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How to calculate complex dot products

  1. Apr 16, 2012 #1
    let y1=(1,0,i,0) and y2=(0,i,1,0)

    what is (y1.y2)/(y1.y1)?

    I make it i/(1+i2) = i/0 which seems incorrect.

    My notes seem to give the answer as -i/2 and I don't understand how this was calculated.

    Any help is appreciated. Thanks
     
  2. jcsd
  3. Apr 16, 2012 #2
    hmm.. I get (i/0) as well.

    (1*0 + 0*i + i*1 + 0²) / (1² + 0² + i² + 0² ) =

    (0 + 0 + i + 0 ) / (1 + 0 -1 +0) =

    (y1*y2)/(y1²)= i/0

    Does anyone get a different answer?
     
  4. Apr 16, 2012 #3
    I suspect the answer is to do with the fact that

    <z1,z2>=<(z2)*,z1> where (z2)* is the Hermitian conjugate of z2, but even if this is the case i'm not sure how to carry out the calculation.
     
  5. Apr 16, 2012 #4
    It seems to work with y1 = (i,0,i,0), you probably didn't copy it correctly.
     
  6. Apr 16, 2012 #5


    The usual (euclidean) complex dot product in [itex]\mathbb C^n[/itex] is defined as the sum of the

    products of the first vector's coordinates times the conjugate of second vector's coordinates.

    In your case, [itex]y_1\cdot y_1 = 1\cdot 1 + 0 + i\cdot (-i) +0=2[/itex] , and this is what must appear in the denominator.

    DonAntonio
     
  7. Apr 16, 2012 #6
    Thank you DonAntonio
    I must have missed this definition. When you apply it to the numerator as well, you get the correct answer -i.
    Thanks
     
  8. Apr 16, 2012 #7


    No. It is either [itex]y_1\cdot y_2 = i\,\,or\,\,y_2\cdot y_1=-i[/itex]. Check this.

    DonAntonio
     
  9. Apr 16, 2012 #8
    Yes it was y[2].y[1] = -i . From the Gram-Schmidt Process although I changed the notation slightly.
    Thanks
     
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