Some questions about maximum rank & typical rank

[flamen]

I was wondering some exact explanations of the definition of maximum rank and typical rank-peculiarity of tensors when I read 'Tensor Decomposition and Application' written by Tamara G. Kolda. As his saying, the maximum rank is defined as the largest attainable rank, whereas the typical rank is any rank that occurs with probability greater than zero (i.e., on a set with positive Lebesgue measure). It sounds very plain, but I'm totally confused after check some reference and relevant examples.(~~o(>_<)o ~~)

Could you have offered some advice for me? Thank you very much!

p.s. I'm a beginner in the field of tensor analysis who is new to this forum, and would appreciate your guidance! O(∩_∩)O~

 

[pat_connell]

The maximum rank of a tensor is the highest possible rank that it can have, while the typical rank is any rank that can occur with a non-zero probability. This means that, for a tensor of a particular size and shape, there may be multiple ranks that can occur with a non-zero probability. For example, if you have a 3x3x3 tensor, then the maximum rank is 3 (since this is the highest possible rank that the tensor can attain), but the typical rank could be 1, 2, or 3, depending on the values of the elements in the tensor.