multilinear form

English

Noun

multilinear form (plural multilinear forms)

(linear algebra, multilinear algebra) Given a vector space V over a field K of scalars, a mapping V^k → K that is linear in each of its arguments;
(more generally) a similarly multiply linear mapping M^r → R defined for a given module M over some commutative ring R.
- 1985, Jack Peetre, Paracommutators and Minimal Spaces, S. C. Power (editor) Operators and Function Theory, Kluwer Academic (D. Reidel), page 163,
  Finally, in the short Lecture 5 we make some remarks on multilinear forms over Hilbert spaces, a theory which is still in a rather embryonic state, motivated by the observation that paracommutators (and Hankel operators too) really should be viewed as forms, not operators.
- 1994, Hessam Khoshnevisan, Mohamad Afshar, Mechanical Elimination of Commutative Redundancy, Baudouin Le Charlier (editor), Static Analysis: 1st International Static Analysis Symposium, Proceedings, Volume 1, Springer, LNCS 864, page 454,
  A multilinear form is said to be degenerate if all its function variables are identical. Thus a degenerate ${\\displaystyle m}$ -multilinear form can more concisely be written as ${\\displaystyle M\\!f}$ .
- 2003, Vladislav V. Goldberg (translator), Maks A. Akivis, Vladislav V. Goldberg, Tensor Calculus with Applications, World Scientific, page 55,
  Since the coordinates of a vector change in transforming to a new basis, the same is true of the coefficients of a multilinear form (since the form itself is to remain invariant).

Usage notes

A multilinear form ${\\displaystyle V^{k}\\rightarrow K}$ (which has ${\\displaystyle k}$ variables) is called a multilinear ${\\displaystyle k}$ -form.
A multilinear ${\\displaystyle k}$ -form on ${\\displaystyle V}$ over ${\\displaystyle \\mathbb {R} }$ is called a (covariant) ${\\displaystyle k}$ -tensor, and the vector space of such forms is usually denoted ${\\displaystyle {\\mathcal {T}}^{k}(V)}$ or ${\\displaystyle {\\mathcal {L}}^{k}(V)}$ . (But note that many authors use an opposite convention, writing ${\\displaystyle {\\mathcal {T}}^{k}(V)}$ for the contravariant ${\\displaystyle k}$ -tensors on ${\\displaystyle V}$ and ${\\displaystyle {\\mathcal {T}}_{k}(V)}$ for the covariant ones.)
A multilinear form differs from a multilinear map in that the former maps to a field of scalars, whereas the latter maps, in the general case, to a cross product of vector spaces.