Normalizing eigenvector with complex entries

[R.Harmon]

Homework Statement

Hi, I'm having a bit of a problem normalizing eigenvectors with complex entries. Currently the eigenvector I'm looking at is $$\[\vec{v}=<br /> \left(\begin{array}{c}<br /> -2+i\\<br /> 1<br /> \end{array}\right)\]$$

Homework Equations

The Attempt at a Solution

If the eigenvectors don't have complex elements I can do this, for example if i have $$\[\vec{v}=<br /> \left(\begin{array}{c}<br /> 3\\<br /> 1<br /> \end{array}\right)\]$$ and I want to normalize I know that this is the same as $$\[\vec{v}=\left(\begin{array}{c}<br /> 3a\\<br /> 1a<br /> \end{array}\right)\]$$ and $$(3a)^2+a^2=1$$ so the normalized eigenvector is $$\vec{v}=\frac{1}{\sqrt{10}}<br /> \left(\begin{array}{c}<br /> 3\\<br /> 1<br /> \end{array}\right)$$. However with the first eigenvector using the same method I get $$(a(-2+i))^2+a^2=1$$ or $$a=\frac{1}{\sqrt{4-4i}}$$ giving the normalized eigenvector as $$\vec{v}=\frac{1}{\sqrt{4-4i}}<br /> \left(\begin{array}{c}<br /> -2+i\\<br /> 1<br /> \end{array}\right)$$ where as the solution should be $$\vec{v}=\frac{1}{\sqrt{6}}<br /> \left(\begin{array}{c}<br /> -2+i\\<br /> 1<br /> \end{array}\right)$$. Could someone please point out where I'm going wrong? Any help is appreciated.

 

[nicksauce]

I believe you should just divide by

sqrt( |-2 + i|^2 + |1|^2)
=
sqrt( 5 + 1) = sqrt(6)

 

[R.Harmon]

Unless my algebras gone out the window:
$$(-2+i)^2=(-2+i)(-2+i)=4-4i+(i^2)=3-4i$$
not 5?

 

[nicksauce]

|a + bi|^2 = (a + bi)(a - bi) = a^2 + b^2
or
|z|^2 = zz* (where z* is the complex conjugate)

so

|-2 + i|^2 = 4 + 1 = 5

 

[R.Harmon]

Ahh ok makes sense. Thanks a lot, that was bugging me for hours!