In the N-mode tensor product (x_n), what are the differences between x_1 and x_2?

Hi, I tried looking this up and got into a weird recursive search through engineering literature.

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If you want to explain it to me in an easy example I can try to translate the langauge.

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In the meantime I can conjecture what I think it will end up being: If V,W are vector-spaces or free modules then an element of V* \otimes W can be described by a matrix if I wish to choose bases for V and W (but this is not true if V and W are only locally free modules or coherent sheaves etc). Even in general, the 'contraction' or 'trace' map from V* \otimes V to the trivial (free of rank one) module or vector-space or sheaf together with multilinearity create a lot of contraction mappings. So for instance there is a contraction map V\otimes V* \otimes W -> W. It is denoted  by the symbol trace \otimes 1  where 1 denotes the identity endomorphism of W.  And if X is yet another object (vector-space, module etc) then there is also the contraction V\otimes V* \otimes W \otimes X -> W \otimes X  which is sometimes denoted as trace \otimes 1 \otimes 1. Note that an element of V \otimes X together with an element of V* \otimes W can be thought of as a pair consisting of a tensor and a matrix, and when we put them together we get an element of our domain V\otimes V*\otimes W \otimes X to which we can apply the contraction to end up with an element of W \otimes X, so this does constitute 'something you can do with a matrix and a tensor'

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It is my guess that you are looking at one or another contracting map but it is frustrating to go through the SIAM literature and figure out the notation.

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Here is roughly how the notation departs.
In engineering language (also sometime in physics) people sometimes use context to speak of ' a tensor' as if there were some big universal 'set of all tensors' in whatever context they happen to be working, and may say in physics "A tensor is just for example a differential form or a vector-field", or in engineering "A tensor is just a tuple A\^I\_J with subscripts and superscripts indexing its entries."

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By contrast in math language there isn't a definition of 'a tensor' just like there is no definition of 'an element of the inverse image' without specifying, the inverse image of what function?

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In the world of rings and modules, there is an adjunction Hom\_R(A\otimes\_S B, C) \cong Hom\_R(A, Hom\_S(B,C)) when A is an R-S bimodule and C is an S module which characterises tensor product categorically, A\otimes B is 'the module whose maps to C could be defined to be maps from A to Hom(B,C)'  and in the case of free modules like if B is a copy of S\^n then A \otimes B is just a copy of A\^n.

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TL;DR It might be  (trace \otimes 1\otimes 1) but if you give me a real-world example I'll know for sure.

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