While sor was a big step forward over the algorithms

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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 j Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 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 meyer@ncsu.edu 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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