rank(MultidimensionalMatrix) -- about the border rank of a multidimensional matrix

Synopsis

Function: rank
Usage:

rank M
Inputs:
- M, a multidimensional matrix
Outputs:
- an integer, a lower bound for the (border) rank of M, which is calculated as the maximum rank of all the flattenings of M

Description

In some cases, we know that the returned integer r = rank M is exactly the border rank of M. For instance, this is the case if r<3 by a result of C. Raicu. In general, however, we obtain only a lower bound and it is not known how to calculate this rank exactly without resorting to elimination.

i1 : M = randomMultidimensionalMatrix(2,4,3,2,MaximalRank=>2)

o1 = {{{{1206, 1305}, {2604, 2730}, {3216, 3480}}, {{1596, 1218}, {3640,
     ------------------------------------------------------------------------
     2716}, {4256, 3248}}, {{1914, 1527}, {4340, 3374}, {5104, 4072}},
     ------------------------------------------------------------------------
     {{1332, 1494}, {2856, 3108}, {3552, 3984}}}, {{{792, 1125}, {1608,
     ------------------------------------------------------------------------
     2265}, {2112, 3000}}, {{456, 537}, {968, 1109}, {1216, 1432}}, {{624,
     ------------------------------------------------------------------------
     768}, {1312, 1576}, {1664, 2048}}, {{936, 1341}, {1896, 2697}, {2496,
     ------------------------------------------------------------------------
     3576}}}}

o1 : 4-dimensional matrix of shape 2 x 4 x 3 x 2 over ZZ

i2 : rank M

o2 = 2

i3 : M' = randomMultidimensionalMatrix(2,4,2,1,3,CoefficientRing=>ZZ/65521,MaximalRank=>4)

o3 = {{{{{12579, 2823, -26362}}, {{-20238, 21337, -29331}}}, {{{12274, 29536,
     ------------------------------------------------------------------------
     -1718}}, {{-25383, -8569, -19467}}}, {{{32013, -11709, 1349}}, {{-661,
     ------------------------------------------------------------------------
     18457, -14503}}}, {{{25956, 4878, -11219}}, {{-25462, 3917, -21610}}}},
     ------------------------------------------------------------------------
     {{{{-7660, 24925, 19945}}, {{-16591, 30427, 29978}}}, {{{-8163, -32551,
     ------------------------------------------------------------------------
     14558}}, {{4616, -11937, -26503}}}, {{{12391, -10012, -10837}}, {{11758,
     ------------------------------------------------------------------------
     -3760, -17677}}}, {{{9920, -29378, 29772}}, {{25674, -4729, 12002}}}}}

                                                            ZZ
o3 : 5-dimensional matrix of shape 2 x 4 x 2 x 1 x 3 over -----
                                                          65521

i4 : rank M'

o4 = 4

Ways to use this method: