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### MA2213 Lecture 8

Eigenvectors

Application of Eigenvectors

Vufoil 18, lecture 7 : The Fibonacci sequence satisfies

Another Biomathematics Application

Leonardo da Pisa, better known as Fibonacci, invented his famous sequence to compute the reproductive success of rabbits* Similar sequences describe frequencies in males, females of a sex-linked gene.

For genes (2 alleles) carried in the X chromosome**

The solution has the form

where

*page i, ** pages 10-12 in The Theory of Evolution and Dynamical Systems ,J. Hofbauer and K. Sigmund, 1984.

Eigenvector Problem (pages 333-351)

is a square matrix then a nonzero

Recall that if

vector

is an eigenvector corresponding to the

eigenvalue

if

Eigenvectors and eigenvalues arise in biomathematics where they describe growth and population genetics

They arise in numerical solution of linear equations because they determine convergence properties

They arise in physical problems, especially those that involve vibrations in which eigenvalues are related to vibration frequencies

Example 7.2.1 pages 333-334

For

the eigenvalue-eigenvector pairs are

and

We observe that every (column) vector

where

Example 7.2.1 pages 333-334

Therefore, since x Ax is a linear transformation

and since

are eigenvectors

We can repeat this process to obtain

?

Question What happens as

Example 7.2.1 pages 333-334

General Principle : If a vector v can be expressed as a linear combination of eigenvectors of a matrix A, then it is very easy to compute Av

It is possible to express every vector as a linear combination of eigenvectors of an n by n matrix A iff either of the following equivalent conditions is satisfied :

(i) there exists a basis consisting of eigenvectors of A

(ii) the sum of dimensions of eigenspaces of A = n

Question Does this condition hold for

?

Question What special form does this matrix have ?

Example 7.2.1 pages 333-334

The characteristic polynomial of

is

2 is the (only) eigenvalue, it has algebraicmultiplicity 2

so the eigenspace

for eigenvalue 5

has dimension 1

the eigenvalue 5 is said to have geometricmultiplicity 1

Question What are alg.&geom. mult. in Example 7.2.7 ?

Characteristic Polynomials pp. 335-337

Example 7.22 (p. 335) The eigenvalue-eigenvector pairs

in Example 7.2.1 are

of the matrix

corresponding eigenvectors

Question What is the equation for

?

Eigenvalues of Symmetric Matrices

The following real symmetric matrices that we studied

have real eigenvalues and eigenvectors correspondingto distinct eigenvectors are orthogonal.

Question What are the eigenvalues of these matrices ?

Question What are the corresponding eigenvectors ?

Question Compute their scalar products

Eigenvalues of Symmetric Matrices

Theorem 1. All eigenvalues of real symmetric matrices

are real valued.

Proof For a matrix

with complex (or real) entries

let

denote the matrix whose entries are the

complex conjugates of the entries of

Question Prove

is real (all entries are real) iff

Question Prove

Assume that

and observe that

therefore

and

Eigenvalues of Symmetric Matrices

Theorem 2. Eigenvectors of a real symmetric matrix that

correspond to distinct eigenvalues are orthogonal.

Proof Assume that

Then compute

and observe that

Orthogonal Matrices

Definition A matrix

is orthogonal if

If

is orthogonal then

therefore either

or

so

is nonsingular and has an inverse

hence

so

Examples

Permutation Matrices

Definition A matrix

is called a permutation

matrix if there exists a function (called a permutation)

that is 1-to-1 (and therefore onto) such that

Examples

Question Why is every permutation matrix orthogonal ?

Eigenvalues of Symmetric Matrices

Theorem 7.2.4 pages 337-338 If

is symmetric

of

then there exists a set

eigenvalue-eigenvector pairs

Proof Uses Theorems 1 and 2 and a little linear algebra.

Choose eigenvectors so that

construct matrices

and observe that

MATLAB EIG Command

>> help eig

EIG Eigenvalues and eigenvectors.

E = EIG(X) is a vector containing the eigenvalues of a square

matrix X.

[V,D] = EIG(X) produces a diagonal matrix D of eigenvalues and a

full matrix V whose columns are the corresponding eigenvectors so

that X*V = V*D.

[V,D] = EIG(X,'nobalance') performs the computation with balancing

disabled, which sometimes gives more accurate results for certain

problems with unusual scaling. If X is symmetric, EIG(X,'nobalance')

is ignored since X is already balanced.

E = EIG(A,B) is a vector containing the generalized eigenvalues

of square matrices A and B.

[V,D] = EIG(A,B) produces a diagonal matrix D of generalized

eigenvalues and a full matrix V whose columns are the

corresponding eigenvectors so that A*V = B*V*D.

EIG(A,B,'chol') is the same as EIG(A,B) for symmetric A and symmetric

positive definite B. It computes the generalized eigenvalues of A and B

using the Cholesky factorization of B.

EIG(A,B,'qz') ignores the symmetry of A and B and uses the QZ algorithm.

In general, the two algorithms return the same result, however using the

QZ algorithm may be more stable for certain problems.

The flag is ignored when A and B are not symmetric.

See also CONDEIG, EIGS.

MATLAB EIG Command

Example 7.2.3 page 336

>> A = [-7 13 -16;13 -10 13;-16 13 -7]

A =

-7 13 -16

13 -10 13

-16 13 -7

>> [U,D] = eig(A);

>> U

U =

-0.5774 0.4082 0.7071

0.5774 0.8165 -0.0000

-0.5774 0.4082 -0.7071

>> D

D =

-36.0000 0 0

0 3.0000 0

0 0 9.0000

>> A*U

ans =

20.7846 1.2247 6.3640

-20.7846 2.4495 -0.0000

20.7846 1.2247 -6.3640

>> U*D

ans =

20.7846 1.2247 6.3640

-20.7846 2.4495 -0.0000

20.7846 1.2247 -6.3640

Positive Definite Symmetric Matrices

is [lec4,slide24]

Theorem 4 A symmetric matrix

(semi) positive definite iff all of its eigenvalues

Proof Let

be the orthogonal, diagonal

matrices on the previous page that satisfy

Then for every

where

Since

is nonsingular

is (semi) positive definite iff

therefore

Clearly this condition holds iff

Singular Value Decomposition

then there

and

Theorem 3 If

exist orthogogonal matrices

has the form

such that

where

Singular Values

= sqrt eig

Proof Outline Choose

so

and

are diagonal, then

satisfies

try to finish

MATLAB SVD Command

>> help svd

SVD Singular value decomposition.

[U,S,V] = SVD(X) produces a diagonal matrix S, of the same dimension as X and with nonnegative diagonal elements in decreasing order, and unitary matrices U and V so that X =U*S*V'.

S = SVD(X) returns a vector containing the singular values.

[U,S,V] = SVD(X,0) produces the "economy size“ decomposition. If X is m-by-n with m > n, then only the first n columns of U are computed and S is n-by-n.

See also SVDS, GSVD.

MATLAB SVD Command

>> M = [0 1; 0.5 0.5]

M =

0 1.0000

0.5000 0.5000

>> [U,S,V] = svd(M)

U =

-0.8507 -0.5257

-0.5257 0.8507

S =

1.1441 0

0 0.4370

V =

-0.2298 0.9732

-0.9732 -0.2298

>> U*S*V'

ans =

0.0000 1.0000

0.5000 0.5000

Square Roots

Theorem 5 A symmetric positive definite matrix

has a symmetric positive definite ‘square root’.

Proof Let

be the orthogonal, diagonal

matrices on the previous page that satisfy

Then construct the matrices

and observe that

is symmetric positive definite

and satisfies

Polar Decomposition

Theorem 6 Every nonsingular matrix

can be factored as

where

is symmetric

and positive definite and

is orthogonal.

Proof Construct

and observe that

is symmetric and positive definite. Let

be symmetric

and construct

positive definite and satisfy

Then

and clearly

Löwdin Orthonormalization

(1) Per-Olov Löwdin, On the Non-Orthogonality Problem Connected with the use of Atomic Wave Functions in the Theory of Molecules and Crystals, J. Chem. Phys. 18, 367-370 (1950).

http://www.quantum-chemistry-history.com/Lowdin1.htm

in an inner product space

Proof Start with

(assumed to be linearly independent), compute the

Gramm matrix

Since

is symmetric and positive definite, Theorem 5

gives (and provides a method to compute) a matrix

that is symmetric and positive definite and

are orthonormal.

Then

The Power Method pages 340-345

Finds the eigenvalue with largest absolute value of a

whose eigenvalues satisfy

matrix

Step 1 Compute a vector with random entries

Step 2 Compute

and

and

Step 3 Compute

( recall that

)

Step 4 Compute

and

and

Repeat

Then

with

The Inverse Power Method

Result If

is an eigevector of

then

corresponding to eigenvalue

and

is an eigenvector of

corresponding to

eigenvalue

Furthermore, if

then

corresponding to

is an eigenvector of

eigenvalue

Definition The inverse power method is the power

method applied to the matrix

It can find the eigenvalue-eigenvector pair if there

is one eigenvalue that has smallest absolute value.

Inverse Power Method With Shifts

Computes eigenvalue

of

closest to

and a corresponding eigenvector

Step 1 Apply 1 or more interations of the power method

to estimate an eigenvalue

using the matrix

- eigenvector pair

- better estimate of

Step 2 Compute

Step 3 Apply 1 or more interations of the power method

using the matrix

to estimate an eigenvalue

and iterate. Then

- eigenvector pair

with cubic rate of convergence !

Unitary and Hermitian Matrices

Definition The adjoint of a matrix

is the matrix

Example

Definition A matrix

is unitary if

Definition A matrix

is hermitian if

is (semi) positive definite

Definition A matrix

if

(or self-adjoint)

Super Theorem : All previous theorems true for complex

matrices if orthogonal is replaced by unitary, symmetric by hermitian, and old with new (semi) positive definite.

Homework Due Tutorial 5 (Week 11, 29 Oct – 2 Nov)

1. Do Problem 1 on page 348.

2. Read Convergence of the Power Method (pages 342-346)

and do Problem 16 on page 350.

3. Do problem 19 on pages 350-351.

4. Estimate eigenvalue-eigenvector pairs of the

matrix M using the power and inverse power

methods – use 4 iterations and compute errors

5. Compute the eigenvalue-eigenvector

pairs of the orthogonal matrix O

6. Prove that the vectors

defined at the bottom of

slide 29 are orthonormal by computing their inner products

We have discussed several matrix decompositions :

LU

Eigenvector

Singular Value

Polar

Find out about other matrix decompositions. How are

they derived / computed ? What are their applications ?

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