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Math Help - Matrix Symmetry Proof

  1. #1
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    Matrix Symmetry Proof

    I am to prove that if A is antisymmetric, then x^TAx = 0 for any column vector x. I know that by definition A^T = -A, so (x^TAx) = -(x^TAx). I have tried taking the transpose of both sides but this does not seem to help prove the above.

    Any help would be appreciated.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Use that

    x^tAx\in \mathbb{R}^{1\times 1}

    so, it is symmetric.
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  3. #3
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    Quote Originally Posted by FernandoRevilla View Post
    Use that

    x^tAx\in \mathbb{R}^{1\times 1}

    so, it is symmetric.
    This completes the proof very nicely. However, how do I show formally that this is true?
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  4. #4
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    xᵀ is 1xn, A is nxn, x is nx1, so Ax is nx1, so xᵀ(Ax) is 1x1 (it's just the inner product of x and Ax).
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