# Linear systems modeling dynamics

## Homework Statement

R' = aJ
J' = bR

What happens to the graphs of R(t) and J(t)?

## The Attempt at a Solution

I made the matrix {{0, a}{b, 0}} and then got the equation (L=lambda) L^2 - 0L + (a+b) after computing the trace and determinant of that matrix. I then solved for the eigenvalues using the quadratic and got L1 = i sqrt(a+b) and L2 = -i sqrt(a+b)

But I didn't think I was supposed to get imaginary numbers when dealing with eigenvalues because then I can't graph the eigenvectors on the real plane. Did I do something wrong?

Thanks.

Dick
Homework Helper
Yes, the determinant of [[-L,a],[b,-L]] is L^2-ab. How did you get this (a+b) stuff?

Good call. I have no idea how I got (a+b) when it should be -ab.

Last edited:
Another question related to this problem. For the eigenvectors I got

(L=lambda)

e1 = {{b},{L-a}} = {{0},{b}}
e2 = {{L-d},{c}} = {{-b},{a}}

Now how do I use those eigenvectors to sketch a phase portrait of the system? My book doesn't explain it well.

Thanks.

Dick
Homework Helper
I don't think those are the eigenvectors either. Let's go back to the first question. What are the eigenvalues?

My bad. My last post was referring to a different problem.

Here's the other problem I'm working on:

R' = 0
J' = aR + bJ

{{0, 0},{a,b}}

L^2 - bL

L1 = (b+sqrt(b^2))/2 = b
L2 = (b-b)2/ = 0

e1 = {{b},{L-a}} = {{0},{b}}
e2 = {{L-d},{c}} = {{-b},{a}}

Now how do I go from those eigenvectors to drawing asymptotes and graphing it? I thought the eigenvectors were used to find the slope of the asymptotes, but they contain variables in this case. Then how do I know where to draw the curves and how are points moving along them in time?

Thanks

HallsofIvy