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Normalising Imaginary Eigenvector

  1. Nov 3, 2015 #1
    Hello,

    whilst solving a system of coupled differential equations I came across an eigen vector of ##\vec{e_{1}} = (^{1}_{i})##.

    Assuming that this is a correct eigenvector, how do I normalise it? I want to say that ##\vec{e_{1}} = \frac{1}{\sqrt{2}} (^{1}_{i})## but if I sum ##1^{2} + i^{2}## I get zero.

    It seems sensible to me that the vector's length is root two, but how do I justify this, if at all?

    Thank you.
     
  2. jcsd
  3. Nov 3, 2015 #2

    Orodruin

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    In a complex vector space, you have to introduce an inner product which satisfies ##\langle x, y\rangle = \overline{\langle y,x\rangle}##.
     
  4. Nov 3, 2015 #3
    The squared length of a complex vector v is defined by v.v(bar) where v(bar) is the complex conjugate, i believe. That will give you sqrt2
     
  5. Nov 4, 2015 #4
    Ah of course. A complex number is essentially a vector.

    Thank you.
     
  6. Nov 4, 2015 #5

    Orodruin

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    I agree if you take away the "essentially". It is an element in a complex one-dimensional vector space. :)
     
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