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How can I prove that the product of two rank-2 tensors, one of which is symmetric and one is antisymmetric, must =0 when their indices are summed over?

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  • $\begingroup$ For the expression $A^i _j S^j _i$ swap the indices in two ways, once by renaming $i$ and $j$(we are allowed to do this since these indices appear twice), second by using the symmetry properties of the two tensors. $\endgroup$ – Daron Sep 10 '13 at 23:35
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Let $\omega$ be antisymmetric and $g$ be symmetric. Then applying both symmetries gives

$$ \sum_{i,j} \omega^{ij} g_{ij} = -\sum_{i,j} \omega^{ji} g_{ji}. $$

Now since $i,j$ are both summed over the same range, we can swap them without changing the meaning of this expression, and thus we have shown the sum is equal to its negative; i.e. it is zero.

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