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 productouter-product" is sometimes used. The condition for of degree and of degree is sometimes called graded commutativity.