Complex confusion

chickster · 13-04-2009: 13th April 2009 16:57

Hi,

Whilst trying to work out eigenvalues i have got $\lambda ^2 - \lambda +1 =0$ .
Now i know this involves complex numbers and using the quadratic formula I have got $\lambda = \frac{1}{2} \pm \frac{\sqrt {3}}{2}i$ . But what is this actually saying about the root, as i know it equals $e^{\frac{\pi i}{3}} =w$ but I am not sure how to write it in brackets. as I would have just put $\lambda ^2 - \lambda +1 = (\lambda - w) =0$ . But i think there are meant to be other brackets involved but i am not even sure what root w is, as i presume it has to be lambda subtracting the different powers of w but i don't know which powers are the relevant ones.

Sorry if i sound a bit stupid but have been revising all day and am confuzzled!

Thanks for any help you can offer.

manderlay in flames · 13-04-2009: 13th April 2009 17:10

geometrically, i think it's saying that on a flat cartesian plane the transformation will change the direction of any vector,

chickster · 13-04-2009: 13th April 2009 18:01

yerr but how many roots are there meant to be? Or do i just use the one i have?

ghostwalker · 13-04-2009: 13th April 2009 19:45

Originally Posted by chickster

Hi,

Whilst trying to work out eigenvalues i have got $\lambda ^2 - \lambda +1 =0$ .
Now i know this involves complex numbers and using the quadratic formula I have got $\lambda = \frac{1}{2} \pm \frac{\sqrt {3}}{2}i$ . But what is this actually saying about the root, as i know it equals $e^{\frac{\pi i}{3}} =w$ but I am not sure how to write it in brackets. as I would have just put $\lambda ^2 - \lambda +1 = (\lambda - w) =0$ . But i think there are meant to be other brackets involved but i am not even sure what root w is, as i presume it has to be lambda subtracting the different powers of w but i don't know which powers are the relevant ones.

Sorry if i sound a bit stupid but have been revising all day and am confuzzled!

Thanks for any help you can offer.

There are two roots, as you've correctly worked out.

If you want to call one of them "w", that's fine, as long as you're not trying to convince me that it's a cube root of unity (which it isn't). If w is one root, then the other is clearly its conjugate.

So, you have:

$(\lambda-w)(\lambda - \bar{w}) = 0$

where $w=\frac{1}{2}+\frac{\sqrt{3}}{2}i$

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Complex confusion

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