Page 380 - A Course in Linear Algebra with Applications

364 Chapter Nine: Advanced Topics

as in Example 8.3.3. However this method can be laborious
for large n and Jordan form provides a simpler alternative
method.
Returning to the system Y' = AY, we know that there is
1
a non-singular matrix S such that N = S~ AS is in Jordan
normal form: say
0
0
7V =

0 jj
Here Ji is a Jordan block, say with di on the diagonal. Of
1
course the di are the eigenvalues of A. Now put U = S~~ Y,
so that the system Y' — AY becomes (SU)' = ASU, or
U' = NU,

1
since N = S~ AS. To solve this system of differential equa-
tions it is plainly sufficient to solve the subsystems U[ = JiUi
1
for i — ,..., k where Ui is the column of entries of U corre-
sponding to the block Ji in N.
This observation effectively reduces the problem to one
in which the coefficient matrix is a Jordan block, let us say
(d 1 0 ••• 0 0^
0 d 1 ••• 0 0
A =
0 0 d 1
\o 0 0 0 d)
Now the equations in the corresponding system have a
much simpler form than in the general triangular case:
du\ + u 2
du 2 + u 3

du n.
u. \- u r
du r
u r