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Thread: Levi-Civita Alternating Symbol

  1. #1
    Apr 2010

    Levi-Civita Alternating Symbol


    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.

    Last edited by purakanui; Mar 23rd 2012 at 07:13 PM.
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  2. #2
    Mar 2012

    Re: Levi-Civita Alternating Symbol

    Hello purakanui,

    the Levi-Civita-Symbol is generally a tensor:

    $\displaystyle \epsilon_{ijk} : \mathcal{V} \times \cdots \times \mathcal{V} \rightarrow \mathbb{K} $

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

    $\displaystyle \epsilon_{ijk} = \left\{ \begin{matrix} 1 & , \quad \text{for even permutations} \\ -1& , \quad \text{for odd permutations}\\ 0&, \quad \text{else} \end{matrix}

    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:

    $\displaystyle \epsilon_{ij} := 1 \ \text{for ij=12,} -1 \ \text{for ij=21,} \ 0 $ else.

    Then a matrix representing this would look like this: $\displaystyle E = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} $

    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,

    $\displaystyle \epsilon_{ijk} = a^p_i b^q_j c^r_k \epsilon_{pqr} $ , using the summation convention, where

    $\displaystyle a^p_i, \ b^q_j, \ c^r_k $ are now really matrices,

    e.g. $\displaystyle a^p_i = \begin{pmatrix} a^1_1 & \cdots & a^3_1 \\ \vdots & \ddots & \vdots \\ a^3_1 & \cdots & a^3_3 \end{pmatrix} $

    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
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