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

1. Consider the equation
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
1a What are the eigenvalues of A ? How many linear independent eigenvectors can you
find?
.
.
29
Solve (1) with the given initial condition (express your answer in terms of the constant a
1e. What is the solution of (1) if
1 2
0 3 ,
0 0
a
A b a b c
c
 
 
= ≠ ≠ ?
).
Hint: The solution has the form:
1 2
At At
u e c e c = +
r r r
where
2
( ) c N A bI ∈ -
r
and
( 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