**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
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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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