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Thread: Distinct eigenvalues of a normal matrix

  1. #1
    Nov 2011

    Distinct eigenvalues of a normal matrix

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

    Here are some things that I already know:

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

    $\displaystyle XDX^T=\sum_{j=1}^na_jx_jx_j^T$.

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

    $\displaystyle \lambda_i=\sum_{j=1}^na_j(x_j^Tv_i)^2$,

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

    Because $\displaystyle XDX^T$ 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 $\displaystyle x_j$. 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; Nov 26th 2011 at 08:07 PM.
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