The Eigenvalue Problems  8.8
1. Definition of Eigenvalues and Eigenvectors:
Let
A
be an
n
!
n
matrix. A scalar
!
is said to be an
eigenvalue
of
A
if the linear system
A
v
"
!
v
has a
nonzero
solution vector
v
. The solution vector
v
is said to be an
eigenvector
of
A
corresponding to the
eigenvalue
!
. The pair
!
,
v
is called an eigenpair of
A
. Note that the left side of the equation is a matrix
vector multiplication and the right side is a scalar vector multiplication.
Example
Let A
"
2
!
1
!
1
2
,
u
"
1
2
,
v
1
"
1
1
,
v
2
"
1
!
1
.
Determine if each of
u
,
v
1
,
v
2
is an eigenvector of A
.
If so, what is its eigenvalue? Sketch the graphs of
u
,
v
1
,
v
2
and A
u
,
Av
1
,
A
v
2
.
A
u
"
0
3
"
"
1
2
, so
u
is not an eigenvector of
A
.
A
v
1
"
1
1
"
v
1
"
!
1
"
v
1
, so
v
1
is an eigenvector of
A
corresponding to the eigenvalue
!
"
1.
A
v
2
"
3
!
3
"
3
1
!
1
"
3
v
2
, so
v
1
is an eigenvector of
A
corresponding to the eigenvalue
!
"
3
3
2
1
0
1
2
3
3
2
1
1
2
3

u
,
...
Au
3
2
1
0
1
2
3
3
2
1
1
2
3

v
1
,
...
Av
1
3
2
1
0
1
2
3
3
2
1
1
2
3

v
2
,
...
Av
2
How can we find all pairs of eigenvalues and eigenvectors of a given
A
? Observe that an eigenpair
!
,
v
satisfies the equation:
Av
"
!
v
, that implies that
Av
!
!
v
"
!
A
!
!
I
"
v
"
0.
Since
v
"
0, the homogeneous system
!
A
!
!
I
"
v
"
0 has a nonzero solution if and only if the matrix
A
!
!
I
is singular or equivalently det
!
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 Spring '12
 MUNK
 Linear Algebra, Matrices, Eigenvalue, eigenvector and eigenspace, Diagonal matrix, Det

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