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Thread: Matrix Proof

  1. #1
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    Matrix Proof

    Hey, I can't figure out how to do this. I'd really appreciate any help.

    Question:

    A is a non-zero matrix of order m by k ($\displaystyle m \geq k$). Show that $\displaystyle A^T A$ is positive semidefinite.



    My thoughts:

    It's quite easy to show that $\displaystyle A^T A$ is symmetric, and I tried to use that fact and let $\displaystyle y = Ax$, and I tried to show $\displaystyle x^T y \geq 0 $, but didn't end up getting anywhere.

    Any ideas?

    Many thanks.
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  2. #2
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    Quote Originally Posted by WWTL@WHL View Post
    $\displaystyle x^T y \geq 0 $, but didn't end up getting anywhere.

    Any ideas?
    .
    Show that this is a sum of squares.
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  3. #3
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    Quote Originally Posted by ThePerfectHacker View Post
    Show that this is a sum of squares.
    $\displaystyle x^T (A^T A) x = x^T ( \sum^k_{i=1} a^2_i ) x $

    Does that hold?

    Sorry, I'm quite clueless on this as you can probably tell. Any more guidance, please?
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