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Unitary Matrix mutually orthonormal vectors

  1. Nov 25, 2012 #1
    1. The problem statement, all variables and given/known data

    Demonstrate that the columns of a unitary matrix form a set of mutually orthonormal vectors.

    2. Relevant equations

    hint - form the vectors [tex]u_i = {U_{ji}}[/tex] and [tex]u_k={U_{jk}}[/tex] from the [tex]i^{th}[/tex] and [tex]j^{th}[/tex] columns of [tex]U[/tex] and make use of the relationship [tex]U^{\dagger}U=I[/tex]

    3. The attempt at a solution

    I thought my work was following the hint...but not sure...I know I need to end up with an identity matrix from the hint, in which it shows the 3 columns are (1,0,0) , (0,1,0), and (0,0,1), respectively...to show that I have a set of basis vectors in the unitary matrix...

    [url=http://postimage.org/][PLAIN]http://s9.postimage.org/sw7ugplvz/photo_3.jpg[/url] upload[/PLAIN]
     
    Last edited: Nov 25, 2012
  2. jcsd
  3. Nov 25, 2012 #2

    Dick

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    It looks like you've got it. The Kronecker delta symbol ## \delta_{jk} ## is 1 if j=k and 0 if j≠k. Doesn't that give you your identity matrix?
     
  4. Nov 25, 2012 #3
    thanks,

    I'm just a little confused because It looks like I only have one vector.

    Do I need to turn that vector into a column, and for column 1 let i=k=1 , and for column 2 let i=k=3 and for column 3 let i=k=3, that would give:
    [url=http://postimage.org/][PLAIN]http://s7.postimage.org/fj2bek7rf/photo_4.jpg[/url] free photo hosting[/PLAIN]

    but I'm just not sure if my previous math boiled down to that, I feel like I'm missing something in the proof
     
  5. Nov 25, 2012 #4
    thanks,

    would this be the next step? I feel like my notation is wrong or I did something wrong in the process

    [url=http://postimage.org/][PLAIN]http://s16.postimage.org/4f3mski91/photo_5.jpg[/url] upload pics[/PLAIN]
     
  6. Nov 25, 2012 #5

    Dick

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    Yes, it's wrong. You want to show that the inner products (your sums) are 1 or 0. That's doesn't show that the original matrix U is the identity. What does 'orthonormal' mean?
     
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