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Math 20 Midterm 2 Review (Solutions)
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
,wesay
~
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:
✓
50
05
◆
Solution
:
Every vector is an eigenvector with eigenvalue of 5
✓
01
10
◆
Solution
:
Any vector on the line
y
=
x
is an eigenvector with
eigenvalue of 1. Any vector on the line
y
=

x
is an eigenvector with eigenvalue
1.
Finding Eigenvalues
A
~
v
=
v
)
A
~
v

v
=0
)
A
~
v

λ
I
~
v
)
(
A

λ
I
)
~
v
We don’t want
~
v
,bu
tw
ewan
t
(
A

λ
I
)
~
v
.F
o
rt
h
i
st
ob
et
r
u
e
,t
h
e
columns of
(
A

λ
I
)
must be linearly dependent, which implies:
1
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
=
✓
12
43
◆
Solution
:
1, 5 both have algebraic multiplicity of 1
Finding Eigenvectors
For each eigenvalue
λ
,wecan±ndtheassoc
iated
eigenvectors
by solving:
(
A

λ
I
)
~
v
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
=
✓
◆
Solution
:
✏

1
=
t
✓
1

1
◆
✏
5
=
t
✓
1
2
◆
both have geometric multiplicity
of 1
Eigenbases and Diagonalization
An
eigenbasis
is a basis made exclusively of linearly independent eigenvectors.
Matrix
A
has an eigenbasis if the sum of the geometric multiplicities =
n
,the
dimension of the matrix.
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