Tue Oct 21

MAIN POINTS
The premise of the reading is that we want to solve a differential equation in terms of a matrix, i.e y'=Ay. The text defines an eigenvalue, a constant l such that Av=lv for some vector v. Then, A solutin to x'=Ax is x(t)=e^(lt)v.

We can find eigenvalues, first by solving 0=Av-lv=(A-lI)v, which requires that det(A-lI)=0. We solve this ("characteristic") polynomial for 0, and the roots are the values of l that are eigenvalues.

Linear systems of dimension 2 are planar systems. The eigenvalues of the matrix A that represents a planar system can be calculated simply by using the quadratic equation (to find the roots of the characteristic polynomial of A). Because we use the quadratic equation, we can see that we could have two distinct real roots, or two complex roots, or one root of multiplicity 2. The text explores these three cases. In the case of a root of multiplicity 2, there is a unique solution that exhausts all exponential solutions.

CHALLENGES
What are the qualitative changes we see in a planar system when it has complex roots, or roots of multiplicity 2? It would be good to see an illustration of a planar system, because we haven't really thought about diff eq's above 1 dimension.

REFLECTIONS
I'm assuming that you can rewrite a system of differential equations as a matrix, and then use eigenvalues of that matrix to solve for that system. Otherwise, what is the difference between a differential equation defined in terms of matrices, and one defined by a system of equations?
Again, it would be cool to see some examples so we can visualize what's actually happening.