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Math Help - transpose and symmetry

  1. #1
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    transpose and symmetry

    Prove if (transpose of A)*A = A, then A is symmetric and A = A^2:


    I think I understand it, but Im not sure if I can put it into proper terms
    Since the transpose of A = A (if symmetric), then if you multiply both sides of the equation, you get (Transpose of A)*A = A^2

    so therefore A must = A^2 (using the original statement...if (transpose of A)*A = A...)

    Any clarification please?
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  2. #2
    Senior Member TheAbstractionist's Avatar
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    A\ =\ A^\mathrm TA

    \implies\ A^\mathrm T\ =\ (A^\mathrm TA)^\mathrm T\ =\ A^\mathrm T(A^\mathrm T)^\mathrm T\ =\ A^\mathrm TA\ =\ A

    Hence A is symmetric and A=A^\mathrm TA=AA=A^2.
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