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

## Synopsis

• Function: rank
• Usage:
rank M
• Inputs:
• 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