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.