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Homework Help: Hermitian matrix

  1. Oct 13, 2007 #1
    Its quantum computing but related to math:
    1. The problem statement, all variables and given/known data

    show every hermitian matrix can be diagonalized by unitary matrix. Prove this using. N x N matrix.

    2. Relevant equations
    H= hermitian matrix. U = unitary matrix
    show U-1(inverse)HU = D (diagonal) using N x N matrix.

    3. The attempt at a solution
    I don't know how to start.

    Any help would be helpful.
  2. jcsd
  3. Oct 13, 2007 #2
    This is fairly nontrivial, but easy if you know what to do. It is a special case of the finite dimensional spectral theorem, as hermitian matrices are normal.

    Sketch: Suppose A is normal. Then by Schur's Lemma we know that A is unitarily similar to an upper triangular matrix. That is, [tex]U^HAU=T[/tex].

    Now, since T is both triangular and normal we see that T is diagonal. Hence, [tex]A=UDU^H[/tex] upon block multiplication.

    I will leave it to you to show that T is normal and if T triangular and normal then T is diagonal.
    Last edited: Oct 13, 2007
  4. Oct 13, 2007 #3
    No, still not clear. I'm good with kind of proving. Can you be more specific on how to answer the question. I'm really really thankfu.
    It is possible to show using the Spectral Decomposition theorem that every hermitian matrix cab be diagonalzed by a unitary matrix. Let H be a an hermitian matrix and set S be unitary where S(inverse)HS= D where D is diagonal.
    Show that S(inverse) H S and hence D is hermitian
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