416
Chapter 8
CHAPTER 8
MATRIX EXPONENTIAL METHODS
SECTION 8.1
In Problems 1–8 we first use the eigenvalues and eigenvectors of the coefficient matrix
A
to
find first a
fundamental matrix
Φ
(
t
)
for the homogeneous system
x
′
=
Ax
.
Then we apply the
formula
x
(
t
)
=
Φ
(
t
)
Φ
(0)
-
1
x
0
,
to find the solution vector
x
(
t
)
that satisfies the initial condition
x
(0)
=
x
0
.
Formulas (11) and
(12) in the text provide inverses of
2-by-2
and
3-by-3
matrices.
1.
Eigensystem:
T
T
1
1
2
2
1,
[1
1] ;
3,
[1
1]
λ
λ
=
=
−
=
=
v
v
1
2
3
1
2
3
( )
t
t
t
t
t
t
e
e
t
e
e
e
e
λ
λ
Φ
=
=
−
v
v
3
3
3
3
1
1
3
5
1
1
( )
1
1
2
2
2
5
t
t
t
t
t
t
t
t
e
e
e
e
t
e
e
e
e
−
+
=
⋅
⋅
=
−
−
−
+
x
2.
Eigensystem:
T
T
1
1
2
2
0,
[1
2] ;
4,
[1
2]
λ
λ
=
=
=
=
−
v
v
1
2
4
1
2
4
1
( )
2
2
t
t
t
t
e
t
e
e
e
λ
λ
Φ
=
=
−
v
v
4
4
4
4
2
1
2
1
3
5
1
1
( )
2
1
1
4
4
2
2
6
10
t
t
t
t
e
e
t
e
e
+
=
⋅
⋅
=
−
−
−
−
x
3.
Eigensystem:
T
4 ,
[1
2
2]
i
i
λ
=
=
+
v
cos4
2sin4
2cos4
sin4
( )
Re(
)
Im(
)
2cos4
2sin 4
t
t
t
t
t
t
t
e
e
t
t
λ
λ
−
+
Φ
=
=
v
v
cos4
2sin4
2cos4
sin4
0
2
0
5sin 4
1
1
( )
2cos4
2sin4
2
1
1
4cos4
2sin4
4
4
t
t
t
t
t
t
t
t
t
t
−
+
−
=
⋅
⋅
=
−
−
x
Section 8.1
417
4.
Eigensystem:
T
T
1
2
1
2
2, 2;
{
,
} with
[1
1] ,
[1
0]
λ
=
=
=
v
v
v
v
2
1
1
2
1
1
( )
(
)
1
t
t
t
t
t
e
t
e
e
t
λ
λ
+
Φ
=
+
=
v
v
v
2
2
1
1
0
1
1
1
( )
1
1
1
0
t
t
t
t
t
e
e
t
t
+
+
=
⋅
⋅
=
−
x
5.
Eigensystem:
T
3 ,
[ 1
3]
i
i
λ
=
= − +
v
cos3
sin3
cos3
sin3
( )
Re(
)
Im(
)
3cos3
3sin3
t
t
t
t
t
t
t
e
e
t
t
λ
λ
−
−
−
Φ
=
=
v
v
cos3
sin3
cos3
sin3
0
1
1
3cos3
sin3
1
1
( )
3cos3
3sin3
3
1
1
3cos3
6sin3
3
3
t
t
t
t
t
t
t
t
t
t
t
−
−
−
−
=
⋅
⋅
=
−
−
+
x
6.
Eigensystem:
T
5
4 ,
[1
2
2]
i
i
λ
=
+
=
+
v
5
cos4
2sin4
2cos4
2sin4
( )
Re(
)
Im(
)
2cos4
2sin 4
t
t
t
t
t
t
t
t
e
e
e
t
t
λ
λ
−
+
Φ
=
=
v
v
5
5
cos4
2sin4
2cos4
2sin4
0
2
2
cos4
sin 4
1
( )
2
2cos4
2sin 4
2
4
0
sin4
4
t
t
t
t
t
t
t
t
t
e
e
t
t
t
−
+
+
=
⋅
⋅
=
−
x
7.
Eigensystem:
T
T
T
1
1
2
2
3
3
0,
[6
2
5] ;
1,
[3
1
2] ;
1,
[2
1
2]
λ
λ
λ
=
=
=
=
= −
=
v
v
v
3
1
2
1
2
3
6
3
2
( )
2
5
2
2
t
t
t
t
t
t
t
t
t
e
e
t
e
e
e
e
e
e
e
λ
λ
λ
−
−
−
Φ
=
=
v
v
v
6
3
2
0
2
1
2
12
12
2
( )
2
1
2
2
1
4
4
5
2
2
1
3
0
0
10
8
2
t
t
t
t
t
t
t
t
t
t
t
t
e
e
e
e
t
e
e
e
e
e
e
e
e
−
−
−
−
−
−
−
−
+
+
=
⋅
−
⋅
=
−
+
+
−
