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Unitary matrices

  1. Feb 15, 2010 #1
    Hi guys

    I have been sitting here for a while thinking of why it is that for a unitary matrix U we have that UijU*ji = |Uij|2. What property of unitary matrices is it that gives U this property?

    Best,
    Niles.
     
  2. jcsd
  3. Feb 15, 2010 #2

    quasar987

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    This is true of any matrix: [tex]U^*=\overline{U}^T[/tex]. So [tex]U^*_{ji}=\overline{U_{ij}}[/tex]. And of course, for any complex number z, we have [itex]z\overline{z}=|z|^2[/itex]
     
  4. Feb 16, 2010 #3
    By an asterix I meant complex conjugation, so [tex]
    (U^\dagger )_{ij} = (U_{ji})^*
    [/tex]. Is it still valid then?
     
    Last edited: Feb 16, 2010
  5. Feb 16, 2010 #4

    quasar987

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    It is not valid in this case. Take for example the rotation matrix

    cos(t) -sin(t)
    sin(t) cos(t)

    It is orthogonal, hence unitary. But for any sin(t) different from 0, we have [itex]U_{12}U^*_{21}=-\sin^2(t)\neq |\sin(t)|^2=|U_{12}|^2[/itex].
     
  6. Feb 17, 2010 #5
    Hmm, I have a problem then. I have a transformation

    [tex]
    \mathbf{m} = S\mathbf{a},
    [/tex]

    which has the components

    [tex]
    m_i = \sum_j S_{ij}a_j.
    [/tex]

    Now I want to find the Hermitian conjugate (I denote this by a dagger, and complex conjugation is denoted by an asterix), and we have

    [tex]
    \mathbf{m}^\dagger = \mathbf{a}^\dagger S^\dagger,
    [/tex]

    which has the components

    [tex]
    m^\dagger_i &= \sum_j a_j^\dagger (S^\dagger)_{ij} \\
    &=\sum_j a_j^\dagger (S^*)_{ji}.
    [/tex]

    My teacher says the last step is wrong, but I cannot see why. Can you help me spot the error?
     
    Last edited: Feb 17, 2010
  7. Feb 17, 2010 #6

    quasar987

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    Your mistake is when you say

    [tex]m^\dagger_i &= \sum_j a_j^\dagger (S^\dagger)_{ij}[/tex]

    According to the definition of matrix multiplication, correct is

    [tex]m^\dagger_i &= \sum_j a_j^\dagger (S^\dagger)_{ji}[/tex]
     
  8. Feb 17, 2010 #7
    Ahh, I see it now. Of course the column has to be fixed, not the row. Thanks.
     
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