A question for real mathematicians out there (or at least the math rigor curious): Do folks in #math maintain consistent distinctions between the meaning of the terms "outer product" and "tensor product" (and for bonus points throw in "Kronecker product")?
I learned these concepts mostly from physicists (which is a bit like learning manners from being raised by wolves), and there was a tendency not to use consistent terminology or draw clear distinctions, though sometimes they were being used to refer to slightly different, but related, things. I could generally follow the sense in which terms were being used in a given application by context, so I didn't worry about it too much. A cursory look online also suggests that usage is heterogeneous, but I'm curious if mathematicians are, in fact, a bit more consistent.
#math #maths #mathematics
Since I didn't get any bites on this question before, let my make it easier by putting in the form of a poll (but comments welcome): When people use the terms "outer product" and "tensor product" how do the typical intended meanings relate?
I'm greatly amused that so far the answers to my poll suggest an utter lack of unanimity as far as use of the term "outer product".
@internic If there's a move to use "outer product" to mean "exterior product" I think that needs to be shut down as fast as possible!
As for your poll. "Same" is a difficult word in mathematics. But there's a cheat way to answer the question. Tensor products over fields of non-zero characteristic are pretty different to outer products. So I have to go with the last option even though this might not be in the spirit intended.
@dpiponi I totally agree with your first statement! As for the second, that's a fair point; I really specifically meant to consider things only over the real or complex numbers, not other fields.
@dpiponi ...plus I figured that asking mathematicians to choose between various imprecise answers was a good away to bait them into commenting (which is what I was more interested in to begin with). 😉
@dpiponi Apparently, yes, referring to the exterior product as the "outer product" may be somewhat common among people who do geometric algebra.
https://en.wikipedia.org/wiki/Geometric_algebra#cite_note-16
@internic Oh, those people.
@dpiponi I hadn't bumped Geometric Algebra under that name until yesterday (though I have encountered Clifford algebras before). The video I watched had much the same flavor of things I read way back when arguing the differential forms were the natural way to express and teach physics (and obviously there's a relationship). But misusing "outer product" like that just seems like it will cause loads of confusion.
In my own experience I've seen people use "outer product" as a synonym for "tensor product" or sometimes to specifically refer to an operation that takes two vectors and outputs the tensor product of one of the vectors with the dual of the other (to make a tensor of type (1,1)). And then I literally just watched a video where someone used "outer product" to mean what I would call an "exterior product"/"wedge product". I hadn't run into that before. Is that common as well?