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Relationship between Trace and Determinant of Unitary Matrices

  1. Mar 24, 2012 #1
    1. The problem statement, all variables and given/known data

    If U is a 2 x 2 unitary matrix with detU=1. Show that |TrU|≤2. Write down the explicit form ofU when TrU=±2

    2. Relevant equations

    Not aware of any particular equations other than the definition of the determinant and trace.

    3. The attempt at a solution

    I have attempted this problem in several different ways to no avail. I'm pretty sure that the explicit form of U when the trace is 2 is simply the 2 x 2 identity matrix.

    I do know that

    Tr(AB) ≠ Tr(A)Tr(B)

    and

    U*U = I

    but not sure where to go with this.


    Any help is greatly appreciated.
     
    Last edited: Mar 24, 2012
  2. jcsd
  3. Mar 24, 2012 #2
    remember that for diagonal matrices the determinant is given by the product of the eigenvalues
     
  4. Mar 24, 2012 #3

    I like Serena

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    Perhaps you can use one of the equivalent definitions of a unitary matrix?
    See: http://en.wikipedia.org/wiki/Unitary_matrix
    I'm thinking that the columns of a unitary matrix form an orthonormal basis...
     
  5. Mar 25, 2012 #4
    Sgd37 I am aware that the product of the eigenvalues has to be 1, but there are an infinite number of combinations that I can create. I'm not sure if the eigenvalues have to be real or not since I don't think that the unitary matrix is necessarily Hermitian.

    I'm probably missing something simple.

    Thanks so much.
     
  6. Mar 25, 2012 #5

    I like Serena

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    In an orthonormal basis, the vectors have length 1.
    So the highest value in such a vector is 1.
     
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