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Homework Help: Unitary matrices

  1. Jun 1, 2010 #1
    1. The problem statement, all variables and given/known data
    Hi

    Is it correct that when I have a unitary 3x3 matrix U, then

    |Un,1|2+|Un,2|2+|Un,3|2=|U1,n|2+|U2,n|2+|U3,n|2,

    since UH=U? Here n denotes some integer between 1 and 3.
     
  2. jcsd
  3. Jun 1, 2010 #2

    MathematicalPhysicist

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    Gold Member

    U=U* is called hermitian matrix not unitary, a unitary matrix satisifies: UU*=I.
    If you multiply what do you get?
     
  4. Jun 1, 2010 #3
    My book says that a unitary matrix satisfies UHU=I, i.e. UH=U-1.
     
  5. Jun 1, 2010 #4

    Cyosis

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    Homework Helper

    I don't think so. That is not an example of a unitary matrix that is Hermitian. You just wrote the definition of a unitary matrix in another form.

    Definition of a unitary matrix: [tex]UU^\dagger=I[/tex]. Then we multiply both sides with the inverse of U, which gives us [tex](U^{-1}U)U^\dagger=IU^\dagger=U^\dagger=U^{-1}[/tex].

    The definition of a Hermitian matrix is:

    [tex]U=U^\dagger[/tex]

    note that it is not the same as the equality you wrote in post #3.

    Use the definition of the conjugate transpose [tex](A^\dagger)_{ij}=\overline{A}_{ji}[/tex].
     
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