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 : V 1 × ⋯ × V n → W , {\displaystyle f\colon V_{1}\times \cdots \times V_{n}\to W{\text{,}}}
where V 1 , … , V n {\displaystyle V_{1},\ldots ,V_{n}} ( n ∈ Z ≥ 0 {\displaystyle n\in \mathbb {Z} _{\geq 0}} ) and W {\displaystyle W} are vector spaces (or modules over a commutative ring), with the following property: for each i {\displaystyle i} , if all of the variables but v i {\displaystyle v_{i}} are held constant, then f ( v 1 , … , v i , … , v n ) {\displaystyle f(v_{1},\ldots ,v_{i},\ldots ,v_{n})} is a linear function of v i {\displaystyle v_{i}} . One way to visualize this is to imagine two orthogonal vectors; if one of these vectors is scaled by a factor of 2 while the other remains unchanged, the cross product likewise scales by a factor of two. If both are scaled by a factor of 2, the cross product scales by a factor of 2 2 {\displaystyle 2^{2}} . A multilinear map of one variable is a linear map, and of two variables is a bilinear map. More generally, for any nonnegative integer k {\displaystyle k} , 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 two 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 R {\displaystyle \mathbb {R} } -vector space is a multilinear map, as is the cross product of vectors in R 3 {\displaystyle \mathbb {R} ^{3}} . The determinant of a square matrix is a multilinear function of the columns (or rows); it is also an alternating function of the columns (or rows). If F : R m → R n {\displaystyle F\colon \mathbb {R} ^{m}\to \mathbb {R} ^{n}} is a Ck function, then the k {\displaystyle k} th derivative of F {\displaystyle F} at each point p {\displaystyle p} in its domain can be viewed as a symmetric k {\displaystyle k} -linear function D k F : R m × ⋯ × R m → R n {\displaystyle D^{k}\!F\colon \mathbb {R} ^{m}\times \cdots \times \mathbb {R} ^{m}\to \mathbb {R} ^{n}} .
Coordinate representation Let
f : V 1 × ⋯ × V n → W , {\displaystyle f\colon V_{1}\times \cdots \times V_{n}\to W{\text{,}}}
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