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    Eigenvalues and Eigenvectors of a Hermitian Matrix

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    Consider the Hermitian matrix

    omega = 1/2
    [2 0 0
    0 3 -1
    0 -1 3]

    (1) Show that omega1 = omega2 = 1; omega3 = 2.

    (2) Show that |omega = 2> is any vector of the form

    1/((2a^2)^(1/2)) [0 a -a]

    (3) Show that the omega = 1 eigenspace contains all vectors of the form

    1/((b^2 + 2x^2)^(1/2)) [b c c]

    either by feeding omega = 1 into the equations or by requiring that the omega = 1 eigenspace be orthogonal to |omega = 2>.

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    The characteristic polynomial of is:

    The eigenvalues satisfy:

    Thus we get:

    So the eigenvalues are (with indices now assigned sequentially):

    To find the ...

    Solution Summary

    The solution describes how to find the eigenvalues and eigenvector of a 3 x 3 Hermitian matrix.