10
Eigenvalues
and
Eigenvectors
Fall 2003
Introduction
To introduce the concepts of
eigenvalues
and
eigenvectors
, we consider first a three
dimensional space with a Cartesian coordinate system.
Consider a vector from the origin
O
to a point
P
;
call this vector
a
.
The components of
a
are
(
a
1
,
a
2
,
a
3
).
Alternatively,
we could say that point
P
has coordinates
(
a
1
,
a
2
,
a
3
).
We can apply a linear transformation to vector
a
to obtain another vector
z
For any
linear transformation, each component of
z
is some linear combination of the
components of
a
This relationship can be expressed as
z
S a
S a
S a
z
S a
S a
S a
z
S a
S a
S a
1
11
1
12
2
13
3
2
21
1
22
2
23
3
3
31
1
32
2
33
3
=
+
+
=
+
+
=
+
+
,
,
.
or
z
S a
i
i j
j
j
=
=
∑
.
1
3
(1)
This can be expressed more compactly by use of matrix language.
We define:
a
z
S
=
F
H
G
G
I
K
J
J
=
F
H
G
G
I
K
J
J
=
F
H
G
G
I
K
J
J
a
a
a
z
z
z
S
S
S
S
S
S
S
S
S
1
2
3
1
2
3
,
,
(2)
Then
z
z
z
S
S
S
S
S
S
S
S
S
a
a
a
S a
S a
S a
S a
S a
S a
S a
S a
S a
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
F
H
G
G
I
K
J
J
=
F
H
G
G
I
K
J
J
F
H
G
G
I
K
J
J
=
+
+
+
+
+
+
F
H
G
G
I
K
J
J
,
(3)
or simply
z
=
S
a
(4)
In general, the
direction
of vector
z
is different from that of
a
But there may be special
cases where
z
has the
same
direction as
a.
For example, suppose the transformation
S
represents a rotation of vector
a
about some fixed axis.
If the direction of
a
happens
to coincide with this axis, then
a
is not changed by this transformation, and
z
=
a
.
More generally, if
z
has the same
direction
(but not necessarily the same magnitude) as
a
then
z
must be a scalar multiple of
a.
That is,
z
=
λ
a,
where
λ
is a
scalar
In that
case,
Sa =
λ
a,
(5)
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10
Eigenvalues and Eigenvectors
or
S
S
S
S
S
S
S
S
S
a
a
a
a
a
a
a
a
a
11
12
13
21
22
23
31
32
33
1
2
3
1
2
3
1
2
3
F
H
G
G
I
K
J
J
F
H
G
G
I
K
J
J
=
F
H
G
G
I
K
J
J
=
F
H
G
G
I
K
J
J
λ
λ
λ
λ
.
(6)
In this case, the result of the transformation
S
applied to the vector
a
is another vector
having the
same
direction
as
a
.
A vector
a
for which
Eqs. (5) and (6)
are valid
is
called an
eigenvector
of the transformation
S
,
and the scalar
λ
is the corresponding
eigenvalue
.
Finding
Eigenvectors
Several questions immediately arise:
(1) How do we know that eigenvectors
exist
, for any given transformation
S
?
(2) If eigenvectors (and their corresponding eigenvalues)
do
exist, how can we find
them?
(3) Can there be more than one eigenvector for a given transformation
S
?
If so, how
are the different eigenvectors and eigenvalues related?
We'll now try to answer these questions.
First, Eq. (6) can be combined with Eq. (3) and
rearranged as follows:
S a
S a
S a
S a
S a
S a
S a
S a
S a
a
a
a
11
1
12
2
13
3
21
1
22
2
23
3
31
1
32
2
33
3
1
2
3
+
+
+
+
+
+
F
H
G
G
I
K
J
J
=
F
H
G
G
I
K
J
J
λ
λ
λ
(7)
Equating corresponding elements in Eq. (7) and rearranging, we obtain the set of three
scalar equations:
(
)
,
(
)
,
(
)
.
S
a
S a
S a
S a
S
a
S a
S a
S a
S
a
1
2
3
1
2
3
1
2
3
0
0
0

+
+
=
+

+
=
+
+

=
λ
λ
λ
.
(8)
If an eigenvector
a
exists, its components
(
a
1
,
a
2
,
a
3
)
and its eigenvalue
λ
must satisfy
Eqs. (8).
This is a set of three simultaneous, linear,
homogeneous
equations.
(I.e., every
term contains an
a
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 Fall '08
 Clarke
 Linear Algebra, Eigenvectors, Heat, Eigenvalues, JJ GG

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