Multilinear form

In mathematics, a - multilinear form is a function that assigns a value to arguments from - vector spaces and is linear in every component. In the more general case, that the image space itself is a vector space, or image and target spaces are modules , one speaks of a multilinear mapping . ${\displaystyle p}$ ${\displaystyle \omega }$ ${\displaystyle p}$ ${\displaystyle v_{i}\in V_{i},\;i\in \{1,\ldots ,p\}}$ ${\displaystyle K}$ ${\displaystyle V_{1},\ldots ,V_{p}}$ ${\displaystyle \omega (v_{1},\ldots ,v_{p})\in K}$

definition

An illustration

${\displaystyle {\begin{aligned}\omega :\ V_{1}\times \cdots \times V_{p}&\rightarrow K\\(v_{1},\ldots ,v_{p})\ &\mapsto \omega \left(v_{1},\dots ,v_{p}\right)\end{aligned}}}$

is called multilinear form if the following two conditions are met for all and all : ${\displaystyle v_{j}\in V_{j},j\in \{1,\ldots ,p\}}$ ${\displaystyle i\in \{1,\ldots ,p\}}$

For all true ${\displaystyle \lambda \in K}$

${\displaystyle \omega \left(v_{1},\ldots ,\lambda \;v_{i},\ldots ,v_{p}\right)=\lambda \;\omega \left(v_{1},\ldots ,v_{i},\ldots ,v_{p}\right)}$

and for everyone ${\displaystyle w\in V_{i}}$

${\displaystyle \omega \left(v_{1},\ldots ,v_{i}+w,\ldots ,v_{p}\right)=\omega \left(v_{1},\ldots ,v_{i},\ldots ,v_{p}\right)+\omega \left(v_{1},\ldots ,w,\ldots ,v_{p}\right)}$.

The set of all multilinear maps forms a vector space. In the case you write . ${\displaystyle {\mathcal {J}}^{p}(V_{1},\ldots ,V_{p})}$ ${\displaystyle K}$ ${\displaystyle V_{1}=\cdots =V_{p}=:V}$ ${\displaystyle {\mathcal {J}}^{p}(V):={\mathcal {J}}^{p}(V,\ldots ,V)}$

Alternating multilinear forms

A multilinear form is called alternating if it results in zero when the same vector is used twice, i.e. H. ${\displaystyle \omega \in {\mathcal {J}}^{p}(V)}$

${\displaystyle \omega \left(\dots ,v,\dots ,v,\dots \right)=0}$

for everyone . ${\displaystyle v\in V}$

In this case it also follows that the form is skew-symmetrical, that is, that it changes its sign if any two arguments are interchanged , i.e.

${\displaystyle \omega \left(v_{1},\dots ,v_{i},\dots ,v_{j},\dots ,v_{p}\right)=-\omega \left(v_{1},\dots ,v_{j},\dots ,v_{i},\dots ,v_{p}\right)}$

for everyone and . The reverse implication - that all skew-symmetric multilinear forms alternately are - but applies only if the characteristic of not 2, so for example . ${\displaystyle v_{k}\in V,\;k\in \{1,\ldots ,p\}}$ ${\displaystyle i,j\in \{1,\ldots ,p\},\;i\neq j}$ ${\displaystyle K}$ ${\displaystyle K=\mathbb {R} }$

If, more generally, is any permutation of the indices, then applies ${\displaystyle \pi \in S_{p}}$

${\displaystyle \omega \left(v_{\pi (1)},\dotsc ,v_{\pi (p)}\right)=\operatorname {sign} (\pi )\cdot \omega \left(v_{1},\dotsc ,v_{p}\right)}$,

where denotes the sign of the permutation. ${\displaystyle \operatorname {sign} (\pi )}$

The set of all alternating multilinear forms is a subspace of . The structure of an algebra can also be defined on this set . This algebra is called Graßmann algebra . The special case is important . Then is a one-dimensional subspace of , and its elements are called determinant functions . ${\displaystyle \Omega ^{p}(V)}$ ${\displaystyle {\mathcal {J}}^{p}(V)}$ ${\displaystyle \ p=\dim V}$ ${\displaystyle \Omega ^{p}(V)}$ ${\displaystyle {\mathcal {J}}^{p}(V)}$

Examples

1. Linear forms are exactly the 1-multilinear forms.
2. Bilinear forms are exactly the 2-multilinear forms. Antisymmetric bilinear forms are alternating multilinear forms (if the characteristic of is not 2). ${\displaystyle K}$
3. If a square matrix is formed from vectors by combining them, the determinant of this matrix is ​​an alternating, normalized multilinear form. In the three-dimensional case it is defined by an alternating 3-multilinear form. The vectors are as follows shown in coordinates: . ${\displaystyle n}$ ${\displaystyle \omega }$
   ${\displaystyle \omega \left(v_{1},v_{2},v_{3}\right):=\det {\begin{pmatrix}v_{1x}&v_{2x}&v_{3x}\\v_{1y}&v_{2y}&v_{3y}\\v_{1z}&v_{2z}&v_{3z}\end{pmatrix}}}$
   ${\displaystyle v_{1},v_{2},v_{3}}$
   ${\displaystyle v_{1}={\begin{pmatrix}v_{1x}\\v_{1y}\\v_{1z}\end{pmatrix}},\quad \quad v_{2}={\begin{pmatrix}v_{2x}\\v_{2y}\\v_{2z}\end{pmatrix}},\quad \quad v_{3}={\begin{pmatrix}v_{3x}\\v_{3y}\\v_{3z}\end{pmatrix}}}$
4. Covariant tensors are multilinear forms: In the case that all vector spaces are identical (i.e. ), the -multilinear form is also a covariant tensor -th level. In the same case, the alternating -Multilinearforms are also totally antisymmetric tensors -th order. ${\displaystyle V_{i}}$ ${\displaystyle V_{i}=V}$ ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle p}$
5. A differential form assigns an alternating multilinear form to a point of a manifold on the associated tangent space .

literature

• Gerd Fischer : Linear Algebra. Vieweg-Verlag, ISBN 3-528-03217-0 .
• Hans-Joachim Kowalsky , Gerhard O. Michler : Lineare Algebra. De Gruyter, Berlin 2003, ISBN 978-3-11-017963-7 .

Individual evidence

1. ↑ ^{a b} Arkady L'vovich Onishchik : multilinear mapping . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).