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Diagonalize matrix by unitary transformation

  1. Mar 6, 2014 #1
    In an exercise I am asked to find the eigenvalues of a matrix A by demanding that a unitary matrix (see the attached file) diagonalizes it. I know I could just solve the eigenvalue equation but I think I am supposed to do it this rather tedious way.
    Now I have introduced an arbitrary unitary matrix and performed the matrix multiplication:
    U^(dagger)AU = D, where D is a diagonalized matrix.
    From this last relation I should somehow be able to find the eigenvalues, but I haven't been able to so far. Have I done it wrong or is there actually a way through this messy algebra. What should I do?
    Note that the unitarity demands that lu_kl^2+lv_kl^2 = 1, which I suspect could be useful, but are there any other relations, which I am missing?
     

    Attached Files:

  2. jcsd
  3. Mar 8, 2014 #2
    To do this you somehow have to know the eigenvectors of A; these eigenvectors v_i would form your matrix U, i.e. U = [v_1 v_2 ... v_n]. Not really sure about the unitary matrix assumption..it could be just that by making D be unitary, then D is said to diagonalizable and therefore your equation U^T*A*U = D holds.
     
  4. Mar 8, 2014 #3
    Oh you should check the wiki page for unitary matrices...there are some extra properties. The eigenvectors must be orthogonal.
     
  5. Mar 9, 2014 #4
    But I am guessing you can write up a general unitary matrix U and then by demanding that it diagonalizes A, you can find the eigenvalues. At least that is how I think the exercise is thought, but it is much easier to just solve the eigenvalue equation.
     
  6. Mar 9, 2014 #5
    yeah so from the pdf you uplinked thats exactly what is done. you have formulae for the eigenvalues in terms of the matrix elements from A and U. I assume you know A immediately, and maybe you can eliminate the eigenvector terms using the cross diagonal equations. The notation seems to be a bit excessive.
     
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