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MA 36600 FINAL REVIEW
5
§
7.5: Homogeneous Linear Systems with Constant Coeﬃcients.
•
Consider the homogeneous linear system with constant coeﬃcients
d
dt
x
=
Ax
.
We may guess a solution in the form
x
(
t
)=
ξ
e
rt
for some constant vector
ξ
and constant scalar
r
.
(The symbol “
ξ
” is the letter “xi,” and is pronounced “zai” or “ksi.”). Then (
A
−
r
I
)
ξ
=
0
, so that
r
is an eigenvalue of
A
and
ξ
is an eigenvector of
A
.
§
7.6: Complex Eigenvalues.
•
Consider a 2
×
2 matrix
A
=
°
a
11
a
12
a
21
a
22
±
=
⇒
det (
A
−
r
I
)=
r
2
−
(tr
A
)
r
+(det
A
)
in terms of the
trace
and
determinant
of
A
:
tr
A
=
a
11
+
a
22
,
det
A
=
a
11
a
22
−
a
12
a
21
.
The discriminant of this polynomial is
disc
A
=(tr
A
)
2
−
4(det
A
)=(
a
11
−
a
22
)
2
+4
a
12
a
21
.
If the discriminant is positive, then the eigenvalues
r
are real. If the discriminant is negative, then
the eigenvalues
r
are imaginary.
•
We consider a homogeneous system
d
dt
x
=
Ax
where
A
is a 2
×
2 matrix with constant coeﬃcients with disc
A
<
0. Its eigenvalues
r
and eigenvectors
ξ
are imaginary. We know that one solution of the system is
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This note was uploaded on 11/30/2011 for the course MATH 366 taught by Professor Edraygoins during the Spring '09 term at Purdue UniversityWest Lafayette.
 Spring '09
 EdrayGoins
 Linear Systems, Scalar

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