to find eigenvectors cannot be used to find eigenvalues.

Note: The set of all solutions to (A-λI)x =0 is called the eigenspace of A corresponding to λ.

Example

Let

An eigenvalue of A is λ=3. Find a basis for the corresponding eigenspace.

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123

221

A

Theorem

The eigenvalues of a triangular matrix are the entries on its main diagonal.

Proof of 3x3 case

Let

So (A-λI) =

33

2322

131211

00

0

a

aa

aaa

A

00

00

00

00

0

33

2322

131211

a

aa

aaa

A

33

2322

131211

00

0

a

aa

aaa

Proof of 3x3 case

By definition λ is an eigenvalue iff (A-λI)x=0 has non-trivial solutions so a free variable must exist.

This occurs when a11=λ or a22=λ or a33=λ

33

2322

131211

00

0

a

aa

aaa

ExampleConsider the lower triangular matrix below.

λ= 4 or 0 or -3

301

000

004

A

Addtion to IMT

Anxn is invertible iff

s. The number 0 is not an eigenvalue

t. det A≠0 (not sure why author waits until not to add this)

Theorem

If eigenvectors have distinct eigenvalues then the eigenvectors are linearly independent

This can be proven by the IMT

Section 5.2The Characteristic Equation

Finding Eigenvalues

1. We know (A-λI)x=0 must have non-trivial solutions and x is non-zero. That is free variables exist.

2. So (A-λI) is not invertible by the IMT

3. Therefore det(A-λI)=0 by IMT

Characteristic Equation

det(A-λI)=0

Solve to find eigenvalues

Note: det(A-λI) is the characteristic polynomial.

Previous Example:

•Let find eigenvalues

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ExampleFind Eigenvalues

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050

121

A

ExampleFind characteristic polynomial and eigenvalues

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1060

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A

Examplea. Find the characteristic polynomialb. Find all eigenvaluesc. Find multiplicity of each eigenvalue

1521

0319

0035

0002

A

Recapa. λ is an eigenvalue of A if (A-λI)x=0 has

non-trivial solutions (free variables exist).b. Eigenvectors (eigenspace )are found by

row reducing (A-λI)x=0.

c. Eigenvalues are found by solving det(A-λI)=0.

Section 5.3Diagonalization

Diagonalization•The goal here is to develop a useful factorization A=PDP-1, when A is nxn. •We can use this to compute Ak quickly for large k.•The matrix D is a diagonal matrix (i.e. entries off the main diagonal are all zeros).

Example

Find a formula for Ak given A=PDP-1 &

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16A

21

11P

40

05D

11

121P

Diagonalizable

Matrix “A” is diagonalizable if A=PDP-1 where P is invertible and D is a diagonal matrix.

Note: AP=PD

When is a matrix diagonalizable?

Let’s examine eigenvalues and eigenvectors of A

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The Diagonalization Theorem

If Anxn & has n linearly independent eigenvectors.

Then1. A=PDP-1

2. Columns of P are eigenvectors3. Diagonals of D are

eigenvalues.

Example

Diagonalize

We need to find P & D

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121

002

A

Theorem

If Anxn has n distinct eigenvalues then A is diagonalizable.

Example

Diagonalize

400

220

642

A

Example

Diagonalize

123

062

002

A