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:
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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:
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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
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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.
Exterior product. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Exterior_product&oldid=46889