Worksheet: generalized eigenvectors

Let

V

be a finite-dimensional vector space, and let

T : V \to V

be a linear operator. Assume that

T

has all real eigenvalues (alternatively, assume we’re working over the complex numbers). Let

A

be the matrix of

T

with respect to some standard basis

B_{0}

of

V .

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Our goal will be to replace the basis

B_{0}

with a basis

B

such that the matrix of

T

with respect to

B

is as simple as possible. (Where we agree that the "simplest" possible matrix would be diagonal.)

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Recall the following results that we’ve observed so far:

The characteristic polynomial $c_{T} (x)$ of $T$ does not depend on the choice of basis.
The eigenvalues of $T$ are the roots of this polynomial.
The eigenspaces $E_{λ} (T)$ are $T$ -invariant subspaces of $V .$
The matrix $A$ can be diagonalized if and only if there is a basis of $V$ consisting of eigenvectors of $T .$
Suppose

$c_{T} (x) = (x - λ_{1})^{m_{1}} (x - λ_{2})^{m_{2}} \dots (x - λ_{k})^{m_{k}} .$

Then $A$ can be diagonalized if and only if $\dim E_{λ_{i}} (T) = m_{i}$ for each $i = 1, \dots, k .$

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In the case where

A

can be diagonalized, we have the direct sum decomposition

V = E_{λ_{1}} (T) \oplus E_{λ_{2}} (T) \oplus \dots \oplus E_{λ_{k}} (T) .

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The question is: what do we do if there aren’t enough eigenvectors to form a basis of

V ?

When that happens, the direct sum of all the eigenspaces will not give us all of

V .

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The idea: replace

E_{λ_{j}} (T)

with a generalized eigenspace

G_{λ_{j}} (T)

whose dimension is

m_{i} .

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Our candidate: instead of

E_{λ} (T) = \ker (T - λ I),

we use

G_{λ} (T) = \ker ((T - λ I)^{m}),

where

m

is the multiplicity of

λ .

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1.

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Recall that in class we proved that

\ker (T)

and

im (T)

are

T

-invariant subspaces. Let

p (x)

be any polynomial, and prove that

\ker (p (T))

and

im (p (T))

are also

T

-invariant.

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Hint: first show that

p (T) T = T p (T)

for any polynomial

T .

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Applying the result of Problem 1 to the polynomial

p (x) = (x - λ)^{m}

shows that

G_{λ} (T)

is

T

-invariant. It is possible to show that

\dim G_{λ} (T) = m

but I won’t ask you to do that. (A proof is in the book by Nicholson if you really want to see it.)

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Instead, we will try to understand what’s going on by exploring an example.

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Consider the following matrix.


    
        
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from sympy import *
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init_printing()
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A=Matrix([[2,0,0,1,0],[-1,0,1,2,3],[0,1,2,0,-1],[-2,-3,2,5,3],[0,-1,0,1,4]])
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A

    
    
    
    
        
            
                Language:
                
            
        
    
    




    
    
        
        Messages

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2.

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Find (and factor) the characteristic polynomial of

A .

For the commands you might need, refer to the textbook ¹.

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3.

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Find the eigenvectors. What are the dimensions of the eigenspaces? Based on this observation, can

A

be diagonalized?

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4.

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Prove that for any

n \times n

matrix

A,

we have

{0} \subseteq null (A) \subseteq null (A^{2}) \subseteq \dots \subseteq null (A^{n}) .

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It turns out that at some point, the null spaces stabilize. If

null (A^{k}) = null A^{k + 1}

for some

k,

then

null (A^{k}) = null (A^{k + l})

for all

l \geq 0 .

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5.

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For each eigenvalue found in Worksheet Exercise 5.5.2, compute the nullspace of

A - λ I,

(A - λ I)^{2},

(A - λ I)^{3},

etc. until you find two consecutive nullspaces that are the same.

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By Worksheet Exercise 5.5.4, any vector in

null (A - λ I)^{m}

will also be a vector in

null (A - λ I)^{m + 1} .

In particular, at each step, we can find a basis for

null (A - λ I)^{m}

that includes the basis for

null (A - λ I)^{m - 1} .

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For each eigenvalue found in Worksheet Exercise 5.5.2, determine such a basis for the corresponding generalized eigenspace. You will want to list your vectors so that the vectors from the basis of the nullspace for

A - λ I

come first, then the vectors for the basis of the nullspace for

(A - λ I)^{2},

and so on.

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6.

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Finally, let’s see how all of this works. Let

P

be the matrix whose columns consist of the vectors found in Problem 4. What do you get when you compute the matrix

P^{- 1} A P ?

Linear Algebra: A second course, featuring proofs and Python

Worksheet 5.5 Worksheet: generalized eigenvectors
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