Section 9.4 is entitled “The Trace-Determinant Plane.”
Let’s do a review of what we learned in 9.2 again!
According to the fundamental theorem of algebra
(http://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra),
the characteristic polynomial must factor to be
As usual, λ1 and λ2 are the
eigenvalues of the matrix. There are important conclusions that we can observe
now:
This means that the eigenvalues determine the trace and
determinant of A.
So now we have a new plane we can observe stuff in: the
trace-determinant plane. There’s nothing tricky about this plane; it’s just a
coordinate plane with coordinates of trace and determinants (T and D).
So you might be wondering why this is important. Well, this
is why.
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| (pruffle.mit.edu) |
Yet again, T2 – 4D becomes important (that's what the actual curve is graphing). Depending
on the sign of T2 – 4D we have different classifications for
equilibrium points in the trace determinant plane. Hooray!
One final thing to leave you with (I know, this is a short
section!), but first, let us recall five of our six cases for equilibrium
points:
1. Saddle point,
where A has two real eigenvalues opposite in sign.
2. Nodal sink,
where A has two real and negative eigenvalues.
3. Nodal source,
where A has two real and positive eigenvalues.
4. Spiral sink,
where A has two complex eigenvalues with negative real parts.
5. Spiral source,
where A has two complex eigenvalues with positive real parts.
Each of these five cases corresponds to a section on the
trace-determinant plane.
Therefore, these five cases are known as generic. The
equilibrium points that are exceptions to these five cases are known as
nongeneric. Obviously, this makes the center (our sixth case) a nongeneric
case.
That’s it for 9.4! Obviously, I’m trying to go in order for
chapter 9, just so I don’t have to jump back and forth in the chapter. We just
started chapter 8 in class, so I’m feeling pretty confident with going in
order.
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