eigenvalue

eigenvalue

Let

V

be a

vector space

over a field

k

, and let

A

be an

endomorphism

of

V

(meaning a

linear mapping

of

V

into itself).
A scalar

λ

∈

k

is said to be an

eigenvalue

of

A

if there is a nonzero

x

∈

V

for which

A

⁢

x

=

λ

⁢

x

.

(1)

Geometrically, one thinks of a vector whose

direction

is unchanged
by the action of

A

, but whose magnitude is multiplied by

λ

.

If

V

is finite dimensional,

elementary

linear algebra

shows that
there are several

equivalent

definitions

of an eigenvalue:

(2) The linear mapping

B

=

λ

⁢

I

-

A

i.e.

B

:

x

↦

λ

⁢

x

-

A

⁢

x

, has no

inverse

.

(3)

B

is not

injective

.

(4)

B

is not

surjective

.

(5)

det

⁡

(

B

)

=

0

, i.e.

det

⁡

(

λ

⁢

I

-

A

)

=

0

.

But if

V

is of

infinite

dimension

, (5) has no meaning and the
conditions (2) and (4) are not equivalent to (1).
A scalar

λ

satisfying (2) (called a

spectral value

of

A

) need not be an eigenvalue. Consider for example the

complex
vector space

V

of all

sequences

(

x

n

)

n

=

1

∞

of

complex
numbers

with the

obvious

operations

, and the map

A

:

V

→

V

given by

A

⁢

(

x

1

,

x

2

,

x

3

,

…

)

=

(

0

,

x

1

,

x

2

,

x

3

,

…

)

.

Zero is a spectral value of

A

, but clearly not an eigenvalue.

Now suppose again that

V

is of finite dimension, say

n

.
The

function

χ

⁢

(

λ

)

=

det

⁡

(

B

)

is a

polynomial

of degree

n

over

k

in the

variable

λ

, called the

characteristic polynomial

of the
endomorphism

A

. (Note that some writers define the characteristic
polynomial as

det

⁡

(

A

-

λ

⁢

I

)

rather than

det

⁡

(

λ

⁢

I

-

A

)

, but the
two have the same zeros.)

If

k

is

ℂ

or any other algebraically closed field, or if

k

=

ℝ

and

n

is odd, then

χ

has at least one zero, meaning that

A

has at least one eigenvalue. In no case does

A

have more than

n

eigenvalues.

Although we didn’t need to do so here, one can compute the coefficients
of

χ

by introducing a basis of

V

and the corresponding matrix for

B

. Unfortunately, computing

n

×

n

determinants

and finding roots
of polynomials of degree

n

are computationally messy procedures
for even moderately large

n

, so for most practical purposes
variations on this naive scheme are needed. See the

eigenvalue
problem

for more information.

If

k

=

ℂ

but the coefficients of

χ

are real (and in particular if

V

has a basis for which the matrix of

A

has only real entries), then
the non-real eigenvalues of

A

appear in

conjugate

pairs. For example,
if

n

=

2

and, for some basis,

A

has the matrix

A

=

(

0

-

1

1

0

)

then

χ

⁢

(

λ

)

=

λ

2

+

1

, with the two zeros

±

i

.

Eigenvalues are of relatively little importance in connection with
an infinite-dimensional vector space, unless that space is endowed with
some additional

structure

, typically that of a

Banach space

or

Hilbert space

. But in those cases the notion is of great value in
physics, engineering, and mathematics proper. Look for “

spectral theory

”
for more on that subject.

Title

eigenvalue

Canonical

name

Eigenvalue

Date of creation

2013-03-22 12:11:52

Last modified on

2013-03-22 12:11:52

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

15

Author

Koro (127)

Entry type

Definition

Classification

msc 15A18

Related topic

EigenvalueProblem

Related topic

SimilarMatrix

Related topic

Eigenvector

Related topic

SingularValueDecomposition

Defines

eigenvalue

Defines

spectral value

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