generalized eigenvector

generalized eigenvector

Let

V

be a

vector space

over a field

k

and

T

a

linear transformation

on

V

(a

linear operator

). A non-zero vector

v

∈

V

is said to be a

generalized eigenvector

of

T

(corresponding to

λ

) if there is a

λ

∈

k

and a positive integer

m

such that

(

T

-

λ

⁢

I

)

m

⁢

(

v

)

=

0

,

where

I

is the

identity operator

.

In the equation above, it is easy to see that

λ

is an

eigenvalue

of

T

. Suppose that

m

is the least such integer satisfying the above equation. If

m

=

1

, then

λ

is an eigenvalue of

T

. If

m

>

1

, let

w

=

(

T

-

λ

⁢

I

)

m

-

1

⁢

(

v

)

. Then

w

≠

0

(since

v

≠

0

) and

(

T

-

λ

⁢

I

)

⁢

(

w

)

=

0

, so

λ

is again an eigenvalue of

T

.

Let

v

be a generalized eigenvector of

T

corresponding to the eigenvalue

λ

. We can form a

sequence

v

,

(

T

-

λ

⁢

I

)

⁢

(

v

)

,

(

T

-

λ

⁢

I

)

2

⁢

(

v

)

,

…

,

(

T

-

λ

⁢

I

)

i

⁢

(

v

)

,

…

,

(

T

-

λ

⁢

I

)

m

⁢

(

v

)

=

0

,

0

,

…

The set

C

λ

⁢

(

v

)

of all non-zero terms in the sequence is called a

cycle of generalized eigenvectors

of

T

corresponding to

λ

. The

cardinality

m

of

C

λ

⁢

(

v

)

is its . For any

C

λ

⁢

(

v

)

, write

v

λ

=

(

T

-

λ

⁢

I

)

m

-

1

⁢

(

v

)

.

Below are some

properties

of

C

λ

⁢

(

v

)

:

•

v

λ

is the only

eigenvector

of

λ

in

C

λ

⁢

(

v

)

, for otherwise

v

λ

=

0

.

•

C

λ

⁢

(

v

)

is

linearly independent

.

Proof.

Let

v

i

=

(

T

-

λ

⁢

I

)

i

-

1

⁢

(

v

)

, where

i

=

1

,

…

,

m

. Let

0

=

∑

i

=

1

m

r

i

⁢

v

i

with

r

i

∈

k

. Induct on

i

. If

i

=

1

, then

v

1

=

v

≠

0

, so

r

1

=

0

and

{

v

1

}

is linearly independent. Suppose the property is true when

i

=

m

-

1

. Apply

T

-

λ

⁢

I

to the equation, and we have

0

=

∑

i

=

1

m

r

i

⁢

(

T

-

λ

⁢

I

)

⁢

(

v

i

)

=

∑

i

=

1

m

-

1

r

i

⁢

v

i

+

1

. Then

r

1

=

⋯

=

r

m

-

1

=

0

by

induction

. So

0

=

r

m

⁢

v

m

=

r

m

⁢

v

λ

and thus

r

m

=

0

since

v

λ

is an eigenvector and is non-zero.
∎

•

More generally, it can be shown that

C

λ

⁢

(

v

1

)

∪

⋯

∪

C

λ

⁢

(

v

k

)

is linearly independent whenever

{

v

1

⁢

λ

,

…

,

v

k

⁢

λ

}

is.

•

Let

E

=

span

⁡

(

C

λ

⁢

(

v

)

)

. Then

E

is a

(

m

+

1

)

-dimensional

subspace

of the

generalized eigenspace

of

T

corresponding to

λ

. Furthermore, let

T

|

E

be the

restriction

of

T

to

E

, then

[

T

|

E

]

C

λ

⁢

(

v

)

is a

Jordan block

, when

C

λ

⁢

(

v

)

is ordered (as an

ordered basis

) by setting

(

T

-

λ

⁢

I

)

i

⁢

(

v

)

<

(

T

-

λ

⁢

I

)

j

⁢

(

v

)

whenever

i

>

j

.

Indeed, for if we let

w

i

=

(

T

-

λ

⁢

I

)

m

+

1

-

i

⁢

(

v

)

for

i

=

1

,

…

⁢

m

+

1

, then

T

⁢

(

w

i

)

=

(

T

-

λ

⁢

I

+

λ

⁢

I

)

⁢

(

T

-

λ

⁢

I

)

m

+

1

-

i

⁢

(

v

)

=

{

λ

⁢

w

i

if

⁢

i

=

1

,

w

i

-

1

+

λ

⁢

w

i

otherwise.

so that

[

T

|

E

]

C

λ

⁢

(

v

)

is the

(

m

+

1

)

×

(

m

+

1

)

matrix given by

(

λ

1

0

⋯

0

0

λ

1

⋯

0

0

0

λ

⋯

0

⋮

⋮

⋮

⋱

⋮

0

0

0

⋯

1

0

0

0

⋯

λ

)

•

A cycle of generalized eigenvectors is called

maximal

if

v

∉

(

T

-

λ

⁢

I

)

⁢

(

V

)

. If

V

is finite dimensional, any cycle of generalized eigenvectors

C

λ

⁢

(

v

)

can always be extended to a maximal cycle of generalized eigenvectors

C

λ

⁢

(

w

)

, meaning that

C

λ

⁢

(

v

)

⊆

C

λ

⁢

(

w

)

.

•

In particular, any eigenvector

v

of

T

can be extended to a maximal cycle of generalized eigenvectors. Any two maximal cycles of generalized eigenvectors extending

v

span the same subspace of

V

.

References

1

Friedberg, Insell, Spence.

Linear Algebra

. Prentice-Hall Inc., 1997.

Title

generalized eigenvector

Canonical

name

GeneralizedEigenvector

Date of creation

2013-03-22 17:23:13

Last modified on

2013-03-22 17:23:13

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

13

Author

CWoo (3771)

Entry type

Definition

Classification

msc 65F15

Classification

msc 65-00

Classification

msc 15A18

Classification

msc 15-00

Related topic

GeneralizedEigenspace

Defines

cycle of generalized eigenvectors

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