Monday March 23
−
Lecture 31 :
Eigenvectors associated to an eigenvalue.
(Refers to
6.2)
Expectations:
1.
Define the family of all eigenvectors associated to an eigenvalue.
2.
Find all eigenvectors associated to an eigenvalue.
3.
Define an eigenspace.
4.
Recognize that
λ
1
is an eigenvalue of
A
iff 1/
λ
1
is an eigenvalue of
A
1
.
5.
Define algebraic and geometric multiplicity of an eigenvalue.
31.1
Theorem
−
Let
λ
1
be a number. The number
λ
1
is an eigenvalue of
A
iff the the
system
A
x
=
λ
1
x
has a nontrivial solution.
Proof :
The number
λ
1
is an eigenvalue of
A
⇔
is a solution to the characteristic equation det(
A
−
λ
I
) = 0
⇔
det(
A
−
λ
1
I
) = 0
⇔
the matrix
A
−
λ
1
I
is not invertible
⇔
(
A
−
λ
1
I
)
x
= 0 has a nontrivial solution (infinitely many solutions)
⇔
A
x
=
λ
1
x
has a nontrivial solution
as required.
31.1.1
Example
−
Show that 1 is an eigenvalue of the following matrix
A
⎡
2
0
0
⎤
⏐
0
3
1
⏐
⎣
1
0
1
⎦
Solution: By the above theorem we need only verify that the following homogeneous
system has a nontrivial solution
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⎡
2 – 1
0
0
⎤
⏐
0
3 –1
1
⏐
x
= 0
⎣
1
0
1 – 1
⎦
This coefficient matrix easily rowreduces to
⎡
1
0
0
⎤
⏐
0
2
1
⏐
⎣
0
0
0
⎦
Since (
A
– (1)
I
)
x
= 0 has a nontrivial solution then 1 is an eigenvalue of
A
.
So if
λ
1
is
an eigenvalue of
A
then there exists infinitely many vectors
x
such that
A
x
=
λ
1
x
.
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 Winter '08
 All
 Linear Algebra, Eigenvectors, Vectors, Eigenvalue, eigenvector and eigenspace

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