**Unformatted text preview: **ocess eventually evolves given the initial state
of the process. For example, the population migration problem in Example 7.3.5
(p. 531) produces a 2 × 2 system of homogeneous linear diﬀerence equations
(7.3.14), and the long-run (or steady-state) population distribution is obtained
by ﬁnding the limiting solution. More sophisticated applications are given in
Example 7.10.8 (p. 635) and Example 8.3.7 (p. 683). 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 617 Solving the equations in (7.10.3) is easy. Direct substitution veriﬁes that
x(k ) = Ak x(0) for and k = 1, 2, 3, . . . k−1 x(k ) = Ak x(0) + (7.10.4)
Ak−j −1 b(j ) for k = 1, 2, 3, . . . It is illegal to print, duplicate, or distribute this material
Please report violations to [email protected] j =0 are respective solutions to (7.10.3). So rather than ﬁnding x(k ) for any ﬁnite k, the real problem is to understand the nature of the limiting solution
limk→∞ x(k ), and this boils down to analyzing limk→∞ Ak . We begin this analysis by establishing conditions under which Ak → 0.
For scalars α we know that αk → 0 if and only if |α| < 1, so it’s natural
to ask if there is an analogous statement for matrices. The ﬁrst inclination is to
replace | | by a matrix norm
, but this doesn’t work for the standard D
E T
H norms. For example, if A = 0 2 , then Ak → 0 but A = 2 for all of the
00
standard matrix norms. Although it’s possible to construct a rather goofy-looking
k
matrix norm
g such that A g < 1 when limk→∞ A = 0, the underlying
mechanisms governing convergence to zero are better understood and analyzed
by using eigenvalues and the Jordan form rather than norms. In particular, the
spectral radius of A deﬁned as ρ(A) = maxλ∈σ(A) |λ| (Example 7.1.4, p. 497)
plays a central role. IG
R Y
P Convergence to Zero For A ∈ C n×n , O
C Proof. lim Ak = 0 k→∞ if and only if ρ(A) < 1. (7.10.5) If P−1 AP = J is the Jordan form for A, then Ak = PJk P...

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