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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 : where and are vector spaces (or modules over a commutative ring), with the following property: for each , if all of the variables but are held constant, then is a linear function of .〔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 . * The determinant of a matrix is an antisymmetric multilinear function of the columns (or rows) of a square matrix. * If is a ''Ck'' function, then the th derivative of at each point in its domain can be viewed as a symmetric -linear function . * 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
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