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) ...
View
Full Document
 Fall '11
 Dr.Gwartney
 Economics, Linear Algebra, Determinant, Vector Space, linear homogeneous systems

Click to edit the document details