Exterior product

A fundamental operation in the exterior algebra of tensors defined on an -dimensional vector space over a field .

Let be a basis of , and let and be - and -forms:

The exterior product of the forms and is the -form obtained by alternation of the tensor product . The form is denoted by ; its coordinates are skew-symmetric:

where are the components of the generalized Kronecker symbol. The exterior product of covariant tensors is defined in a similar manner.

The basic properties of the exterior product are listed below:

1) , (homogeneity);

2) (distributivity);

3) (associativity).

4) ; if the characteristic of is distinct from two, the equation is valid for any form of odd valency.

The exterior product of vectors is said to be a decomposable -vector. Any poly-vector of dimension is a linear combination of decomposable -vectors. The components of this combination are the ()-minors of the ()-matrix , , , of the coefficients of the vectors . If their exterior product has the form

Over fields of characteristic distinct from two, the equation is necessary and sufficient for vectors to be linearly dependent. A non-zero decomposable -vector defines in an -dimensional oriented subspace , parallel to the vectors , and the parallelotope in formed by the vectors issuing from one point, denoted by . The conditions and are equivalent.

For references see Exterior algebra.

Comments

Instead of exterior product the phrase "outer product" is sometimes used. The condition for of degree and of degree is sometimes called graded commutativity.