Linear Algebra, Theory And Applications, 2012a

372 NUMERICAL METHODS FOR FINDING EIGENVALUES
Consider one of the blocks in the Jordan form.
∑r k
( ) n λ
Jk n = λ n n−i
k
1
i λ n Nk i ≡ λ n 1 K (k, n)
1
Then from the above,
A n
s n s n−1 ···s 1
= P
i=0
⎛
λ n 1 ⎜
⎝
s n s n−1 ···s 1
K (1,n)
⎞
. ..
⎟
⎠ P −1
K (m, n)
Consider one of the terms in the sum for K (k, n) fork>1. Letting the norm of a matrix
be the maximum of the absolute values of its entries,
) n λ
n−i
∣
k
∣
∣∣( i λ n Nk
i 1
∣∣ ≤ nr k
λ k ∣∣∣
n
∣ p r k
C
λ 1
where C depends on the eigenvalues but is independent of n. Then this converges to 0
because the infinite sum of these converges due to the root test. Thus each of the matrices
K (k, n) converges to 0 for each k>1asn →∞.
Now what about K (1,n)? It equals
( n
r 1
) r1
∑
i=0
(( ) ( )) n n
/ λ −i
1 N1
i i r 1
=
( n
r 1
) (λ
−r 1
1 N r 1
1 + m (n))
where lim n→∞ m (n) =0. This follows from
(( ) ( )) n n
lim / =0,i