Linear Algebra, Theory And Applications, 2012a

10.5. THE JORDAN CANONICAL FORM 261
The matrix J s is an m s × m s matrix which is of the form
⎛
⎞
α ··· ∗
⎜
J s = ⎝
.
. .. .
⎟
⎠ (10.8)
0 ··· α
which can be written in the form
J s = D + N
for D a multiple of the identity and N an upper triangular matrix with zeros down the main
diagonal. Therefore, by the Cayley Hamilton theorem, N m s
= 0 because the characteristic
equation for N is just λ ms
= 0. (You could also verify this directly.) Now since D is just a
multiple of the identity, it follows that DN = ND. Therefore, the usual binomial theorem
may be applied and this yields the following equations for k ≥ m s .
J k s = (D + N) k =
=
j=0
k∑
j=0
( k
j)
D k−j N j
∑m s
( k
D
j)
k−j N j , (10.9)
the third equation holding because N ms =0. Thus Js k is of the form
⎛
⎞
α k ··· ∗
Js k ⎜
= ⎝
.
. .. .
⎟
⎠ .
0 ··· α k
Lemma 10.5.3 Suppose J is of the form J s described above in (10.8) where the constant
α, on the main diagonal is less than one in absolute value. Then
(
J
k ) =0. ij
lim
k→∞
Proof: From (10.9), it follows that for large k, and j ≤ m s ,
( k
≤
j)
k (k − 1) ···(k − m s +1)
.
m s !
Therefore, letting C be the largest value of ∣ ( N j) pq∣ for 0 ≤ j ≤ m s ,
∣ ( J k) ( )
k (k − 1) ···(k − ms +1)
pq∣ ≤ m s C
|α| k−m s
m s !
which converges to zero as k →∞. This is most easily seen by applying the ratio test to
the series
∞∑
( )
k (k − 1) ···(k − ms +1)
|α| k−ms
m s !
k=m s
and then noting that if a series converges, then the k th term converges to zero.