# System of DiffEq Problem!

• Apr 22nd 2010, 10:43 AM
Mec
System of DiffEq Problem!
Hey guys,

First time poster here, love the site so far though. Thanks in advance for any help received. I'm really lost!

I'm doing a very simple system of differential equations problem here and I've run into a situation I don't know how to handle.

The differential equation is:

x' = $\left(\begin{array}{cc}1&i\\-i&1\end{array}\right)$ x

Now, I have the eigenvalues as 0 and 2. Two distinct real eigenvalues, should be easy, right!?

But my eigenvectors are complex... Namely:

$E_0$ = $\left(\begin{array}{cc}-i\\1\end{array}\right)$

$E_2$ = $\left(\begin{array}{cc}i\\1\end{array}\right)$

(Complex conjugates as expected with complex eigenvectors)

So how do I handle that? I don't know how to get a solution with real eigenvalues and complex eigenvectors... Help please!
• Apr 22nd 2010, 02:56 PM
Mec
Hi, just wanted to give this a friendly bump because I feel like its a rather simple answer that I'm missing.

For example, is it possible that the answer is unaffected by the complex eigenvectors?

Therefore, y = $c_1$ $\left(\begin{array}{cc}-i\\1\end{array}\right)$ $e^{0t}$ + $c_2$ $\left(\begin{array}{cc}i\\1\end{array}\right)$ $e^{2t}$

Does that look correct?