Talk:Interior product: Difference between revisions

The reason I think this article describes a symmetrical interior product is because of the last paragraph of the "definition" section. I think there may be an error there where it says "we can take the wedge product m∨W from the front". I think what it's showing is actually an example of the interior product, and this is the one place in the article where I see the input on the left having a smaller grade than the thing on the right. And I think the last clause of this sentence "this can only lead to a difference in sign" could be continued with the phrase "compared to W∨m", i.e. the inputs reordered so that the lower-grade input is on the right. So if it's possible to reorder the inputs like this, then that implies this article assumes the symmetrical interior product. The change in sign is due to how the wedge product is sometimes anticommutative. It always is for (mono)vectors or (mono)covectors, but I think it's commutative for even grades. I think the statement means "it can, at most" lead to a mere change in signs, but won't necessarily; for example, I did [-3 2 -1 2 -1⟩ ⨼ ⟨⟨⟨1 2 -3 -2 1 -4 -5 12 9 -19]]] and [-3 2 -1 2 -1⟩ ⨼ ⟨⟨⟨1 2 -3 -2 1 -4 -5 12 9 -19]]] and they both give me ⟨⟨6 -7 -2 15 -25 -20 3 15 59 49].

So if we simply wanted to take what is on the page now and help it conform better with established mathematical usages, I would recommend we remove the line about interior being the wedge's dual, and change the symbol the wiki uses for the interior product. Above I used •, the fat dot, which came up on that Facebook post recently as a reasonable choice for this operation. I'm not attached to it though so if anyone has other suggestions I'm not opposed at all to considering them. I'm just concerned that ∨ probably should only be used for the regressive product. If we went this direction, then ∨ would also need to be replaced with • on the following pages, too:

d-((d-r)+n) which simplifies to r-n.

Dimension 7. Left input rank 6. Right input nullity 2. Output rank = r-n =4

(greater than right grade)

I don't have a strong preference which way people want to go with this information. And certainly some combination of these recommendations could be done. I am very open to hearing that there are cases where interior product does exactly what you want and regressive wouldn't cut it. I haven't gone through every use on the wiki (and outside it) to check. Also there's the argument that this interior product defined as it is has been around for a while and maybe even if it has some issues we should preserve it for backwards compatibility. All fine.

And here's a summary table that has helped me get my head around this situation:

|+

!operations

a ∧ b

examples given where grade(a) ≥ grade(b)

examples given where grade(a) < grade(b)

a • b = if grade(a) ≥ grade(b), a ⨽ b; else a ⨼ b

|-

!grade(a) + grade(b)

!grade(a) + grade(b) - dimensionality

|(in terms of other two interior products)

|}