**Unformatted text preview: **1 ⎛ ⎜
= P⎝ .. ⎞
. ⎟ −1
⎠P , Jk
.. . ⎛
where J =⎝ λ ⎞
1
. . . . . ⎠ (7.10.6)
.
λ denotes a generic Jordan block in J. Clearly, Ak → 0 if and only if Jk → 0
for each Jordan block, so it suﬃces to prove that Jk → 0 if and only if |λ| <
1. Using the function f (z ) = z n in formula (7.9.2) on p. 600 along with the
convention that k = 0 for j > k produces
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618
Chapter 7
Eigenvalues and Eigenvectors
http://www.amazon.com/exec/obidos/ASIN/0898714540
⎞
⎛ k k k −1 k k −2
k
k−m+1
1 λ 2 λ ··· λk λ ⎜
⎜
⎜
⎜
⎜
k
J =⎜
⎜
⎜
⎜
⎜
⎝ k
1 λk−1 .. .. .. . m−1 λ .
.
. . k
2 λk λk−2 k
1 . λk−1 ⎟
⎟
⎟
⎟
⎟
⎟.
⎟
⎟
⎟
⎟
⎠ It is illegal to print, duplicate, or distribute this material
Please report violations to [email protected] λk (7.10.7) m×m It’s clear from the diagonal entries that if Jk → 0, then λk → 0, so |λ| < 1.
Conversely, if |λ| < 1 then limk→∞ k λk−j = 0 for each ﬁxed value of j
j
because
k
j = k (k − 1) · · · (k − j + 1)
kj
≤
j!
j! =⇒ D
E kj
k k−j
≤ |λ|k−j → 0.
λ
j
j! T
H You can see that the last term on the right-hand side goes to zero as k → ∞
either by applying l’Hopital’s rule or by realizing that k j goes to inﬁnity with
polynomial speed while |λ|k−j is going to zero with exponential speed. Therefore, if |λ| < 1, then Jk → 0, and thus (7.10.5) is proven. IG
R Intimately related to the question of convergence to zero is the convergence
∞
k
of the Neumann series
k=0 A . It was demonstrated in (3.8.5) on p. 126
that if limn→∞ An = 0, then the Neumann series converges, and it was argued
in Example 7.3.1 (p. 527) that the converse holds for diagonalizable matrices.
Now we are in a position to prove that the converse is true for all square matrices
and thereby produce the following complete statement regarding the convergence...

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