Results 1 to 1 of 1

Math Help - Distinct eigenvalues of a normal matrix

  1. #1
    Newbie
    Joined
    Nov 2011
    Posts
    14

    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

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

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

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

    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.

    Thanks
    Last edited by uasac; November 26th 2011 at 09:07 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 10
    Last Post: December 19th 2011, 10:34 AM
  2. Eigenvalues of a 3 x 3 Matrix
    Posted in the Algebra Forum
    Replies: 6
    Last Post: November 24th 2011, 06:18 PM
  3. Replies: 2
    Last Post: November 27th 2010, 04:07 PM
  4. Finding a value in a matrix with distinct eigenvalues
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: November 28th 2009, 02:16 PM
  5. Matrix eigenvalues
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: January 29th 2009, 08:27 AM

Search Tags


/mathhelpforum @mathhelpforum