multilinear form

English

Noun

multilinear form (plural multilinear forms)

1. (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 $m$-multilinear form can more concisely be written as $M\!f$.

   • 2003, Maks A. Akivis, Vladislav V. Goldberg, translated by 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 $V^{k}\rightarrow K$ (which has $k$ variables) is called a multilinear $k$-form.
• A multilinear $k$-form on $V$ over $\mathbb {R}$ is called a (covariant) $k$-tensor, and the vector space of such forms is usually denoted ${\mathcal {T}}^{k}(V)$ or ${\mathcal {L}}^{k}(V)$. (But note that many authors use an opposite convention, writing ${\mathcal {T}}^{k}(V)$ for the contravariant $k$-tensors on $V$ and ${\mathcal {T}}_{k}(V)$ 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.

Synonyms

• (multiply linear mapping to a field of scalars): multicovector

Hyponyms

• (multiply linear mapping to a field of scalars): bilinear form, covector, linear form

Derived terms

• alternating multilinear form, multilinear k-form

Translations

multiply linear mapping to a field of scalars

• French: forme multilinéaire f
• German: Multilinearform f
• Spanish: forma multilineal f, función multilineal f

See also

• covariant
• differential form
• linear
• linear form
• multilinear algebra
• multilinear map
• multivector
• tensor

Further reading

• Multilinear algebra on Wikipedia.Wikipedia
• Multilinear map on Wikipedia.Wikipedia