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?
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?
@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.