What is the result of multiplying a vector by its complex conjugate?

[hassouna]

I found that the equation is expressed by

there is outer product ...what I really don't get it is if j is a vector then the outer product of j and j is is obtained by multiplying each element of j by the complex conjugate of each element of j which is basically a matrix not a vector

 

[Chestermiller]

I found that the equation is expressed by

there is outer product ...what I really don't get it is if j is a vector then the outer product of j and j is is obtained by multiplying each element of j by the complex conjugate of each element of j which is basically a matrix not a vector

But when you take the divergence of the outer product of j and j, this yields a vector.

 

[hassouna]

divergence is a vector operator we can't operate it on matrix can't we??

 

[Chestermiller]

divergence is a vector operator we can't operate it on matrix can't we??

Yes. The divergence of a vector is a scalar. The divergence of a 2nd order tensor is a vector. You need to check the literature to see how to take the divergence of a tensor (basically a dyad). See Appendix A of Transport Phenomena by Bird, Stewart, and Lightfoot to see how to work with dyadics and other 2nd order tensors.

 

[hassouna]

thank you for your help