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Unformatted text preview: Systems of ﬁrstorder differential equations 157 Then systems 4.4 and 4.5 are simply speciﬁc examples of the homogeneous linear
system
x = Ax
˙ (4.13) while system 4.6 is a speciﬁc example of the nonhomogeneous linear system
x = Ax + b
˙ (4.14) For linear homogeneous systems, if the determinant of A is not zero, then the
∗
∗
only solution, the only ﬁxed point, is x∗ = 0, i.e., (x1 = 0 and x2 = 0). On the
other hand, for nonhomogeneous linear systems, the equilibrium can be found, so
long as A is nonsingular, from
0 = Ax∗ + b
x∗ = −A−1 b (4.15) When considering the issue of stability/instability it is useful to note that linear
nonhomogeneous systems can always be reduced to linear homogeneous systems
in terms of deviations from equilibrium if an equilibrium exists. For
x = Ax + b
˙
0 = Ax∗ + b
subtracting we immediately have in deviation form
x = A(x − x∗ )
˙ (4.16)
∗ ∗∗
(x1 , x2 ). which is homogeneous in terms of deviations from the ﬁxed point x =
There will be no loss of generality, therefore, if we concentrate on linear homogeneous systems.
The matrix A is of particular importance in dealing with stability and instability.
Two important properties of such a square matrix are its trace, denoted tr(A), and
its determinant, denoted det(A), where4
tr(A) = a11 + a22
det(A) = a11
a21 a12
= a11 a22 − a12 a21
a22 (4.17) It should be noted that both the trace and the determinant are scalars. The matrix
A is nonsingular if det(A) = 0.
There is another property of the matrix A that arises for special linear systems.
Consider the following general linear system
y = Ax
This can be viewed as a transformation of the vector x into the vector y. But
suppose that x is transformed into a multiple of itself, i.e., y = λx, where λ is a
scalar of proportionality. Then
Ax = λx
or
(A − λI)x = 0
4 See Chiang (1984) or any book on linear algebra. (4.18) ...
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 Fall '11
 Dr.Gwartney
 Economics

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