162
Economic Dynamics
4.6 Solutions with repeating roots
In chapter 2 we used
ce
λ
t
and
cte
λ
t
for a repeated root. If
λ
=
r
which is a repeated root, then either there are two
independent eigenvectors
v
1
and
v
2
which will lead to the general solution
x
=
c
1
e
rt
v
1
+
c
2
e
rt
v
2
or else there is only
one
associated eigenvector, say
v
. In this latter case we use
the result
x
=
c
1
e
rt
v
1
+
c
2
(
e
rt
t
v
+
e
rt
v
2
)
(4.23)
In this latter case the second solution satisFes
e
rt
t
v
+
e
rt
v
2
and is combined with
the solution
e
rt
v
1
to obtain the general solution (see Boyce and DiPrima 1997,
pp. 390–6). We shall consider two examples, the Frst with a repeating root, but
with two linearly independent eigenvectors, and a second with a repeating root but
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 Fall '11
 Dr.Gwartney
 Economics, Linear Algebra, Generalized eigenvector, Det, Economic Dynamics, ert v2

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