Lecture Notes in Differential Equations - Bruce E. Shapiro

311
The Jordan Form
Let A be a square n × n matrix.
A is diagonalizable if for some matrix U, U −1 AU = D is diagonal.
A is diagonalizable if and only if it has n linearly independent eigenvectors
v 1 , ..., v n , in which case U = ( v 1 · · · v n
)
will diagonalize A.
If λ 1 , ..., λ k are the eigenvalues of A with multiplicities m 1 , ..., m k (hence
n = m 1 + · · · + m k ) then the Jordan Canonical Form of A is
⎛
⎞
B 1 0 · · · 0
J =
0 B 2 .
⎜
⎝
. ⎟
. .. 0 ⎠
0 · · · 0 B k
(31.73)
where (a) if m i = 1, B i = λ i (a scalar or 1 × 1 matrix; and (b) if m i ≠ 1,
B i is an m i ×m i submatrix with λ i repeated in every diagonal element, the
number 1 in the supra-diagonal, and zeroes everywhere else, e.g., if m i = 3
then
⎛
λ i 1 0
⎞
0 0 λ i
B i = ⎝ 0 λ i 1 ⎠ (31.74)
The B i are called Jordan Blocks.
For every square matrix A, there exists a matrix U such J = U −1 AU
exists.
If J is the Jordan form of a matrix, then
Thus e A = Ue J U −1 .
⎛
⎞
e B1 0 0
e J ⎜
= ⎝
.
0 ..
⎟
0 ⎠ (31.75)
0 0 e B k
Example 31.3. Evaluate the solution
y = e A(t−t0) y 0 =
{ [( 1 1
exp
4 −2
) ]} ( 6
t
5
)
(31.76)
found in the previous example.