104 are indeed the solutions to the dierence

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Unformatted text preview: 22 , . . . , ann ) , and (7.10.28) implies that each aii > 0. Note: Comparing properties of M-matrices with those of positive definite matrices reveals many parallels, and, in a rough sense, an M-matrix often plays the role of “a poor man’s positive definite matrix.” Only a small sample of M-matrix theory has been presented here, but there is in fact enough to fill a monograph on the subject. For example, there are at least 50 known equivalent conditions that can be imposed on a real matrix with nonpositive off-diagonal entries (often called a Z-matrix) to guarantee that it is an M-matrix—see Exercise 7.10.12 for a sample of such conditions in addition to those listed above. 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 629 We now focus on broader issues concerning when limk→∞ Ak exists but may be nonzero. Start from the fact that limk→∞ Ak exists if and only if limk→∞ Jk exists for each Jordan block in (7.10.6). It’s clear from (7.10.7) that limk→∞ Jk cannot exist when |λ| > 1, and we already know the story for |λ| < 1, so we only have to examine the case when |λ| = 1. If |λ| = 1 with λ = 1 (i.e., λ = eiθ with 0 < θ < 2π ), then the diagonal terms λk oscillate indefinitely, and this prevents Jk (and Ak ) from having a limit. When λ = 1, It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] ⎛ k 1 ⎜ ⎜ Jk = ⎜ ⎝ .. ··· .. . . . . . . ⎞ k m−1 .. 1 k 1 ⎟ ⎟ ⎟ ⎠ 1 D E (7.10.30) m×m T H has a limiting value if and only if m = 1, which is equivalent to saying that λ = 1 is a semisimple eigenvalue. But λ = 1 may be repeated p times so that there are p Jordan blocks of the form J = [1]1×1 . Consequently, limk→∞ Ak exists if and only if the Jordan form for A has the structure Ip×p 0 0 K J = P−1 AP = IG R , where p = alg mult (1)...
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