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Linear Algebra

  1. Sep 27, 2006 #1
    Find the diagonal form of the Hermitian matrix

    [tex]A=\left(
    \begin{array}{cc}
    2 & 3i\\
    -3i & 2
    \end{array}
    \right)
    [/tex]

    The spectral theorem could be used with PAP*=D where D is diagonal matrix and P is a unitary matrix.

    I put the columns of P as the eigenvectors (with unit length) of A,

    [tex]P=\frac{1}{\sqrt{2}}\left(
    \begin{array}{cc}
    i & -i\\
    1 & 1
    \end{array}
    \right)
    [/tex]

    I have checked that P is unitary with [tex]P^{-1}=P^{*}[/tex] and the diagonal entries of D should be 5 and -1. But I got

    [tex]D=\left(
    \begin{array}{cc}
    2 & -3\\
    -3 & 2
    \end{array}
    \right)
    [/tex]

    which clearly isn't correct.
     
    Last edited: Sep 27, 2006
  2. jcsd
  3. Sep 28, 2006 #2

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    Yes, you're right it isn't correct. I'm, nots sure what you want anyone here to do. You have the method correct, so just make sure you'renot making any dumb mistakes in multiplying out matrices.
     
  4. Sep 28, 2006 #3
    hey
    rigth method but wrong eigenvalues

    P = 1/sqrt(2) [i -1;i 1]

    this will help
     
  5. Sep 28, 2006 #4

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    I can never remember for sure, but isn't it supposed to be A = PDP*? (and thus P*AP = D?)
     
  6. Sep 28, 2006 #5
    That could be my mistake.
     
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