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 productouter-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=14457