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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.
    Question:
    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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