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Complex matrix problem

  1. May 17, 2004 #1
    Let A and B be Hermitian matrices with AB = BA and let N = A + iB.

    1) Show that N is normal.

    2) Show that A = 1/2(N+N*) (* = conjugate transpose) and find a formula for B.

    3) Let U be a unitary matrix such that U*NU is a diagonal matrix. Show that U*AU and U*BU is diagonal matrices.

    I had no problems with 1) and 2) but I simply can't figure out 3)... Please help.
  2. jcsd
  3. May 17, 2004 #2

    matt grime

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    You can recover A form N, and if U diagonalizes N, does it diagonalize N*?
  4. May 17, 2004 #3


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    Clearly, U*NU= U*AU+ i U*BU. Since U*NU is a diagonal matrix, all non-diagonal elements are 0. That is, All non-diagonal elements of U*AU and iU*BU must cancel. What does that tell you about them individually (and don't forget the "i").
  5. May 17, 2004 #4
    I honorstly don't know... I don't think the fact that two matrices P and Q sum up to a diagonalmatrix D implies that they are diagonalmatrices themselves - it just means that their non-diagonal elements cancel - as you say yourself... Or what?
  6. May 17, 2004 #5

    matt grime

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    I don't think Hall's method works since it doesn't use at any point the properties of A, B and N, and would thus appear to be 'true' for all matrices, which isn't possible.

    However, U*NU diagonal implies (U*NU)*=U*N*U is diagonal, and you may recover U*AU from these two diagonal matrices using part 2
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