Multilinear mapping

In the mathematical sub-area of linear algebra and related areas, the concept of linear mapping is generalized by multilinear mapping . An important example of a multilinear mapping is the determinant .

definition

If a commutative ring with one and are and for modules above the ring , then a multilinear map is a map defined on the product space , which is a linear map with respect to each of its arguments . More precisely: if an integer, then a - (multi) linear mapping has the property ${\displaystyle R}$ ${\displaystyle F}$ ${\displaystyle E_{i}}$ ${\displaystyle i\in \{1,...,p\}}$ ${\displaystyle R}$ ${\displaystyle f\colon E_{1}\times \cdots \times E_{p}\to F}$ ${\displaystyle p>0}$ ${\displaystyle p}$

${\displaystyle \forall a\in E_{1}\times \cdots \times E_{p},\forall i\in \{1,...,p\}:f_{i}(a)\in L(E_{i};F)}$,

where the partial mapping ${\displaystyle f_{i}(a)}$

${\displaystyle f_{i}(a)\colon E_{i}\to F~;~~x\mapsto f(a_{1},...,a_{i-1},x,a_{i+1},...,a_{p})~}$

and denotes the set of linear mappings from to . ${\displaystyle L(E;F)}$ ${\displaystyle E}$ ${\displaystyle F}$

If so, one speaks of a - multilinear form . ${\displaystyle F=R}$ ${\displaystyle R}$

The set of all linear mappings from to is given by ${\displaystyle p}$ ${\displaystyle E_{1}\times \cdots \times E_{p}}$ ${\displaystyle F}$

${\displaystyle L_{p}(E_{1},...,E_{p};F)}$

designated; if all are the same, one also notes ${\displaystyle E_{i}=E}$

${\displaystyle L_{p}(E,...,E;F)=:L_{p}(E;F)}$and finally . ${\displaystyle L_{p}(E,...,E;R)=:L_{p}(E)}$

Examples

• Each linear map is a 1-linear map.
• For , the zero mapping is the only linear mapping that is also -linear. (To prove it, write from what and use that is because of linearity as soon as one of the arguments is.) ${\displaystyle p>1}$ ${\displaystyle p}$ ${\displaystyle (x,y,...)=(0,y,...)+(x,0,...)}$ ${\displaystyle f(x,y,...)=f(0,y,...)+f(x,0,...)}$ ${\displaystyle f(...)=0}$ ${\displaystyle 0}$
• Every bilinear map is a 2-linear map.
• The late product im is a 3-linear map, i.e. H. . ${\displaystyle [x,y,z]=x\cdot (y\times z)}$ ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle [\cdot ,\cdot ,\cdot ]\in L_{3}(\mathbb {R} ^{3})=L_{3}(\mathbb {R} ^{3};\mathbb {R} )=L_{3}(\mathbb {R} ^{3},\mathbb {R} ^{3},\mathbb {R} ^{3};\mathbb {R} )}$
• All commonly used products are 2-linear mappings: the multiplication in a field (real, complex , rational numbers ) or a ring ( whole numbers , matrices), but also the vector or cross product , scalar product .
• The determinant in an n -dimensional vector space is an n -linear multilinear form.

Other properties

The symmetric group of permutations of defining an operation on , ${\displaystyle \{1,\ldots ,p\}}$ ${\displaystyle L_{p}(E;F)}$

${\displaystyle S_{p}\times L_{p}(E;F)\to L_{p}(E;F)~;~~(\sigma ,f)\mapsto \sigma f:(x_{1},...,x_{p})\mapsto (x_{\sigma (1)},...,x_{\sigma (p)})}$

that is, by permutating the arguments of the -linear mapping. (One shows that by showing this first for two transpositions .) ${\displaystyle p}$ ${\displaystyle \sigma (\tau f)=(\sigma \circ \tau )f,}$ ${\displaystyle (ij),(ik)}$

A figure is then called ${\displaystyle f\in L_{p}(E;F)}$

• symmetrical if applies to all . ${\displaystyle \sigma f=f}$ ${\displaystyle \sigma }$
• antisymmetric if applies to all , where is the sign of the permutation. ${\displaystyle \sigma f=\epsilon (\sigma )f}$ ${\displaystyle \sigma }$ ${\displaystyle \epsilon (\sigma )}$
• alternating if as soon as two of the arguments are equal. ${\displaystyle f(x_{1},\ldots ,x_{p})=0}$

Conversely, one defines the symmetrizer

${\displaystyle S\colon f\mapsto Sf=\sum _{\sigma \in S_{p}}\sigma f}$

and the antisymmetrizer

${\displaystyle S\colon f\mapsto Sf=\sum _{\sigma \in S_{p}}\varepsilon (\sigma )\,\sigma f}$,

which any multilinear mapping symmetrically resp. "make" antisymmetric. (Some authors divide by a factor in order to make these operators idempotent ( i.e. to projectors onto the corresponding subspaces), which is not always possible in fields with finite characteristics .) ${\displaystyle f}$ ${\displaystyle p!}$

One simply shows that an alternating mapping is antisymmetric, while an antisymmetric mapping is alternating if , and is otherwise symmetric. ${\displaystyle 1+1\neq 0}$

For example, the cross product and the late product are antisymmetric maps.

Determinant forms are examples of alternating multilinear forms (by definition).

Tensors

Multilinear mappings are required to define the tensor product by means of the following universal property , and they are thus classified at the same time: For every multilinear map there is exactly one homomorphism , so that the following diagram commutes: ${\displaystyle A_{1}\times \cdots \times A_{n}\to B}$ ${\displaystyle A_{1}\otimes _{R}\cdots \otimes _{R}A_{n}\to B}$

literature

• AL Onishchik: Multilinear mapping . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).