ch8 - CHAPTER 8 MATRIX EXPONENTIAL METHODS SECTION 8.1 In...

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
− −
− −
− −
    − − + +
   
   
   
= ⋅ − ⋅ = − + +
   
   
   
−