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

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

But my eigenvectors are complex... Namely:

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

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

(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!