This preview shows pages 1–2. Sign up to view the full content.
EIGENVECTORS
Method for solving systems of first order differential equations
using eigenvalues and eigenvectors
9.
p288
x
x
x
1
1
2
2
5
'
=

x
x
x
2
1
2
4
2
'
=

x
x
1
2
0
2
0
3
( )
,
( )
=
=
The problem can be rewritten
x
x
'
=


2
5
4
2
Definition:
An
eigenvalue
of matrix
A
is a number
l
such that
A
I

=
l
0
To find the eigenvalues we rewrite the matrix
2
5
4
2




l
l
and find its determinant
(
)(
)
(
)( )
2
2
5
4

 
 
l
l
which we simplify and set equal to zero
l
2
16
0
+
=
Solving for
l
we get
l
=
±

0
0
64
2
yielding complex eigenvalues
l
=
±
0
4
i
Definition:
An
eigenvector
associated with the eigenvalue
l
is a nonzero vector
v
such that
Av
v
=
l
so that
(
)
A
I v

=
l
0
To find the eigenvectors we set up the equation
2
4
5
4
2
4
0
0




=
(
)
(
)
i
i
a
b
using the eigenvalue
0
4
+
i
Since in this case we have complex eigenvalues it will be necessary to find only one eigenvector to solve the
problem.
Otherwise we would need to repeat these steps using the second eigenvalue to find a second
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
 Spring '08
 comech
 Differential Equations, Eigenvectors, Equations, Vectors

Click to edit the document details