6
MA 36600 FINAL REVIEW
§
7.7: Fundamental Matrices.
•
Consider the initial value problem
d
dt
x
=
P
(
t
)
x
,
x
(
t
0
) =
x
0
.
Say that
x
(1)
,
x
(2)
, . . . ,
x
(
n
)
is a fundamental set of solutions. Then the general solution is
x
(
t
) =
c
1
x
(1)
(
t
) +
c
2
x
(2)
(
t
) +
· · ·
+
c
n
x
(
n
)
(
t
) =
Ψ
(
t
)
c
in terms of the matrices
Ψ
(
t
) =
x
11
(
t
x
12
(
t
)
· · ·
x
1
n
(
t
)
x
21
(
t
)
x
22
(
t
)
· · ·
x
2
n
(
t
)
.
.
.
.
.
.
.
.
.
.
.
.
x
n
1
(
t
)
x
n
2
(
t
)
· · ·
x
nn
(
t
)
and
c
=
c
1
c
2
.
.
.
c
n
.
We call
Ψ
(
t
) =
x
ij
(
t
)
a
fundamental matrix
. It satisfies the matrix equation
d
dt
Ψ
=
P
(
t
)
Ψ
.
Since the Wronskian is
W
= det
Ψ
, we see that
Ψ
(
t
) is a nonsingular matrix. Hence we can choose
the constant
c
in order to solve the initial value problem:
x
0
=
Ψ
(
t
0
)
c
=
⇒
c
=
Ψ
(
t
0
)
−
1
x
0
.
•
Say
P
(
t
) =
A
is an
n
×
n
constant matrix. The solution to the initial value problem is
x
(
t
) =
Φ
(
t
)
x
0
in terms of the
matrix exponential
Φ
(
t
) = exp (
A
t
) =
∞
k
=0
A
k
t
k
k
!
=
I
+
A
t
+
A
2
t
2
2
+
· · ·
.
In general, the initial value problem
d
dt
Ψ
=
A
Ψ
,
Ψ
(0) =
Ψ
0
has the unique solution
Ψ
(
t
) = exp (
A
t
)
Ψ
0
.
The order does matter! The matrix
Ψ
0
exp (
A
t
) is
not
a solution to the initial value problem.
•
We say that an
n
×
n
matrix is a
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 Spring '09
 EdrayGoins
 Linear Algebra, Exponential Function, Matrices, Matrix exponential

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