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**Unformatted text preview: **nd has eigenvalues λ such that |λ| = 1, then it’s
necessary that index (λ) = 1. The condition also is suﬃcient—i.e., if ρ(A) = 1
and each eigenvalue on the unit circle is semisimple, then A is summable. This
follows because each Jordan block associated with an eigenvalue µ such that
|µ| < 1 is convergent (and hence summable) to 0 by (7.10.5), and for semisimple Copyright c 2000 SIAM Buy online from SIAM
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632
Chapter 7
Eigenvalues and Eigenvectors
http://www.amazon.com/exec/obidos/ASIN/0898714540
eigenvalues λ such that |λ| = 1, the associated Jordan blocks are 1 × 1 and
hence summable because (7.10.35) implies It is illegal to print, duplicate, or distribute this material
Please report violations to meyer@ncsu.edu 1 + λ + ··· + λ
k ⎧
⎪
⎨ k−1 = ⎪
⎩ 1 λk
−
k
k 1
1−λ →0 1 for |λ| = 1, λ = 1,
for λ = 1. D
E In addition to providing a necessary and suﬃcient condition for A to be
Ces`ro summable, the preceding analysis also reveals the nature of the Ces`ro
a
a
λ limit because if A is summable, then each Jordan block J = T
H 1
.. ...
.
λ in the Jordan form for A is summable, in which case we have established that I + J + · · · + Jk−1
k→∞
k
lim ⎧
⎪1
⎪
⎨
0
=
⎪
⎪
⎩
0 1×1 if λ = 1 and index (λ) = 1, IG
R
1×1 if |λ| = 1, λ = 1, and index (λ) = 1,
if |λ| < 1. Consequently, if A is summable, then the Jordan form for A must look like Y
P J = P−1 AP = O
C Ip×p
0 0
C , where p = alg multA (λ = 1) , and the eigenvalues of C are such that |λ| < 1 or else |λ| = 1, λ = 1,
index (λ) = 1. So C is summable to 0, J is summable to I + A + · · · + Ak−1
=P
k I + J + · · · + Jk−1
k P−1 → P Ip×p
0 Ip×p
0 0
0 0
0 , and P−1 = G. Comparing this expression with that in (7.10.32) reveals that the Ces`ro limit
a
is exactly the same as the ordinary limit, had it existed. In other words, if A is
summable, then regardless of whether or not A is convergent, A is summable
to the projector on...

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