What Are Eigenvalues and Eigenvectors? 211

What Are Eigenvalues and

Eigenvectors?

What do you say we

start o with a few

problems?

Sure.

H...

Like this?

So close!

Oh, like

this?

Exactly!

So...the answer can

be expreed using

multiples of the

original two vectors?

Okay, first problem.

Find the image of

using the linear

transformation

determined by the

2×2 matrix

3

1

1

2

c

1

+ c

2

8

2

−3

1

(where c

1

and c

2

are real numbers).

Like so.

Oh...

212 Chapter 8 Eigenvalues and Eigenvectors

That’s right! So you

could say that the

linear transformation

equal to the matrix

8

2

−3

1

...transforms All points

on the

x

1

x

2

plane...

Like this?

Coect.

So this solution can

be expreed with

multiples as we...

H

Let's move on to another problem.

Find the image of using

(where

c

1

,

c

2

, and

c

3

are real numbers).

c

1

+ c

2

+ c

3

1

0

0

0

1

0

0

0

1

4

0

0

0

2

0

0

0

−1

the linear transformation

determined by the

3×3

matrix

What Are Eigenvalues and Eigenvectors? 213

214 Chapter 8 Eigenvalues and Eigenvectors

4

0

0

0

2

0

0

0

−1

...transforms every

point in the

x

1

x

2

x

3

space...

So you could

say that the

linear transformation

equal to the matrix

Like this.

I get

it!

4

0

0

0

2

0

0

0

−1

...transforms every

point in the

x

1

x

2

x

3

space...

So you could

say that the

linear transformation

equal to the matrix

Times 4

Lking

back

Times 2

kping those

examples in mind.

Eigenvalues and eigenvectors

a

11

a

21

a

n1

a

12

a

22

a

n2

a

1n

a

2n

a

nn

x

1

x

2

x

n

x

1

x

2

x

n

x

1

x

2

x

n

If the image of a vector through the linear transformation determined by the matrix

is equal to

λ

,

λ

is said to be an eigenvalue to the matrix,

and is said to be an eigenvector corresponding to the eigenvalue

λ

.

The zero vector can never be an eigenvector.

R

n

R

n

x

1

x

2

x

n

λ

x

1

x

2

x

n

So the two

examples could

be suarized

like this?

Exactly!

You can generay

never find more than n

dierent eigenvalues

and eigenvectors for

any n×n matrix.

Oh...

g

s

an

d

Ei

ge

nv

e

c

t

o

rs

Let's have a

lk at the

definition...

Matrix

Eigenvalue

Eigenvector

4

0

0

0

2

0

0

0

−1

8

2

−3

1

λ = 7, 2 λ = 4, 2, −1

the vector

corresponding

to

λ = 7

3

1

the vector

corresponding

to

λ = 2

1

2

the vector

corresponding

to

λ = 4

1

0

0

the vector

corresponding

to

λ = 2

0

1

0

the vector

corresponding

to

λ = 1

0

0

1

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