# Complex Eigenvalues - Proof

1. Mar 10, 2008

### rocketboy

1. The problem statement, all variables and given/known data

This is the second part of a multi-part question. Part (a) shows that:

x'' = Ax = $$\left(\stackrel{-2}{4/3}\stackrel{3/2}{-3}\right)$$x

Part (b): Assume x = $$\epsilon$$e$$^{rt}$$ and show that (A - r$$^{2}$$I)$$\epsilon$$ = 0

x is the solution to the second order differential equation above, and $$\epsilon$$ is an eigenvector corresponding to the eigenvalue r$$^{2}$$ of A.

3. The attempt at a solution

Part (b): Alright, so given the above, I stated that $$\epsilon$$ = e$$^{-rt}$$x

I then substituted everything into the left side of the equation I'm trying to prove to obtain:

(A - r$$^{2}$$I)$$\epsilon$$ = $$\left(\stackrel{-2-r^{2}}{4/3}\stackrel{3/2}{-3-r^{2}}\right)$$$$\left(\stackrel{x_{1}}{x_{2}}\right)$$e$$^{-rt}$$

from here, I can see that I am not going in the right direction... any suggestions to get me moving along?

Thank-you!
-J

EDIT: for clarification, the above are 2x2 matrices... latex put the entries fairly close together. The matrix A has entries (-2 3/2) on the top and (4/3 -3) on the bottom.

Last edited: Mar 10, 2008
2. Mar 10, 2008

### HallsofIvy

Staff Emeritus
What is $\epsilon$? A vector? But since you also don't tell us what x is, what see no way to understand x= $\epsilon e^{rt}$.

3. Mar 10, 2008

### rocketboy

Sorry, thanks, I'll edit my above post to be more complete.

Note: for some reason my $$\epsilon$$ appear to be superscripts... nowhere should this be the case.

Last edited: Mar 10, 2008