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Math Help - Special Eigenvalues

  1. #1
    Senior Member sfspitfire23's Avatar
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    Special Eigenvalues

    Hi all, I have been asked to diagonalize the matrix [-1,3 ; 3,2] and I am supposed to notice something special about the eigenvectors.

    Eigenvectors are as follows:

    [-1.61803 ; 1] and [.61803 ; 1]. They are independent etc....not sure what is so special about them...


    Thanks.
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  2. #2
    MHF Contributor
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    Re: Special Eigenvalues

    Hey sfspitfire23.

    I'm not exactly sure about the context of your question, but those vectors are independent.

    Don't know if that helps or not.
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  3. #3
    MHF Contributor

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    Re: Special Eigenvalues

    it might help if you wrote the ACTUAL values for the eigenvectors, and not just decimal approximations (decimals are BAD! unless you're a physicist. silly them).

    i get as eigenvalues:

    \lambda_1 = \frac{1+3\sqrt{5}}{2},\ \lambda_2 = \frac{1-3\sqrt{5}}{2}

    leading to the following eigenvectors:

    v_1 = \begin{bmatrix}1\\ \frac{1+\sqrt{5}}{2} \end{bmatrix},\ v_2 = \begin{bmatrix}1\\ \frac{1-\sqrt{5}}{2} \end{bmatrix}.

    note that:

    v_1 \cdot v_2 = 1^2 + \left(\frac{1+\sqrt{5}}{2}\right)\left(\frac{1-\sqrt{5}}{2}\right) = 1 + \frac{1 - 5}{4} = 0

    in other words, the eigenvectors are ORTHOGONAL.
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