翻訳と辞書
Words near each other
・ "O" Is for Outlaw
・ "O"-Jung.Ban.Hap.
・ "Ode-to-Napoleon" hexachord
・ "Oh Yeah!" Live
・ "Our Contemporary" regional art exhibition (Leningrad, 1975)
・ "P" Is for Peril
・ "Pimpernel" Smith
・ "Polish death camp" controversy
・ "Pro knigi" ("About books")
・ "Prosopa" Greek Television Awards
・ "Pussy Cats" Starring the Walkmen
・ "Q" Is for Quarry
・ "R" Is for Ricochet
・ "R" The King (2016 film)
・ "Rags" Ragland
・ ! (album)
・ ! (disambiguation)
・ !!
・ !!!
・ !!! (album)
・ !!Destroy-Oh-Boy!!
・ !Action Pact!
・ !Arriba! La Pachanga
・ !Hero
・ !Hero (album)
・ !Kung language
・ !Oka Tokat
・ !PAUS3
・ !T.O.O.H.!
・ !Women Art Revolution


Dictionary Lists
翻訳と辞書 辞書検索 [ 開発暫定版 ]
スポンサード リンク

multilinear map : ウィキペディア英語版
multilinear map
In linear algebra, a multilinear map is a function of several variables that is linear separately in each variable. More precisely, a multilinear map is a function
:f\colon V_1 \times \cdots \times V_n \to W\text
where V_1,\ldots,V_n and W\! are vector spaces (or modules over a commutative ring), with the following property: for each i\!, if all of the variables but v_i\! are held constant, then f(v_1,\ldots,v_n) is a linear function of v_i\!.〔Lang. Algebra. Springer; 3rd edition (January 8, 2002)〕
A multilinear map of one variable is a linear map, and of two variables is a bilinear map. More generally, a multilinear map of ''k'' variables is called a ''k''-linear map. If the codomain of a multilinear map is the field of scalars, it is called a multilinear form. Multilinear maps and multilinear forms are fundamental objects of study in multilinear algebra.
If all variables belong to the same space, one can consider symmetric,
antisymmetric and alternating ''k''-linear maps. The latter coincide if the underlying ring (or field) has a characteristic different from two,
else the former two coincide.
==Examples==

* Any bilinear map is a multilinear map. For example, any inner product on a vector space is a multilinear map, as is the cross product of vectors in \mathbb^3.
* The determinant of a matrix is an antisymmetric multilinear function of the columns (or rows) of a square matrix.
* If F\colon \mathbb^m \to \mathbb^n is a ''Ck'' function, then the k\!th derivative of F\! at each point p\! in its domain can be viewed as a symmetric k\!-linear function D^k\!f(p)\colon \mathbb^m\times\cdots\times\mathbb^m \to \mathbb^n.
* The tensor-to-vector projection in multilinear subspace learning is a multilinear map as well.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
ウィキペディアで「multilinear map」の詳細全文を読む



スポンサード リンク
翻訳と辞書 : 翻訳のためのインターネットリソース

Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.