14add_2 - r a 1 2 du r r 1. Consider the equation = 0 a 3 u...

1. Consider the equation
1a What are the eigenvalues of A ? How many linear independent eigenvectors can you
find?
1b Check that
( 29
[ ] 3
0 zero matrix A aI - = =
1c. Show that
( 29
( 2
2
2
At at
A aI t
e e I A aI t
 
-
  = + - +
 
 
.
1d. Solve (1) with the given initial condition (express your answer in terms of the constant a
1 2
0 3
0 0
a
du
a u Au
dt
a
 
 
= =
 
 
 
r
r r
subjected to the initial condition
[ ]
(0) 0,0,1
T
o
u u = =
r r
.
.
29
).
1e. What is the solution of (1) if ?
1f. What is the solution of (1) if
( 29
(
2
1
c N A aI ∈ -
r
. Write
(
A I t
At
e e e
λ
λ
-
= . Find
1
c
r
and
2
c
r
using the initial condition
[ ]
(0) 0,0,1
T
o
u u = =
r r
.
It
1b. Direct calculation shows that

( 29 ( 29
[ ]
2
2 3
0 1 2 0 0 3 0 1 2 0 0 3
0 0 3 0 0 0 0 0 3 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
A aI A aI
       
       
- = = ⇒ - = =
       
       
       
1c. Using the fact that I and A commutes, we have
(
t A aI
At atI atI At atI atI At atI
e e e e e e e e
-
- -
= = =
Note
atI at
e e I =
(as shown in class) (1a)
( 29
( 29 ( 29 ( 29
[ ]
2 3
2 3
0
[0]
2! 3!
t A aI
t t
e I t A aI A aI A aI
-
= + - + - + - +
14243
(1b)
In (1b), we used (1b), i.e.,
( 29 [ ] ( 29 [ ] 3
0 0
n
A aI A aI - = ⇒ - = for 3 n ≥ . Combining
(1a,b), we have
( 29 ( 29 ( 29 ( 29
2 2
2 2
2! 2
At at at
t t
e e I I t A aI A aI e I t A aI A aI
   
= + - + - = + - + -
   
   
1d. The solution of the initial value problem is
( 29 ( 29
2
2
0 0 0 0
2
0 0 0
2
0 1 2 0 0 3
0 0 3 0 0 0
2
0 0 0 0 0 0
At at
at
t
e u e u t A aI u A aI u
t
e u t u u
 
= + - + -
 
 
 
   
 
   
= + +
 
   
 
   
   
 
r r r r
r r r
Substituting
[ ]
(0) 0,0,1
T
o
u u = =
r r
gives:
2
0 2 3
0 3 0
2
1 0 0
at
t
u e