It can be proved.docx - It can be proved the proof is...

It can be proved (the proof is omitted) that that
for the case of double root, this second equation
must be consistent, though the determinant
det( A − λ
1
I ) = 0 .
Example: Given
.
We have der(
.
The system has a double eigenvalues λ = λ
1
= 2,
m
1
= 2. In this case, for the second solution we
need to solve the systems,
A v ˆ1 , 1 =
λ 1 v ˆ1 , 1 ,

( ) ∗
SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS 2
A v ˆ1 , 0 = λ 1 v ˆ1 , 0
+ v ˆ1 , 1 ,
becomes
A v ˆ1 , 1 = 2 v ˆ1 , 1 ,
A v ˆ1 , 0 = 2 v ˆ1 , 0 + v ˆ1 , 1 ,
We solve v ˆ
1 , 1
= [ c,d ]
T
from the first equation,
.
It is derived that rank{( A − 2 I )} = 1 and
v ˆ
1 , 1
= r [2 , 1] T .
The second equation for v ˆ
1 , 0
= [ a,b ]
T
becomes
.
It is seen that this equation is consistent and has
the solutions:

( ) ∗
SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS 3
v ˆ ,
where r,s are arbitrary constants. Let r = 2 ,s = 0,
we get
v ˆ
1 , 1
= [4 , 2] T , v ˆ
1 , 0
= [1 , 0] T .
Note: For the case under discussion, one may
introduce the matrix
P = [ v ˆ1 , 1 , v ˆ1 , 0] ,
we have
.
Since P is invertible, we may write
.
Setting as before
x = P u

( ) ∗
SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS 4
with
u ,
our system becomes