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I have the following question

"Let $A_{ij}$ be a symmetric tensor and let $B_{ij}$ be an antisymmetric tensor. Prove that the inner product of $A_{ij}$ and $B_{ij}$ is zero."

How would I go about doing this? I know that $A_{ij}=A_{ji}$ and $B_{ij}=-B_{ji}$ but I'm not too sure how this helps.

Any help is greatly appreciated, thank-you in advance.

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    $\begingroup$ Hi, welcome. One way to show a number is zero is to show that it's equal to its negative. Might that help? $\endgroup$ May 4, 2019 at 10:47
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    $\begingroup$ @MatthewLeingang I remember when this result was first shown in my general relativity class, and your argument was pointed out, and I kept thinking to myself "except in characteristic 2", waiting for the professor to say it. Then I realized that this was a physics class, not an algebra class. $\endgroup$
    – Arthur
    May 4, 2019 at 10:52
  • $\begingroup$ @Arthur These physicists, with their quantification of real-valued things you can measure! $\endgroup$
    – J.G.
    May 4, 2019 at 11:01

1 Answer 1

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Contracting these equations, $A_{ij}B_{ij}=-A_{ji}B_{ji}=-A_{ij}B_{ij}$, where the second $=$ sign relabels the indices. Hence $2A_{ij}B_{ij}=0$.

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  • $\begingroup$ Im not getting how come the second sign comes $\endgroup$
    – haider
    Jan 16, 2020 at 9:14
  • $\begingroup$ The first $=$ multiplies $A_{ij}=A_{ji}$ with $B_{ij}=-B_{ji}$; the second $=$ uses $X_{ji}Y_{ji}=X_{ij}Y_{ij}$ for any $X,\,Y$, because $i,\,j$ are just dummy indices whose names can be swapped. You can think of it as $X_{ji}Y_{ji}=X_{lk}Y_{lk}=X_{ij}Y_{ij}$, where first we rename $i$ to $k$ and $j$ to $l$, then we rename $k$ to $j$ and $l$ to $i$. $\endgroup$
    – J.G.
    Jan 16, 2020 at 10:35

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