Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

If inner products ($V$) are generalisations of dot products ($ \mathbb{R}^n$), then are outer products ($V$) also related to cross-products ($ \mathbb{R}^3$) in some way?

A quick search reveals that they are, but yet the outer product of two column vectors in $ \mathbb{R}^3$ is a 3x3 matrix, not another column vector. What's the link? Thanks!

share|improve this question
    
See en.wikipedia.org/wiki/Exterior_product. –  joriki Aug 13 '12 at 10:17
    
Just found this, on the ambiguity of the term "outer product": en.wikipedia.org/wiki/… –  Ryan Aug 13 '12 at 11:28

1 Answer 1

up vote 5 down vote accepted

Cross product is much more related to exterior product which is in fact a far going generalization.

Outer product is a matricial description of tensor product of two vectors.

share|improve this answer
    
Ah, thanks! So are outer products and exterior products related in some way then? In euclideanspace.com/maths/algebra/clifford/theory/extendXProduct/… , the author claims that "Inner product by a vector reduces the grade of a multivector. It is related to the dot product. Outer product by a vector increases the grade of a multivector. It is related to the cross product." and I've seen this claim elsewhere too. Are they mixing up their terminologies? –  Ryan Aug 13 '12 at 10:45
    
I think it is naming conventions ambiguity. In your reference outer product is exactly what I call exterioir product. –  no identity Aug 13 '12 at 10:49
    
en.wikipedia.org/wiki/Interior_product states that "The interior product, named in opposition to the exterior product ... should not be confused with an inner product." So Inner is a generalisation of Dot. Exterior is a generalisation of Cross. Interior is in opposition to Exterior but shld not be confused with Inner. Can someone put these into a simple framework for me? For eg. how is Inner related to Interior then?? –  Ryan Aug 13 '12 at 10:57
    
Interior product is degree descending operation. It looks like inner product, but this is just an analogy. Here is an example of how one can define interior product given some $x\in V$. You define interioir product of vectors $x_1,\ldots,x_{n-1}$ as exterior product of $x, x_1,\ldots, x_n$. Note that interioir product depends on the choice of vector $x$. Interioir products are closely related to annihilation operators on the Fock space in quantum mechanics. –  no identity Aug 13 '12 at 11:15
    
Thanks Norbert. Is this map fair? math.stackexchange.com/q/182024/21813 –  Ryan Aug 13 '12 at 12:13

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.