Thread: Levi-Civita Alternating Symbol

Hi,

I am stuck on the following question:

What is the matrix representation of e_ij...k in an arbitrary basis?

As all the slots for e are for vectors, then you can choose {b_i} as the basis. Then you could write in this basis

e_ij...k b_i b_j...b_k?

I am unsure where to go. Any hints would be great.

Thanks!

Hello purakanui,

the Levi-Civita-Symbol is generally a tensor:



mapping the n-fold cartesian product of a vector space to its underlying field (usually the real or complex numbers).



However if you represented it as matrix, it'd have to have only 2 indices. Of course one can define the Levi-Civita-Symbol with 2 indices:

else.

Then a matrix representing this would look like this:

However, I have to warn you: This E is actually NOT really a matrix.
At least it doesn't behave like a matrix under the usual multiplication of a matrix with vectors.

So better leave the tensor notation of the Levi-Civita-symbol as it is instead of writing it as an array.

If you want to express the Levi-Civita-symbol in another basis, then you have to apply basis transformations on it,

, using the summation convention, where

are now really matrices,


e.g.

But use the summation convention to calculate the values!

I have to remark that it might well be that the Levi-Civita-Symbol is defined as it is, in every basis, but I don't know.
Here is how the Levi-Civita-Symbol with 3 indices may be represented:

Visualization of the Levi-Civita symbol as a 3×... - Visual Hints - Quora