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Linear algebra - eigenvalues and eigenvectors and hermitian

  1. Jul 28, 2010 #1
    1. The problem statement, all variables and given/known data

    I attached the problem in a picture so its easier to see.

    2. Relevant equations



    3. The attempt at a solution

    These are the values i got
    [tex]\lambda[/tex]_ 1 = 1
    [tex]\lambda[/tex]_ 2 = -1

    x_1 = [-i; 1] (x_1)^H = [i 1]
    x_2 = [ i; 1] (x_2)^H = [-i 1]
    * where x_1 and x_2 are 2x1 matricies, and their hermitians are 1x2

    after each multiplication I got
    [tex]\lambda[/tex]_ 1 x_1 (x_1)^H =
    [1 -i
    i 1]

    [tex]\lambda[/tex]_ 2 x_2 (x_2)^H =
    [-1 -i
    i -1]

    When I add these together I get
    [0 -2i
    2i 0]

    which is not the original A_5. I can't figure out where I went wrong in this process. If someone could look over it and let me know that would be great. Thank you
     

    Attached Files:

    Last edited: Jul 28, 2010
  2. jcsd
  3. Jul 28, 2010 #2
    No picture?
     
  4. Jul 28, 2010 #3
    Did you forget the picture, I can't see any.
     
  5. Jul 28, 2010 #4
    I think I forgot to press the upload button after trying to attach it. Sorry about that. I edited it and it's there now
     
  6. Jul 28, 2010 #5
    The spectral decomposition allows you to write a Hermitian matrix as a linear combination of projections onto the orthonormal basis of eigenvectors. So you need to normalize your eigenvectors, and then the 2 will nicely vanish.
     
  7. Jul 28, 2010 #6
    oh! that worked perfectly. Thank you, I would've never gotten that
     
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