Exterior Algebra Notes #4: The Interior Product
Vector spaces are assumed to be finite-dimensional and over . The grade of a multivector will be written , while its magnitude will be written . Bold letters like will refer to (grade-1) vectors, while Greek letters like refer to arbitrary multivectors with grade .
More notes on exterior algebra. This time, the interior product , with a lot more concrete intuition than you’ll see anywhere else, but still not enough.
I am not the only person who has had trouble figuring out what the interior product is for. This is what I have so far…
1. The Interior Product
The last main tool of exterior algebra is the interior product, written or . It subtracts grades () and, conceptually, does something akin to ‘dividing out of ’. It’s also called the ‘contraction’ or ‘insertion’ operator. We use the same symbol as the inner product because we think of it as a generalization of the inner product: when , then .
Its abstract definition is that it is adjoint to the wedge product with respect to the inner product:
In practice this means that it sort of ‘undoes’ wedge products, as we will see.
When we looked at the inner product we had a procedure for computing . We switched from the inner product to the inner product, by writing both sides as tensor products, with the right side antisymmetrized using :1
1Recall that we basically elect to antisymmetrize one side because if we did both we would need an extra factor of for the same result. It might be that there are abstractions of this where you do need to do both sides (for instance if ?) ↩
Interior products directly generalize inner products to cases where the left side has a lower grade2
2It is probably possible to generalize to either side having the lower grade, but it’s not normally done that way. I want to investigate it sometime. ↩
A general formula for the interior product of a vector with a multivector, which can be deduced from the above, is
2. Projection
The intuitive meaning of the interior product is related to projection. We can construct the projection and rejection operators of a vector onto a multivector with:
To understand this, recall that the classic formula for projecting onto a unit vector is:
That is, we find the scalar coordinate along , then multiply by once again. With multivectors, is not a scalar, so we can’t just use scalar multiplication – so it makes some sense that it would be replaced with .3
3the other candidate would be , but we’d like the result to also be a multivector so it makes sense to only consider . ↩
The classic vector rejection formula is
Using the interior product we can write this as
The multivector version is only non-zero if has a component which does not contain – all -ness is removed by the wedge product, leaving something like . Then .
The correct interpretation of , then, is a lot like what it means when : it’s finding the ‘-component’ of . It’s just that, when is a multivector, the ‘-coordinate’ is no longer a scalar.
For example this is the ‘‘-component of a bivector :
Note that the result doesn’t have any factors in it.
So, we might things up like this: For a unit multivector , the meaning of is to find , the ‘ component’ of .
We can remove the stipulation that be a unit multivector, but it requires being a bit careful. To illustrate why, consider an example with just vectors. What should be the value of ? Probably as , so that . Likewise, we need to divide through by the magnitude of , so it’s actually .
Unfortunately the rejection formula doesn’t work if is a multivector. It’s still true that gives the ‘-coordinate’ of , if there is one. But we can only use . The problem is that there are cases where both , such as for and .4
4I think there’s a way to make it work. It looks something like: for each basis multivector of lower grade, remove it from both sides, like . But that’s complicated and will have to be saved for the future. ↩
3. More identities
We can use to prove a few more vector identities. First, note that is just a special case of .5
5It is easier to use the notation for inner products, since after all they are a special case of interior products. But sometimes I use anyway when it makes things clearer. ↩
Since this holds for all :
(1) implies that obeys many of the the same rules as :
Combining these, we have a way to transform applications of :
Since the cross product is :
This lets us unpack cross product identities. Note that in .
Here’s the vector triple product:
The quadruple product:
The Jacobi Identity:
My other articles about Exterior Algebra:
- Oriented Areas and the Shoelace Formula
- Matrices and Determinants
- The Inner product
- The Hodge Star
- The Interior Product
- EA as Linearized Set Theory?
- Oriented Projective Geometry
- Simplex Volumes and Boundaries
- All the Exterior Algebra Operations