3 by 3 by 4 Tensors

We now study the case of tensors.

Visualizations over the Real numbers

Since the maximum rank of tensors is 5, the maximum rank of tensors is at least 5 and at most 8. [C-B-L-C2009] (table 1) states (as know previously) that the typical rank is 5.

A Random Example

We choose , , and randomly in . Since these three tensors are in , any tensor on the line connecting any two of them is in , and a tensor in the plane is generically in , which we is large enough to capture general tensors.

We first produce a visualization of on , and obtain

[data file 3x3x4pr5random.dat] Next, using the same set of random , , and , we produce a visualization of , and obtain

[data file 3x3x4pr6random.dat] Initially the upper-right corner was generally lighter, so we added more points on , which explains the greater density in this region.

Based on these visualizations, we would conjecture that the maximum rank of tensors is probably 5, and at most 6. As noted above, [C-B-L-C2009] says it is known (proven) to be 5.