# Homework Help: Evaluating expression involving permutation symbol and Kronecker delta

1. Sep 9, 2012

### xWaffle

1. The problem statement, all variables and given/known data

Evaluate the following expression:
$\sum_{j}\sum_{k}\epsilon_{ijk}\delta_{jk}$

2. Relevant equations
$\delta_{ij}$ = $[i = j]$

3. The attempt at a solution
I don't have a solution attempt to this one yet, because somehow I completely missed out on what the permutation thing has to do with anything.
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This is the second expression given on this homework assigment. The first one was a little easier, which I did work out, and came up with the solution. I'm going to show you guys this first problem so you know I at least know a little of what I'm doing..

Evaluate expression:
$\sum_{i}\sum_{k}\delta_{ij}\delta_{ji}$

I used my knowlege of the Kronecker delta to say that:
$\delta_{ij}\delta_{ji} = \delta_{ii} = \delta_{jj}$

Then using my knowledge of the trace of an n x n matrix (since I'm only dealing with square matrices), the trace of an n x n matrix is just n. So the final solution to the expression I found to be:
$\sum_{i}\sum_{k}\delta_{ij}\delta_{ji} = \sum_{i}\delta_{ii} = tr(I_{i}) = i$
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So I do have some of the knowledge I'm expected to have, but I really have no idea how to progress further, with the $\epsilon_{ijk}$ thrown in there. Any help is greatly appreciated. Thanks