We now study the case of tensors over both the real and complex numbers. Our purpose is to give visualizations of theoretical work of others.
We follow the work of Vin De Silva and Lek-Heng Lim in [DES-LIM2008], although several of these results appeared earlier in [KRUSKA1989]. They show that all tensors are in but both and have positive volume. The tensors in can be divided into the cases and .
The class consists of tensors not in that are a limit of tensors in . The existence of tensors in shows that is not a closed set. A representative element of is
which can also be viewed as
Using the plane defined by the three tensors
the point is in their center. Since these three tensors are in , any tensor on the line connecting any two of them is in . These lines help us identify the three tensors in the visualization, and thus help approximately locate the tensor.
Plotting in this way, we obtain the visualization
[data file 2x2x2pr2D3.dat] Notice that we obtain black points near , and this lack of a void validates the result that there are tensors in near . It does not allow us to distinguish points in from points in .
The class consists of tensors not in that are not a limit of tensors in . A representative element of is
which can also be viewed as
Using the plane defined by the three tensors
the point is in their center. Since these three tensors are in , any tensor on the line connecting any two of them is in . These lines help us identify the three tensors in the visualization, and thus help approximately locate the tensor.
Plotting in this way, we obtain the visualization
[data file 2x2x2pr2G3.dat] We obtain black points everywhere outside of the triangle, and part of the region inside of the triangle. However, near there is a distinct void, indicating that this tensor cannot be approached as a limit of tensors in .
We also note that we used only one-third the points to make this visualization as compared to the visualization, and yet obtained more black points. This indicates that the ALS routine had more difficulty on the tensors in the plane, perhaps because they are only limit points of .
Randomly choosing , , and , we obtained a visualization
[data file 2x2x2pr2random.dat] In this visualization we see both black regions and voids, suggesting that it is relatively easy to hit both types of behaviors.
Over the complex numbers all tensors are in . We illustrate this by considering the tensor defined above. Using the complex line defined by the two tensors
the point is the (real) midpoint of the line. Plotting in this way on , we obtain the visualization
[data file 2x2x2pr2G3C.dat] We obtain black points everywhere, indicating that is in over the complex numbers (prehaps as a limit point). The ALS routine had more trouble finding black points in the left half of the visualization, which is closer to . This is counterintuitive, since is only rank 1 whereas is rank 2.