J 0 the gramschmidt orthogonalization procedure p 309

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Unformatted text preview: to N (I − A) along R (I − A). Below is a formal summary of our observations concerning Ces`ro summability. a Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 7.10 Difference Equations, Limits, and Summability http://www.amazon.com/exec/obidos/ASIN/0898714540 633 Ces` ro Summability a A ∈ C n×n is Ces`ro summable if and only if ρ(A) < 1 or else a ρ(A) = 1 with each eigenvalue on the unit circle being semisimple. • It is illegal to print, duplicate, or distribute this material Please report violations to meyer@ncsu.edu • When it exists, the Ces`ro limit a I + A + · · · + Ak−1 =G k→∞ k lim D E (7.10.36) is the projector onto N (I − A) along R (I − A). T H • G = 0 if and only if 1 ∈ σ (A) , in which case G is the spectral projector associated with λ = 1. • If A is convergent to G, then A is summable to G, but not conversely. IG R Since the projector G onto N (I − A) along R (I − A) plays a prominent role, let’s consider how G might be computed. Of course, we could just iterate on Ak or (I + A + · · · + Ak−1 )/k, but this is inefficient and, depending on the proximity of the eigenvalues relative to the unit circle, convergence can be slow— averaging in particular can be extremely slow. The Jordan form is the basis for the theoretical development, but using it to compute G would be silly (see p. 592). The formula for a projector given in (5.9.12) on p. 386 is a possibility, but using a full-rank factorization of I − A is an attractive alternative. A full-rank factorization of a matrix Mm×n of rank r is a factorization O C Y P M = Bm×r Cr×n , where rank (B) = rank (C) = r = rank (M). (7.10.37) All of the standard reduction techniques produce full-rank factorizations. For example, Gaussian elimination can be used because if B is the matrix of basic columns of M, and if C is the matrix containing the nonzero rows in the reduced row echelon form EM , then M = BC is a full-rank factori...
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