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Normalizing eigenvector with complex entries

  1. Jan 7, 2008 #1
    1. The problem statement, all variables and given/known data
    Hi, I'm having a bit of a problem normalizing eigenvectors with complex entries. Currently the eigenvector I'm looking at is [tex]\[\vec{v}=
    \left(\begin{array}{c}
    -2+i\\
    1
    \end{array}\right)\][/tex]


    2. Relevant equations



    3. The attempt at a solution

    If the eigenvectors don't have complex elements I can do this, for example if i have [tex]\[\vec{v}=
    \left(\begin{array}{c}
    3\\
    1
    \end{array}\right)\][/tex] and I want to normalize I know that this is the same as [tex]\[\vec{v}=\left(\begin{array}{c}
    3a\\
    1a
    \end{array}\right)\][/tex] and [tex](3a)^2+a^2=1[/tex] so the normalized eigenvector is [tex]\vec{v}=\frac{1}{\sqrt{10}}
    \left(\begin{array}{c}
    3\\
    1
    \end{array}\right)[/tex]. However with the first eigenvector using the same method I get [tex](a(-2+i))^2+a^2=1[/tex] or [tex]a=\frac{1}{\sqrt{4-4i}}[/tex] giving the normalized eigenvector as [tex]\vec{v}=\frac{1}{\sqrt{4-4i}}
    \left(\begin{array}{c}
    -2+i\\
    1
    \end{array}\right)[/tex] where as the solution should be [tex]\vec{v}=\frac{1}{\sqrt{6}}
    \left(\begin{array}{c}
    -2+i\\
    1
    \end{array}\right)[/tex]. Could someone please point out where I'm going wrong? Any help is appreciated.
     
  2. jcsd
  3. Jan 7, 2008 #2

    nicksauce

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    Science Advisor
    Homework Helper

    I believe you should just divide by

    sqrt( |-2 + i|^2 + |1|^2)
    =
    sqrt( 5 + 1) = sqrt(6)
     
  4. Jan 7, 2008 #3
    Unless my algebras gone out the window:
    [tex](-2+i)^2=(-2+i)(-2+i)=4-4i+(i^2)=3-4i[/tex]
    not 5?
     
  5. Jan 7, 2008 #4

    nicksauce

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    Science Advisor
    Homework Helper

    |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
     
  6. Jan 7, 2008 #5
    Ahh ok makes sense. Thanks alot, that was bugging me for hours!
     
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