Three-Dimensional Matrix Multiplication

[marschmellow]

Does this concept exist? Google yields weird results that mostly have to do with programming, and Wikipedia says nothing about it. I always find that I understand tensor math better when I can translate it into matrix notation, but if I'm dealing with tensors of too high a rank, I don't know what to do anymore.

And if it does exist, I have some questions about the function of the extra dimension in the arrays. Column vectors correspond to vectors, and row vectors correspond to 1-forms, so what would vectors "orthogonal" (in the physical notation space) to row vectors and column vectors correspond to?

 

[fresh_42]

What you are asking is, how to multiply $\left( \sum u_\rho \otimes v_\rho \otimes w_\rho \right)\left( \sum u'_\tau \otimes v'_\tau \otimes w'_\tau \right)$ given as vectors in coordinate form and write the result in this form again.

Now here we are at the crucial point: How should this multiplication be defined? There are various possibilities, depending on whether dual vector spaces are involved, what the purpose of such a multiplication is, whether it is in three dimensional space where we have a cross product, whether it is just the multiplication in the tensor algebra etc.

So the answer to your question is: yes, but details depend on details given.