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Math Help - If A is an invertible matrix, then A+A^T is skew-symmetric. (Proof).

  1. #1
    s3a
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    If A is an invertible matrix, then A+A^T is skew-symmetric. (Proof).

    My teacher wrote:

    (A+A^T)^T = A^T + (A^T)^T = A^T + A

    While I get this algebraically, I don't see how it proves skew-symmetry. Like how does this relate to A = -A^T?! Also, I just wanted to confirm if the reason why the stuff (=bolded part) in (stuff)^T is A+A^T because we want to force them to be square matrices since you can only add matrices that have the same dimensions.

    Any input would be greatly appreciated!
    Thanks in advance!
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by s3a View Post
    My teacher wrote:

    (A+A^T)^T = A^T + (A^T)^T = A^T + A

    While I get this algebraically, I don't see how it proves skew-symmetry. Like how does this relate to A = -A^T?! Also, I just wanted to confirm if the reason why the stuff (=bolded part) in (stuff)^T is A+A^T because we want to force them to be square matrices since you can only add matrices that have the same dimensions.

    Any input would be greatly appreciated!
    Thanks in advance!
    I would guess they meant \left(A-A^{\top}\right)^{\top}=A^{\top}-A=-\left(A-A^{\top}\right)
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  3. #3
    s3a
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    So the way it should have been phrased was?: "If A is an invertible matrix, then A-A^T is skew-symmetric."
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by s3a View Post
    So the way it should have been phrased was?: "If A is an invertible matrix, then A-A^T is skew-symmetric."
    That's my guess
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