question regarding complex eigen values.

Ok, so I've read this

http://www.mathhelpforum.com/math-he...tml#post356455

But I'm still a bit confused.

My question is this: How do I know when to use $\lambda$ vs $\bar{\lambda}$ to solve.

Here's an example.

$x'_1 = 2x_1-5x_2$

$x'_2=4x_1-2x_2$

So i take
$0=\begin{vmatrix}2-\lambda &-5\\ 4 &-2-\lambda \end{vmatrix}$

(skip a few steps)
(some quadratic magic)
*poof*

$\lambda = \pm 4i$

Ok, so now, how do I know which is my $\lambda$ which is my $\bar{\lambda}$ ?
Which one do I use to grind through the rest of this?
Does it matter?

If I solve the matrix for the zero vector using $\lambda = 4i$ , I wind up with the vector $\begin{bmatrix}5\\ 2-4i\end{bmatrix}$ .

With $\lambda = -4i$ , however, I get $\begin{bmatrix}5\\2+4i\end{bmatrix}$ .

Given initial conditions, this will effect my result, correct? But it only changes the signs of my constants. Which would be expected since the sign of the equations I end up will will also be opposite?

(sorry if this is a stupid question)