# Inner product of symmetric and anti-symmetric tensors

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.

• Hi, welcome. One way to show a number is zero is to show that it's equal to its negative. Might that help? May 4, 2019 at 10:47
• @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. May 4, 2019 at 10:52
• @Arthur These physicists, with their quantification of real-valued things you can measure!
– J.G.
May 4, 2019 at 11:01

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