eigenvalue problem

eigenvalue problem

The general eigenvalue problem

Suppose we have a

vector space

V

and a

linear operator

A

∈

End

⁡

(

V

)

. Then the

eigenvalue problem

is this:

For what values

λ

does the

equation

A

⁢

x

=

λ

⁢

x

have a nonzero solution

x

? For such a

λ

, what are all the
solution vectors

x

?

Values

λ

admitting a solution are called

eigenvalues

; nonzero
solutions

x

are called

eigenvectors

.

The question may be rephrased as a question about the linear operator

(

A

-

λ

⁢

I

)

, where

I

is the

identity

on

V

. Since

λ

⁢

I

is

invertible

whenever

λ

is nonzero, one might expect that

(

A

-

λ

⁢

I

)

should be invertible for “most”

λ

. As
usual, when dealing with infinite-dimensional spaces, the situation is
more complicated.

A special sitation arises when

V

has an

inner product

under which

A

is

self-adjoint

. In this case,

A

has a discrete set of
eigenvalues, and if

x

λ

1

and

x

λ

2

are
eigenvectors corresponding to distinct eigenvalues, then

x

λ

1

and

x

λ

2

are

orthogonal

. In fact, since
the inner product makes

V

into a normed linear space one can find an

orthonormal basis

for

V

consisting entirely of eigenvectors of

A

.

Differential eigenvalue problems

Many problems in physics and elsewhere lead to

differential

eigenvalue problems, that is, problems where the vector space is some space of

differentiable functions

and where the linear operator involves

multiplication

by

functions

and taking

derivatives

. Such problems arise from the method of

separation of variables

, for example. One class of eigenvalue problems that is well-studied are Sturm-Liouville problems, which always lead to self-adjoint operators. The

sequences

of eigenvectors obtained are therefore orthogonal under a suitable inner product.

An example of a Sturm-Liouville problem is this: Find a function

f

⁢

(

x

)

satisfying

f

′′

⁢

(

x

)

=

-

λ

⁢

f

⁢

(

x

)

and

f

⁢

(

0

)

=

f

⁢

(

1

)

=

0

.

Observe that for most values of

λ

, there is only the solution

f

⁢

(

x

)

=

0

. If

λ

=

(

n

⁢

π

)

2

for some

n

, though,

sin

⁡

(

λ

⁢

x

)

is a solution. Observe that if

n

≠

m

, then

∫

0

1

sin

⁡

(

n

⁢

π

⁢

x

)

⁢

sin

⁡

(

m

⁢

π

⁢

x

)

⁢

𝑑

x

=

0

.

Moreover, recalling the

properties

of

Fourier series

, we see that any function satisfying the

boundary conditions

can be written as an

infinite

linear combination

of eigenvectors of this problem.

Many of the families of

special functions

that turn up throughout applied mathematics do so precisely because they are an orthogonal family of eigenvectors for a Sturm-Liouville problem. For example, the

trigonometric functions

sine and cosine and the

Bessel functions

both arise in this way.

Matrix eigenvalue problems

Matrix eigenvalue problems arise in a number of different situations. The eigenvalues of a matrix describe its behaviour in a coordinate-independent way;

theorems

about

diagonalization

allow computation of

matrix powers

efficiently, for example. As a result, matrix eigenvalues are useful in statistics, for example in analyzing Markov chains and in the fundamental theorem of demography.

Matrix eigenvalue problems also arise as the discretization of differential eigenvalue problems.

An example of where a matrix eigenvalue problem arises is the determination of the main axes of a

second order

surface

Q

=

x

T

⁢

A

⁢

x

=

1

(defined by a

symmetric matrix

A

). The task is to find the places where the normal

∇

⁡

(

Q

)

=

(

∂

⁡

Q

∂

⁡

x

1

,

⋯

,

∂

⁡

Q

∂

⁡

x

n

)

=

2

⁢

A

⁢

x

is

parallel

to the vector

x

, i.e

A

⁢

x

=

λ

⁢

x

.

A solution

x

of the above equation with

x

T

⁢

A

⁢

x

=

1

has the

squared

distance

x

T

⁢

x

=

d

2

from the origin. Therefore,

λ

⁢

x

T

⁢

x

=

1

and

d

2

=

1

/

λ

. The main axes are

a

i

=

1

/

λ

i

(

i

=

1

,

…

,

n

)

.

The matrix eigenvalue problem can be written as

(

A

-

λ

⁢

I

)

⁢

x

=

0

.
A non-trivial solution to this system of

n

linear

homogeneous equations

exists if and only if the

determinant

det

⁡

(

A

-

λ

⁢

I

)

=

|

a

11

-

λ

a

12

⋯

a

1

⁢

n

a

21

a

22

-

λ

⋯

a

2

⁢

n

⋮

⋮

⋱

⋮

a

n

⁢

1

a

n

⁢

2

⋯

a

n

⁢

n

-

λ

|

=

0

This

n

th degree

polynomial

in

λ

is called the

characteristic polynomial

. Its roots

λ

are called the eigenvalues and the corresponding vectors

x

eigenvectors. In the example,

x

is a

right eigenvector

for

λ

; a

left eigenvector

y

is defined by

y

T

⁢

A

=

μ

⁢

y

T

.

Numerical eigenvalue problems

Frequently, one wishes to solve the eigenvalue problem approximately (generally on a

computer

). While one can do this using

generic

matrix methods such as

Gaussian elimination

,

L

⁢

U

factorization, and others, these have problems due to roundoff error when attempting to deal with eigenvalue problems. Other methods are necessary. For example, a

Q

⁢

R

-based method is a much more adequate tool ([Golub89]); it works as follows.
Assume that

A

∈

ℝ

n

×

n

is

diagonalizable

. The

Q

⁢

R

iteration

is given by

A

0

=

A

for

k

=

1

,

2

,

…

A

k

=

:

Q

k

R

k

A

k

+

1

:=

R

k

⁢

Q

k

end

At each step, the matrix

Q

k

is orthogonal and

R

k

is

upper triangular

.

Note that

A

k

+

1

=

(

Q

0

⁢

⋯

⁢

Q

k

)

T

⁢

A

⁢

Q

0

⁢

⋯

⁢

Q

k

.

For a full matrix, the

Q

⁢

R

iteration requires

O

⁢

(

n

3

)

flops per step. This is prohibitively expensive, so we first reduce

A

to an

upper Hessenberg matrix

,

H

, using an orthogonal

similarity transformation

:

U

T

⁢

A

⁢

U

=

H

(

H

is upper Hessenberg if

h

i

⁢

j

=

0

for

i

>

j

+

1

). We will use

Householder transformations

to achieve this. Note that if

A

is

symmetric

then

H

is symmetric, and hence

tridiagonal

.

The eigenvalues of

A

are found by applying iteratively the

Q

⁢

R

decomposition

to

H

. These two matrices have the same eigenvalues as they are

similar

.
In particular:

H

=

H

1

is decomposed into

H

1

=

Q

1

⁢

R

1

, then an

H

2

is computed,

H

2

=

R

1

⁢

Q

1

.

H

2

is similar to

H

1

because

H

2

=

R

1

⁢

Q

1

=

Q

1

-

1

⁢

H

1

⁢

Q

1

, and is decomposed to

H

2

=

Q

2

⁢

R

2

. Then

H

3

is formed,

H

3

=

R

2

⁢

Q

2

, etc. In this way a sequence of

H

i

’s (with the same eigenvalues) is generated, that finally

converges

to (for conditions, see [Golub89])

(

λ

1

*

*

⋯

*

*

0

λ

2

*

⋯

*

*

0

0

λ

3

⋯

*

*

⋮

⋮

⋮

⋱

⋮

⋮

0

0

0

⋯

λ

n

-

1

*

0

0

0

⋯

0

λ

n

)

for the Hessenberg and

(

λ

1

0

0

⋯

0

0

0

λ

2

0

⋯

0

0

0

0

λ

3

⋯

0

0

⋮

⋮

⋮

⋱

⋮

⋮

0

0

0

⋯

λ

n

-

1

0

0

0

0

⋯

0

λ

n

)

for the tridiagonal.

References

DAB

Originally from The Data Analysis Briefbook
(

http://rkb.home.cern.ch/rkb/titleA.html

http://rkb.home.cern.ch/rkb/titleA.html)

Golub89

Gene H. Golub and Charles F. van Loan: Matrix Computations, 2nd edn., The John Hopkins University Press, 1989.

Title

eigenvalue problem

Canonical

name

EigenvalueProblem

Date of creation

2013-03-22 12:11:30

Last modified on

2013-03-22 12:11:30

Owner

archibal (4430)

Last modified by

archibal (4430)

Numerical id

22

Author

archibal (4430)

Entry type

Definition

Classification

msc 65F15

Classification

msc 65-00

Classification

msc 15A18

Classification

msc 15-00

Related topic

Eigenvalue

Related topic

Eigenvector

Related topic

SimilarMatrix

Related topic

SolvingTheWaveEquationByDBernoulli

Related topic

TimeDependentExampleOfHeatEquation

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