3 by 3 by 3 Tensors ==================== We now study the case of :math:`3\times 3 \times 3` tensors over the real and complex numbers. Our goals are * to determine the maximum separation rank :math:`r` of such tensors and * to explore the geometry of :math:`S_r` for some :math:`r` less than the maximum. In principle one can never determine the maximum separation rank :math:`r` using a numerical routine such as this, since one could be unlucky and miss it. Generally one can only hope to determine the typical separation rank. Visualizations over the Real numbers ------------------------------------ In [KRUSKA1989]_, Kruskal claims to have shown rank :math:`r\le 5`, but his results were not published. `Jos ten Berge `_ says that he saw this 20 page proof but discarded it when `Roberto Rocci `_ provided a 4 page proof in 1993, which is also unpublished but is available. A full proof is published in [BRE-HU2013]_. A Random Example ^^^^^^^^^^^^^^^^^ We choose :math:`T_1`, :math:`T_2`, and :math:`T_3` randomly in :math:`S_3`. Since these three tensors are in :math:`S_3`, any tensor on the line connecting any two of them is in :math:`S_6`, and a tensor in the plane is generically in :math:`S_9`, which we suppose is large enough to capture general :math:`3\times 3 \times 3` tensors. We first produce a visualization of :math:`S_4` on :math:`[-1,2]\times[-1,2]`, and obtain .. image:: 3x3x3pr4random.* [data file `<3x3x3pr4random.dat>`_] There are only a few scattered black points. Thus we conclude that the maximum separation rank of a :math:`3\times 3 \times 3` tensor over the real numbers is more than 4. Next, using the same set of random :math:`T_1`, :math:`T_2`, and :math:`T_3`, we produce a visualization of :math:`S_5`, and obtain .. image:: 3x3x3pr5random.* [data file `<3x3x3pr5random.dat>`_] Black points are distributed over the whole plane, with no apparent voids. We interpret this as validating the result that the maximum rank is 5. Finally, using the same set of random :math:`T_1`, :math:`T_2`, and :math:`T_3`, we first produce a visualization for :math:`S_6`, expecting it to be solid black. Since the line connecting any two :math:`T_i` is in :math:`S_6`, these lines appear in the visualization, and help to locate the :math:`T_i`. We obtain the visualization .. image:: 3x3x3pr6random.* [data file `<3x3x3pr6random.dat>`_] The occasional lighter point is attributed to failure in the routine that tries to fit a tensor on the plane with a tensor in :math:`S_6`. A Specific Example ^^^^^^^^^^^^^^^^^^ We now consider a specific example that was suggested as possibly interesting. The target tensor, which has nominal rank 5, is .. math:: X_5 &= [1,0,0]\otimes[1,0,0]\otimes[1,0,0] +[1,0,0]\otimes[0,0,1]\otimes[0,0,1]\\ &+[0,1,0]\otimes[1,0,0]\otimes[0,1,0] +[0,1,0]\otimes[0,1,0]\otimes[0,0,1]\\ &+[0,0,1]\otimes[1,0,0]\otimes[0,0,1] \,, which can also be viewed as .. math:: X_5 = [1,0,0]\otimes\left[\begin{array}{ccc} 1&0&0\\0&0&0\\0&0&1\end{array}\right] + [0,1,0]\otimes\left[\begin{array}{ccc} 0&1&0\\0&0&1\\0&0&0\end{array}\right] + [0,0,1]\otimes\left[\begin{array}{ccc} 0&0&1\\0&0&0\\0&0&0\end{array}\right] \,. Using the plane defined by the three tensors .. math:: T_1&= [1,0,0]\otimes[1,0,0]\otimes[1,0,0] +[1,0,0]\otimes[0,0,1]\otimes[0,0,1]\\ T_2&= [0,1,0]\otimes[1,0,0]\otimes[0,1,0] +[0,1,0]\otimes[0,1,0]\otimes[0,0,1] \quad\text{and}\\ T_3&= [0,0,1]\otimes[1,0,0]\otimes[0,0,1] \,, the point :math:`X_5` is in their center. We first produce a visualization of :math:`S_3` on :math:`[-1,2]\times[-1,2]`, and obtain .. image:: 3x3x3pr3X5.* [data file `<3x3x3pr3X5.dat>`_] Since the region in the vicinity of :math:`X_5` has no black points, we conclude it is not rank 3. We next produce a visualization of :math:`S_4` and obtain .. image:: 3x3x3pr4X5.* [data file `<3x3x3pr4X5.dat>`_] A large number of test points were needed to produce this visualization. We used more points near the center to try to get better resolution there. Although there are not that many black points, they are spread throughout the visualization with no apparent voids. We also tried zooming in on a relatively light region near the center and adding many more points there, and were then able to find dark points. Based on this behavior, we conclude that :math:`X_5` is probably of rank 5 but reachable as a limit of rank 4 tensors. Visualizations over the Complex numbers ---------------------------------------- The rank over the complex numbers is at most the rank over the real numbers, so we now investigate whether or not the maximum rank is strictly smaller for :math:`3\times 3 \times 3` tensors. A Random Example ^^^^^^^^^^^^^^^^ We choose :math:`T_1` and :math:`T_2` randomly in :math:`S_3`, so that a tensor in the plane is generically in :math:`S_6`, which we know is large enough to capture general :math:`3\times 3 \times 3` tensors. We then produce a visualization of :math:`S_4` and obtain .. image:: 3x3x3pr4randomC.* [data file `<3x3x3pr4randomC.dat>`_] Since there are few black points, we conclude the maximum rank over the complex numbers is more than 4, and hence is 5. As an additional check, we use the same :math:`T_1` and :math:`T_2` and produce a visualization of :math:`S_5`, and obtain .. image:: 3x3x3pr5randomC.* [data file `<3x3x3pr5randomC.dat>`_] A Specific Example ^^^^^^^^^^^^^^^^^^ We now reconsider the tensor :math:`X_5`, which we studied in the real case. We use the complex line defined by .. math:: T_1&= [1,0,0]\otimes[1,0,0]\otimes[1,0,0] +[1,0,0]\otimes[0,0,1]\otimes[0,0,1]\quad\text{and}\\ T_2&= [0,1,0]\otimes[1,0,0]\otimes[0,1,0] +[0,1,0]\otimes[0,1,0]\otimes[0,0,1] +[0,0,1]\otimes[1,0,0]\otimes[0,0,1] \,. We first produce a visualization of :math:`S_3` on :math:`[-1,2]\times[-i,i]`, and obtain .. image:: 3x3x3pr3X5C.* [data file `<3x3x3pr3X5C.dat>`_] Since there are essentially no black points, we then produce a visualization of :math:`S_4` and obtain .. image:: 3x3x3pr4X5C.* [data file `<3x3x3pr4X5C.dat>`_] This visualization required quite a lot of points and effort. The scattered dark blue points indicate that rank 4 points are present (or nearby) but that they are hard to construct. We zoomed in to :math:`[0.4,0.6]\times[-0.1i,0.1i]` and added points near :math:`X_5`, and obtain .. image:: 3x3x3pr4X5Czoom.* Based on this behavior, we again conclude that :math:`X_5` is probably of rank 5 but reachable as a limit of rank 4 tensors.