Operation to make an (m+n)th rank tensor of rank-m and rank-n tensors

[bjnartowt]

Homework Statement

We know that c[ij] = ab[j] is a way to make a rank-2 tensor from two rank-1 tensors. We also know that C[abcxyz]=A[abc]B[xyz] is a way to make a rank-6 tensor from two rank-3 tensors. However, is there a matrix representation of this? I know the idea of a 6-dimensional matrix is painful and seemingly-unwieldy, but the question came up when I was writing myself some classical mechanics notes. The Levi-Civita symbol, if re-realized as a 3D "matrix", can be written as a 3D "matrix" whose "layers" are the three rotation matrices Rx, Ry, Rz, and feeding in the argument theta = Pi/2.

Homework Equations

The Attempt at a Solution

 

[lippyka]

I don't think this is possible, as a 6-dimensional matrix would be far too unwieldy and difficult to work with.