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Thread: Distinct eigenvalues of a normal matrix

  1. #1
    Nov 2011

    Distinct eigenvalues of a normal matrix

    Given a matrix X\in\mathbf{R}^{3\times n} and a diagonal matrix D with known positive diagonal entries, what conditions may I impose on the columns of X such that XDX^T has distinct eigenvalues?

    Here are some things that I already know:

    If D=\text{diag}(a) and a\in\mathbb{R}^n, then


    Also, this seems to be the most promising lead I have:


    where \lambda_i and v_i is an eigenvalue/eigenvector of XDX^T.

    Because XDX^T is Hermitian, the eigenvectors define an orthonormal basis. Thus the change from the canonical basis to a basis of the eigenvectors is solely a rotation which does not change the geometry of the vectors x_j. Therefore, it seems that any geometrical properties that I might find before the change of basis might still hold, but I don't know how I can prove this.

    Last edited by uasac; November 26th 2011 at 08:07 PM.
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