Values of ##k## for which ##A_{ij}A_{ij} = |\vec a|^2##?

[Incand]

Homework Statement

The antisymmetric tensor is constructed from a vector ##\vec a## according to ##A_{ij} = k\varepsilon_{ijk}a_k##.
For which values of ##k## is ##A_{ij}A_{ij} = |\vec a|^2##?

Homework Equations

Identity
##\varepsilon_{ijk}\varepsilon_{klm} = \delta_{il}\delta_{jm}-\delta_{im}\delta_{jl}##

The Attempt at a Solution

##A_{ij}A_{ij} = k^2\varepsilon_{ijk}\varepsilon_{ijm}a_ka_m = k^2\varepsilon_{jki}\varepsilon_{ijm}a_ka_m = k^2(\delta_{jj}\delta_{km}-\delta_{jm}\delta_{kj})a_ka_m = k^2(\delta_{km}-\delta_{km})a_ka_m =0##
Which I obviously shouldn't get but I can't see where I'm making an error.

 

[fzero]

You are using the summation convention, so
$$\delta_{jj} \equiv \sum_{j=1}^3 \delta_{jj}.$$
This gives a different numerical factor in front of the first ##\delta_{km}## term.

 

[Incand]

You are using the summation convention, so
$$\delta_{jj} \equiv \sum_{j=1}^3 \delta_{jj}.$$
This gives a different numerical factor in front of the first ##\delta_{km}## term.

Right thanks!