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Math Help - Symmetric Matrix - Eigenvectors, Eigenvalues

  1. #1
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    Symmetric Matrix - Eigenvectors, Eigenvalues

    Hi, just wondering if anyone would be able to assist me with this question.

    Assume that is symmetric. Let be an orthonormal basis of eigenvectors with corresponding (real) eigenvalues .

    Show that the 2-norm of any vector satisfies
    .

    Note, I have shown in part a of the question that:


    Any help is appreciated!
    Last edited by shinn; March 24th 2012 at 08:42 AM.
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  2. #2
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    Re: Symmetric Matrix - Eigenvectors, Eigenvalues

    This isn't my area, but since A is symmetric, isn't v_j=v_j^T? That would get you the \left( v_j^T \right)^2 part. I'm not sure if this helps any or not.
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  3. #3
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    Re: Symmetric Matrix - Eigenvectors, Eigenvalues

    Thank you for your attempt, but I figured it out.

    Setting V as the matrix with columns v_j you can write this in matrix notation. v_j are orthonormal, or in matrix notation V^t V = I which also means VVt = I. |x|^2 = x^t x, and the j'th entry of V^t x is v_j^t x, and:

    |x|^2 = x^t x = x^t I x = x^t V V^t x = (V^t x)^t (V^t x)=|V^t x|^2.

    But, I also have a follow up question to this which is to show that [ ||Ax || \leq |\lambda_1| ||x|| ] where the eigenvalues are ordered such that .

    I'm guessing we need to use the previous part of the question, but after substituting Ax into the LHS, I get stuck showing the inequality.

    Would it possible to give me some idea on this one? Thanks!
    Last edited by shinn; March 24th 2012 at 05:32 PM.
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