13
$\begingroup$

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!

  • 0
    See http://en.wikipedia.org/wiki/Exterior_product.2012-08-13
  • 0
    Just found this, on the ambiguity of the term "outer product": http://en.wikipedia.org/wiki/Talk%3AOuter_product#Outer_product_vs_exterior_product2012-08-13

2 Answers 2

10

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.

  • 0
    Ah, thanks! So are outer products and exterior products related in some way then? In http://www.euclideanspace.com/maths/algebra/clifford/theory/extendXProduct/index.htm , 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?2012-08-13
  • 0
    I think it is naming conventions ambiguity. In your reference outer product is exactly what I call exterioir product.2012-08-13
  • 0
    http://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??2012-08-13
  • 0
    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.2012-08-13
  • 0
    Thanks Norbert. Is this map fair? http://math.stackexchange.com/q/182024/218132012-08-13
-1

The inner product creates a scalar and the outer a skew-symmetric matrix. If you sum like-terms of this matrix you get a vector which also results from the cross product computation.

  • 0
    What do you mean by `sum like-terms of this matrix`? How to sum?2016-06-09