Talk:Interior product: Difference between revisions

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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].

Any one of these products has a formula which can tell you what the output grade will be ~~(assuming empty intersections between the vectors they are wedges of)~~. For the progressive product it's simply g(a) + g(b). For the regressive product it's g(a) + g(b) - d. For the left interior product it's g(a) - g(b), and for the right interior product it's g(b) - g(a). All of these formulas max out at d (can't have higher grade than dimensionality) and min out at 0 (no such thing as negative grade). I can give derivations for these if anyone wants.

Any one of these products has a formula which can tell you what the output grade will be. For the progressive product it's simply g(a) + g(b). For the regressive product it's g(a) + g(b) - d. For the left interior product it's g(a) - g(b), and for the right interior product it's g(b) - g(a). All of these formulas max out at d (can't have higher grade than dimensionality) and min out at 0 (no such thing as negative grade). I can give derivations for these if anyone wants.

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:

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

e.g. ~~Assuming empty intersections~~:

e.g.:

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

(greater than right grade)

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

|-

!resultant grade~~, assuming empty intersections~~

!resultant grade

!grade(a) + grade(b)

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

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 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].
-Any one of these products has a formula which can tell you what the output grade will be (assuming empty intersections between the vectors they are wedges of). For the progressive product it's simply g(a) + g(b). For the regressive product it's g(a) + g(b) - d. For the left interior product it's g(a) - g(b), and for the right interior product it's g(b) - g(a). All of these formulas max out at d (can't have higher grade than dimensionality) and min out at 0 (no such thing as negative grade). I can give derivations for these if anyone wants.
+Any one of these products has a formula which can tell you what the output grade will be. For the progressive product it's simply g(a) + g(b). For the regressive product it's g(a) + g(b) - d. For the left interior product it's g(a) - g(b), and for the right interior product it's g(b) - g(a). All of these formulas max out at d (can't have higher grade than dimensionality) and min out at 0 (no such thing as negative grade). I can give derivations for these if anyone wants.
 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:
@@ Line 83: / Line 83: @@
 d-((d-r)+n) which simplifies to r-n.
-e.g. Assuming empty intersections:
+e.g.:
 Dimension 7. Left input rank 6. Right input nullity 2. Output rank = r-n =4
 (greater than right grade)
@@ Line 123: / Line 123: @@
 a • b = if grade(a) ≥ grade(b), a ⨽ b; else a ⨼ b
 |-
-!resultant grade, assuming empty intersections
+!resultant grade
 !grade(a) + grade(b)
 !grade(a) + grade(b) - dimensionality