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 $\displaystyle \lambda$ vs $\displaystyle \bar{\lambda}$ to solve.

Here's an example.

$\displaystyle x'_1 = 2x_1-5x_2$

$\displaystyle x'_2=4x_1-2x_2$

So i take

$\displaystyle 0=\begin{vmatrix}2-\lambda &-5\\ 4 &-2-\lambda \end{vmatrix}$

(skip a few steps)

(some quadratic magic)

*poof*

$\displaystyle \lambda = \pm 4i$

Ok, so now, how do I know which is my $\displaystyle \lambda$ which is my $\displaystyle \bar{\lambda}$?

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 $\displaystyle \lambda = 4i$, I wind up with the vector $\displaystyle \begin{bmatrix}5\\ 2-4i\end{bmatrix}$.

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

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)

Re: question regarding complex eigen values.

It doesn't matter whether you work with 4i or -4i. You will wind up with the same set of solutions regardless. Try making up initial conditions and working the problem both ways. You will wind up with the same particular solution either way if you don't make any mistakes.