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**Unformatted text preview: **g e−At v produces
d e−At v
= e−At v − e−At Av = 0,
dt Copyright c 2000 SIAM so e−At v is constant for all t. Buy online from SIAM
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542
Chapter 7
Eigenvalues and Eigenvectors
http://www.amazon.com/exec/obidos/ASIN/0898714540
At t = 0 we have e−At v t=0 = e0 v(0) = Ic = c, and hence e−At v = c for
all t. Multiply both sides of this equation by eAt and use (7.4.5) to conclude
v = eAt c. Thus u = eAt c is the unique solution to u = Au with u(0) = c.
Finally, notice that vi = Gi c ∈ N (A − λi I) is an eigenvector associated
with λi , so that the solution to u = Au, u(0) = c, is It is illegal to print, duplicate, or distribute this material
Please report violations to [email protected] u = eλ1 t v1 + eλ2 t v2 + · · · + eλk t vk , (7.4.6) and this solution is completely determined by the eigenpairs (λi , vi ). It turns
out that u also can be expanded in terms of any complete set of independent
eigenvectors—see Exercise 7.4.1. Let’s summarize what’s been said so far. Differential Equations T
H D
E If An×n is diagonalizable with σ (A) = {λ1 , λ2 , . . . , λk } , then the
unique solution of u = Au, u(0) = c, is given by
u = eAt c = eλ1 t v1 + eλ2 t v2 + · · · + eλk t vk (7.4.7) IG
R in which vi is the eigenvector vi = Gi c, where Gi is the ith spectral
projector. (See Exercise 7.4.1 for an alternate eigenexpansion.) Nonhomogeneous systems as well as the nondiagonalizable case are treated in
Example 7.9.6 (p. 608). Y
P Example 7.4.1 An Application to Diﬀusion. Important issues in medicine and biology involve the question of how drugs or chemical compounds move from one cell to
another by means of diﬀusion through cell walls. Consider two cells, as depicted
in Figure 7.4.1, which are both devoid of a particular compound. A unit amount
of the compound is injected into the ﬁrst cell at time t = 0, and as time proceeds
the compound diﬀuses according to the following assumption. O
C α β Cell 1 Cell 2 Figure 7.4.1 Copyright c 20...

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