# The jordan form for a reveals everything suppose that

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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| &lt; 1 is convergent (and hence summable) to 0 by (7.10.5), and for semisimple Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 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| &lt; 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| &lt; 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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## This document was uploaded on 03/06/2014 for the course MA 5623 at City University of Hong Kong.

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