The Eigenvector/Eigenvalue Method
We first recall the following from linear algebra:
Definition
Let
A
be an
n
×
n
matrix. A vector
v
such that
A
v
=
λ
v
for some scalar
λ
is called an
eigenvector
(EV) of
A
with
eigenvalue
(ev)
λ
.
Before we discuss how these objects can be applied to solving the homogeneous equation
˙
y
=
A
y
,
we review the process for finding the eigenvectors and eigenvalues of a given matrix
A
.
Note that
Av
=
λ
v
m
(
A

λI
)
v
= 0
⇓
det(
A

λI
) = 0
where
I
is the
n
×
n
identity matrix (the matrix such that every entry on the main diagonal is 1
and every other entry is zero).
Since
det(
A

λI
)
is a polynomial in the variable
λ
, we need only find the zeros of this polynomial
to find all the eigenvalues of
A
. To find a corresponding eigenvector, we must solve the system
(
A

λI
)
v
= 0
for the components of
v
.
I
Example
Find the eigenvalues and corresponding eigenvectors of the matrix
A
=
4
2

1
1
.
Solution
The eigenvalues are the roots of the polynomial
det(
A

λI
)
. We compute
A

λI
=
4
2

1
1

λ
1
0
0
1
=
4

λ
2

1
1

λ
Then we compute
det(
A

λI
) = (4

λ
)(1

λ
)

(

1)(2) =
λ
2

5
λ
+ 6
Setting this to zero we have
0 =
λ
2

5
λ
+ 6 = (
λ

2)(
λ

3)
so the eigenvalues are
λ
1
= 2
and
λ
2
= 3
.
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To find the eigenvectors
v
=
v
1
v
2
corresponding to
λ
1
= 2
, we need to solve
(
A

λ
1
I
)
v
= 0
that is,
4
2

1
1

2
1
0
0
1
v
1
v
2
=
0
0
.
⇓
2
2

1

1
v
1
v
2
=
0
0
⇓
2
v
1
+ 2
v
2
=
0

v
1

v
2
=
0
Since the second equation is just a scalar multiple of the first, this system has infinitely many
solutions of the form
v
1
=

v
2
.
Then the eigenvectors corresponding to
λ
1
= 2
are all of the form
v
=
c

c
=
c
1

1
where
c
is arbitrary.
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 '08
 staff
 Linear Algebra, Algebra, Eigenvectors, Scalar, homogeneous equation, Det

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