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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 :$f\backslash colon\; V\_1\; \backslash times\; \backslash cdots\; \backslash times\; V\_n\; \backslash to\; W\backslash text$ where $V\_1,\backslash ldots,V\_n$ and $W\backslash !$ are vector spaces (or modules over a commutative ring), with the following property: for each $i\backslash !$, if all of the variables but $v\_i\backslash !$ are held constant, then $f(v\_1,\backslash ldots,v\_n)$ is a linear function of $v\_i\backslash !$.〔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 $\backslash mathbb^3$. * The determinant of a matrix is an antisymmetric multilinear function of the columns (or rows) of a square matrix. * If $F\backslash colon\; \backslash mathbb^m\; \backslash to\; \backslash mathbb^n$ is a ''Ck'' function, then the $k\backslash !$th derivative of $F\backslash !$ at each point $p\backslash !$ in its domain can be viewed as a symmetric $k\backslash !$-linear function $D^k\backslash !f(p)\backslash colon\; \backslash mathbb^m\backslash times\backslash cdots\backslash times\backslash mathbb^m\; \backslash to\; \backslash mathbb^n$. * 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.