multilinear formの意味・使い方・読み方 | Weblio英和辞書

意味

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multilinear form

多重線形形式

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Weblio英和対訳辞書

Multilinear form

多重線型形式

数学、より具体的には抽象代数学と多重線型代数において、多重線型形式（たじゅうせんけいけいしき、英: multilinear form）とは、複数のベクトルを変数とするスカラー値の函数であって、どの変数に関しても（ほかの変数を止めて）線型写像となっているようなものを言う。

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Wiktionary英語版

出典：Wiktionary

multilinear form

名詞

multilinear form (複数形 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 (かつ 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, 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).

使用する際の注意点

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.

同意語

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

下位語

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

派生語

alternating multilinear form, multilinear k-form

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