9.1, or the extension of chapter 8 (plus eigens)

Chapter 9 is entitled “Linear Systems with Constant Coefficients.”

Section 9.1 is entitled “Overview of the Technique.”

Suppose we’re looking for solutions to the system y’ = Ay. A is a matrix with constant entries (it’s as if they planned for this in the chapter title or something). Let’s start with looking at a first-order and homogeneous equation, which would have the form y’ = ay. The solution for this equation is simple (compared to other examples we could have picked). Since it’s separable, the solution would be y(t) = Ce^at, where C is any constant (which we get from the integration and solving for y(t) and stuff).

So it’s pretty reasonable to look for solution to our system y’ = Ay that have exponentials involved. Let’s look for the solutions that have the form y(t) = e^λtv, where v is a vector with constants for entries (much like A) and λ is a constant we have yet to solve for. If we were to substitute this equation into our original system y’ = Ay, we would get

The exponential factor will never be equation to zero, so we can put these two sides of our equation to together, but only if Av = λv.

As is with most unnamed entities that come from seemingly nowhere in mathematics, v and λ have special names and actually have mathematical roots and are helpful.

 “Suppose A is an n × n matrix. A number λ is called an eigenvalue of A if there is a nonzero vector v such that Av = λv. If λ is an eigenvalue, then any vector v satisfying [this equation] is called an eigenvector associated with the eigenvalue λ” (373).

Something to note about this is that any multiple of v will work as well as an eigenvector.

A big thing to take away from all this is that if λ is the eigenvalue of A and v is the eigenvector, then x(t) = e^λtv is a solution to x’ = Ax. It will also satisfy the initial condition x(0) = v.

Now, your next question might be, “How do we find these eigenvalues?”

So we have this equation Av = λv, which we can rewrite as 0 = Av – λv = Av – λIv = [A – λI]v. Just to remind you, I is the identity matrix. This allows us to factor the v out of the equation, since λ is a mere number and A is a mighty matrix and thus it doesn’t make much sense to write A – λ.

So, in conclusion (and with fancy equation text):

Since v is nonzero (because this is how we defined it), then the matrix A – λI has a nontrivial nullspace. Recall that back in section 7.7 that our pressing issue was how to know whether or not a nullspace was nontrivial. We determined that it’s nontrivial if the determinant was equal to zero, i.e.

So because of the way things turned out, λ will turn up on the diagonals and nowhere else (since the identity matrix is merely ones on the main diagonal, i.e., on the n diagonal terms).  So when we take the determinant, we’ll get a polynomial of degree n in terms of our unknown λ.

“If A is an n × n matrix, the polynomial

is called the characteristic polynomial of A, and the equation

is called the characteristic equation” (374).

This means that eigenvalues of the matrix A are the roots of its characteristic polynomial. We have three choices for roots: real ones, complex ones, and repeated ones.

As if you weren’t already overwhelmed by the amount of eigens floating around, here’s another one for you:

“Let A be an n × n matrix, and let λ be an eigenvalue of A. The set of all eigenvectors associated with λ is equal to the nullspace of A – λI. Hence, the eigenspace of λ is a subspace of Rⁿ” (375).

All right, that’s all 9.1 has to offer you. I guarantee you the next section summary won’t be up until at the very least Friday. My test is on Thursday, and I don’t want these eigens messing up my jive with this test. Thus I will put them to rest for now. I’m approximately two weeks ahead of the weekly schedule, so I’m feeling pretty good at where I’m at with everything (at least enough to be on hiatus for four to five days).

Also, something else to note is that the schedule goes out of order a bit with chapter 9. Just in terms of chapter 9, it goes 9.1, 9.2 + 9.5, 9.6, 9.3, 9.4, 9.7, 9.9, 9.8 + chapter 4, so I might just follow that pattern. Section 9.2 is fairly long (in the sense that I’m not excited to summarize it) so depending on how long summarizing that takes, I might just go in order.

I’ll see you when I see you.