Alternating multilinear map

In mathematics, more specifically in multilinear algebra, an alternating multilinear map is a multilinear map with all arguments belonging to the same space (e.g., a bilinear form or a multilinear form) that is zero whenever any two adjacent arguments are equal.

The notion of alternatization (or alternatisation in British English) is used to derive an alternating multilinear map from any multilinear map with all arguments belonging to the same space.

Definition

An -multilinear map of the form is said to be alternating if whenever there exists such that [1][2]

Example

Properties

whenever there exist and such that and

Alternatization

Given an -multilinear map of the form , the alternating multilinear map defined by is said to be the alternatization of .

Properties

See also

Notes

  1. 1 2 Lang 2002, pp. 511-512.
  2. Bourbaki 2007, p. A III.80, §4.
  3. 1 2 Dummit & Foote 2004, p. 436.
  4. Rotman 1995, p. 235.

References

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