Math 20 Midterm 2 Review
Carolyn Stein
Exam Date: November 9, 2011
Eigenvectors and Eigenvalues
What are Eigenvectors and Eigenvalues?
"Eigen" is German for "own" or "characteristic"
If
A
~
v
=
λ~
v
, we say
~
v
is an eigenvector of matrix
A
with an associated eigenvalue
of
λ
If we view
A
as a transformation matrix, the eigenvectors are all vectors that
are only scaled by the transformation
Example:
Find the eigenvectors and eigenvalues of the following matrices us
ing geometric intuition:
✓
5
0
0
5
◆
✓
0
1
1
0
◆
Finding Eigenvalues
A
~
v
=
λ~
v
)
A
~
v

λ~
v
= 0
)
A
~
v

λ
I
~
v
= 0
)
(
A

λ
I
)
~
v
= 0
We don’t want
~
v
= 0
, but we want
(
A

λ
I
)
~
v
= 0
.
For this to be true, the
columns of
(
A

λ
I
)
must be linearly dependent, which implies:
1
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det (
A

λ
I
) = det
0
@
a
11

λ
a
12
a
13
a
21
a
22

λ
a
23
a
31
a
32
a
33

λ
1
A
= 0
Solving this will yield an equation in the form
(
λ

λ
1
)
m
1
(
λ

λ
2
)
m
2
...
(
λ

λ
n
)
m
n
This equation is known as the
characteristic polynomial
λ
1
,
λ
2
, ...,
λ
n
are the
eigenvalues
Each eigenvalue has an associated
algebraic multiplicitiy
m
1
, m
2
, ...m
n
Example:
Find the eigenvalues and algebraic multiplicities of
A
=
✓
1
2
4
3
◆
Finding Eigenvectors
For each eigenvalue
λ
, we can find the associated
eigenvectors
by solving:
(
A

λ
I
)
~
v
= 0
The space of solutions to the system is known as the
eigenspace
of
λ
The dimension of the eigenspace is known as the
geometric multiplicity
of
λ
Example:
Find the eigenspaces and geometric multiplicities of
A
=
✓
1
2
4
3
◆
Eigenbases and Diagonalization
An
eigenbasis
is a basis made exclusively of linearly independent eigenvectors.
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 Fall '11
 RachelEpstein
 Eigenvectors, Derivative, Vectors, Gradient, stationary points

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