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Math Help - Order of eigenvalues

  1. #1
    Forum Admin topsquark's Avatar
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    Order of eigenvalues

    A recent post reminded me of a several year old question I have. I think I know the answer, but I wanted your thoughts on it.

    I was doing a homework assignment in Quantum Physics and one of the problems was to diagonalize a unitary Hermitian matrix. When I did this he claimed that the order of the elements in the diagonal was incorrect.

    Now, I typically don't care about the order of the elements in the diagonal. In Quantum we don't care about the order because the difference in order is represented by a similarity transformation between the two matrices, which has no effect on the Physics. When I pointed this out to my professor he gave me half the points back for the "error."

    However I never did check to see if there is some established paradigm for ordering the diagonal.

    So my question is this:
    When you diagonalize the matrix:
    \left ( \begin{array}{cc} -3 & 0 \\ 2 & -1 \end{array} \right )
    should the answer be
    \left ( \begin{array}{cc} -1 & 0 \\ 0 & -3 \end{array} \right )
    or
    \left ( \begin{array}{cc} -3 & 0 \\ 0 & -1 \end{array} \right )
    or is there no practical difference between the two?

    -Dan
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  2. #2
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    I'm not sure if this is what you're asking, but the order of the elements in the diagonal matrix has to be in the corresponding order to the eigenvectors used. That way, your eigenvalues correspond to the correct eigenvector. I'm not familiar with Quantum Physics at all, so ignore this if it's not what you're asking.
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by AfterShock View Post
    I'm not sure if this is what you're asking, but the order of the elements in the diagonal matrix has to be in the corresponding order to the eigenvectors used. That way, your eigenvalues correspond to the correct eigenvector. I'm not familiar with Quantum Physics at all, so ignore this if it's not what you're asking.
    I think what topsquark is asking is "is there a convention for ordering the
    the eigen values down the diagonal after diagonalising a matrix". In maths
    I think the answer is no, but there may be in Physics where presumably these
    are something like energy levels of eigen states (I'm guessing here as I don't
    recall what they do represent) and so at a guess it might be conventional
    to go from the lowest to the highest down the diagonal.

    RonL
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  4. #4
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by CaptainBlack View Post
    I think what topsquark is asking is "is there a convention for ordering the
    the eigen values down the diagonal after diagonalising a matrix". In maths
    I think the answer is no, but there may be in Physics where presumably these
    are something like energy levels of eigen states (I'm guessing here as I don't
    recall what they do represent) and so at a guess it might be conventional
    to go from the lowest to the highest down the diagonal.

    RonL
    I'm sorry, I should have been a bit more specific. Yes, I was trying to see if there was a convention in Mathematics. And I should have stated it in terms of ordering the basis, which would order the diagonal elements.

    Okay, I didn't think that there was a specific convention. As I said, in Physics we really don't care about the order of the basis elements because two bases are related by a similarity transformation which doesn't alter the Physics. So it seems to be the same for Mathematics as well.

    Thanks to both of you.

    -Dan
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