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610
Chapter 7
Eigenvalues and Eigenvectors
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Example 7.9.7 It is illegal to print, duplicate, or distribute this material
Please report violations to [email protected] Nondiagonalizable Mixing Problem. To make the point that even simple
problems in nature can be nondiagonalizable, consider three V gallon tanks as
shown in Figure 7.9.1 that are initially full of polluted water in which the ith
tank contains ci lbs of a pollutant. In an attempt to ﬂush the pollutant out, all
spigots are opened at once allowing fresh water at the rate of r gal/sec to ﬂow
into the top of tank #3, while r gal/sec ﬂow from its bottom into the top of
tank #2, and so on.
Fresh D
E r gal/sec 3 T
H r gal/sec 2 r gal/sec IG
1 R
Y r gal/sec Figure 7.9.1 Problem: How many pounds of the pollutant are in each tank at any ﬁnite time
t > 0 when instantaneous and continuous mixing occurs? P Solution: If ui (t) denotes the number of pounds of pollutant in tank i at time
t > 0, then the concentration of pollutant in tank i at time t is ui (t)/V lbs/gal,
so the model ui (t) = (lbs/sec) coming in − (lbs/sec) going out produces the nondiagonalizable system:
⎛
⎛
⎞⎛
⎞
⎞
⎛⎞
u1 (t)
−1
1
0
u1 (t)
c1
⎜
⎟⎜
⎟ r⎜
⎟
u2 (t) ⎠ = ⎝ 0 −1
1 ⎠⎝ u2 (t) ⎠, or u = Au with u(0) = c = ⎝ c2 ⎠ .
⎝
V
c3
u (t)
u (t)
0
0 −1
3 O
C 3 This setup is almost the same as that in Exercise 3.5.11 (p. 104). Notice that
A is simply a scalar multiple of a single Jordan block J = −1
0
0 1
−1
0 0
1
−1 , so eAt is easily determined by replacing t by rt/V and λ by −1 in the second
equation of (7.9.18) to produce
⎞
⎛
2
1 rt/V (rt/V ) /2
⎟
⎜
eAt = e(rt/V )J = e−rt/V ⎝ 0
1
rt/V
⎠.
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7.9 Functions of Nondiagonalizable Matrice...

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