Generalized Eigenvectors

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Generalized

Eigenvectors

Math 240

Definition

Computationand Properties

Chains

Generalized Eigenvectors

Math 240 Calculus III

Summer 2013, Session II

Wednesday, July 31, 2013


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Generalized

Eigenvectors

Math 240

Definition


Chains

Agenda

1. Definition

2. Computation and Properties

3. Chains


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Generalized

Eigenvectors

Math 240

Definition


Chains

Definition

Definition

IfA is an n n matrix, a generalized eigenvector ofAcorresponding to the eigenvalue is a nonzero vector x

satisfying

(A I)p x= 0

for some positive integer p. Equivalently, it is a nonzeroelement of the nullspace of(A I)p.

Example

Eigenvectors are generalized eigenvectors with p= 1.

In the previous example we saw that v= (1, 0) and

u= (0, 1) are generalized eigenvectors for

A=

1 10 1

and = 1.


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Generalized

Eigenvectors

Math 240

Definition


Chains

Computing generalized eigenvectors

Example

Determine generalized eigenvectors for the matrix

A=

1 1 00 1 2

0 0 3

.

1. Characteristic polynomial is (3 )(1 )2.2. Eigenvalues are = 1, 3.

3. Eigenvectors are

1= 3 : v1= (1, 2, 2),

2= 1 : v2= (1, 0, 0).

4. Final generalized eigenvector will a vector v3= 0 suchthat

(A 2I)2v3= 0 but (A 2I)v3= 0.

Pick v3= (0, 1, 0). Note that (A 2I)v3= v2.


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Math 240

Definition


Chains

Facts about generalized eigenvectors

How many powers of(A I) do we need to compute in orderto find all of the generalized eigenvectors for ?

FactIfA is an n n matrix and is an eigenvalue with algebraicmultiplicityk, then the set of generalized eigenvectors for

consists of the nonzero elements of nullspace

(A I)k

.

In other words, we need to take at most k powers ofA I to

find all of the generalized eigenvectors for .


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Generalized

Eigenvectors

Math 240

Definition


Chains


Example


A=

1 2 01 1 2

0 1 1

.

1. Single eigenvalue of= 1.

2. Single eigenvector v1= (2, 0, 1).

3. Look at

(A I)2 = 2 0 40 0 01 0 2

to find generalized eigenvector v2= (0, 1, 0).

4. Finally, (A I)3 = 0, so we get v3= (1, 0, 0).


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Generalized

Eigenvectors

Math 240

Definition


Chains

Facts about generalized eigenvectors

The aim of generalized eigenvectors was to enlarge a set oflinearly independent eigenvectors to make a basis. Are there

always enough generalized eigenvectors to do so?

FactIf is an eigenvalue of A with algebraic multiplicityk, then

nullity

(A I)k

=k.

In other words, there are k linearly independent generalized

eigenvectors for .

Corollary

IfA is an n n matrix, then there is a basis forRn consistingof generalized eigenvectors of A.


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Generalized

Eigenvectors

Math 240

Definition


Chains


Example


A=

1 2 0

1 1 20 1 1

.

1. From last time, we have eigenvalue = 1 and eigenvector

v1= (2, 0, 1).

2. Solve (A I)v2= v1 to get v2= (0,1, 0).3. Solve (A I)v3= v2 to get v3= (1, 0, 0).


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Eigenvectors

Math 240

Definition


Chains

Chains of generalized eigenvectors

Let A be an n n matrix and v a generalized eigenvector ofAcorresponding to the eigenvalue . This means that

(A I)p v= 0

for a positive integer p.

If0q < p, then

(A I)pq

(A I)q

v= 0.That is, (A I)qv is also a generalized eigenvectorcorresponding to forq= 0, 1, . . . , p 1.

Definition

Ifp is the smallest positive integer such that (A I)p

v= 0,then the sequence

(A I)p1 v, (A I)p2 v, . . . , (A I)v, v

is called a chain or cycle of generalized eigenvectors. The

integer p is called the length of the cycle.


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Generalized

Eigenvectors

Math 240

Definition


Chains

Chains of generalized eigenvectors

ExampleIn the previous example,

A I=

0 2 01 0 20 1 0

and we found the chain

v=

10

0

, (A I)v=

01

0

, (A I)2v=

20

1

.

FactThe generalized eigenvectors in a chain are linearly

independent.


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Generalized

Eigenvectors

Math 240

Definition

Computation

and Properties

Chains

Jordan canonical form

Whats the analogue of diagonalization for defective matrices?

That is, if{v1,v2, . . . ,vn} are the linearly independentgeneralized eigenvectors ofA, what does the matrix S1AS

look like, where S=v1 v2 vn

?

Generalized Eigenvectors

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