(I'm not sure that I even phrased the question correctly. I will explain more about this below.)

Given a k-tensor $T$, we can define an alternating k-tensor $Alt(T)$ in the following way:

where $\epsilon$ is the sign function.

My first question is, what do we even call the $Alt$? In my title, I called it the "Alternating Operator". I would like to know its formal name.

And here's my main question. Given alternating k- and l- tensors $\theta, \eta$, we have:

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It seems that the "Alternating Operator" distributes to the two tensors in the first equality. Why is this?

I am new to tensors (just starting studying this month). I am aware that the tensor product distributes, but is this in any way related to the reason why the Alternating Operator also distributes? If so, how?


1 Answer 1


I think if you call it the "Alternating operator" people will generally understand what you mean. But at least in my circles it is more frequently called "the (total) antisymmetrization of $T$".

For your main question: the operator $\mathscr{A}$ is a linear mapping from $\mathscr{T}^k$ to itself. For any linear mapping you have the distribution law $L(x - y) = L(x) - L(y)$.

(It is in fact not just a linear mapping but a projection, meaning that $\mathscr{A}^2 = \mathscr{A}$.)

  • $\begingroup$ Thanks. Just curious, what is your circle? $\endgroup$
    – Sally G
    May 20, 2020 at 1:22
  • $\begingroup$ For the purpose of this answer: math physics / differential geometry $\endgroup$ May 20, 2020 at 3:11

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