0
$\begingroup$

The Kronecker delta can be represented by a two dimensional matrix:

\begin{gather} \delta_{ij}=\mathbb{I}= \begin{bmatrix} 1&0&0\\ 0&1&0\\ 0&0&1\\ \end{bmatrix}. \end{gather}

Similarly, can the Levi-Civita tensor \begin{gather} \epsilon_{ijk} \end{gather} be represented by a three dimensional matrix, and if so, what does the matrix look like?

$\endgroup$
  • 4
    $\begingroup$ $\uparrow$ Yes. $\endgroup$ – AccidentalFourierTransform Apr 30 '18 at 14:39
  • $\begingroup$ Wow! Short comment for a short question! Make a sandwich pile of three antisymmetric matrices. Three different antisymmetric matrices : one of bread, one of cheese, and one of peanut butter! ;-) $\endgroup$ – Cham Apr 30 '18 at 14:44
  • 2
    $\begingroup$ I'm reasonably certain that the Wikipedia entry in the LC tensor had exactly what you're looking for... $\endgroup$ – Kyle Kanos Apr 30 '18 at 15:58
3
$\begingroup$

From Wikipedia:

It might be a little messy, but imagine the layers as having the $i$ index, the rows having the $j$ index, and the columns having the $k$ index.

Chiefly, the Levi-Civita Tensor gives $1$ for cyclic permutations and $-1$ for anti-cyclic permutations. In the image above, the front most layer (blue) has index $i=1$. Hence, the indices in the layer is ${1,j,k}$.

For the cyclic combination ${1,2,3}$, that is, at layer $1$, row $2$, column $3$, the value is $1$. Similarly, for the anti-cyclic combination ${1,3,2}$, the value is $-1$ appears at layer $1$, row $3$, column $2$. There are repeats indices in other rows and columns combinations in this layer, and hence the other values are $0$ in this layer.

Similar arguments can be done for the middle layer (red), i.e. for indices ${2,j,k}$ and for the back most layer (green), i.e. for indices ${3,j,k}$, which is left as an exercise for the reader.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.