Multilinear form
-linear form, on a unitary
-module
A multilinear mapping (here
is a commutative associative ring with a unit, cf. Associative rings and algebras). A multilinear form is also called a multilinear function (
-linear function). Since a multilinear form is a particular case of a multilinear mapping, one can speak of symmetric, skew-symmetric, alternating, symmetrized, and skew-symmetrized multilinear forms. For example, the determinant of a square matrix of order
over
is a skew-symmetrized (and therefore alternating)
-linear form on
. The
-linear forms on
form an
module
, which is naturally isomorphic to the module
of all linear forms on
. In the case
(
), one speaks of bilinear forms (cf. Bilinear form) (respectively, trilinear forms).
The -linear forms on
are closely related to
-times covariant tensors, i.e. elements of the module
. More precisely, there is a linear mapping
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such that
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for any ,
. If the module
is free (cf. Free module),
is injective, while if
is also finitely generated,
is bijective. In particular, the
-linear forms on a finite-dimensional vector space over a field are identified with
-times covariant tensors.
For any forms ,
one can define the tensor product
via the formula
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For symmetrized multilinear forms (cf. Multilinear mapping), a symmetrical product is also defined:
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while for skew-symmetrized multilinear forms there is an exterior product
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These operations are extended to the module , where
,
, to the module of symmetrized forms
and to the module of skew-symmetrized forms
respectively, which transforms them into associative algebras with a unit. If
is a finitely-generated free module, then the mappings
define an isomorphism of the tensor algebra
on
and the exterior algebra
on the algebra
, which in that case coincides with the algebra of alternating forms. If
is a field of characteristic
, then there is also an isomorphism of the symmetric algebra
on the algebra
of symmetric forms.
Any multilinear form corresponds to a function
, given by the formula
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Functions of the form are called forms of degree
on
; if
is a free module, then in coordinates relative to an arbitrary basis they are given by homogeneous polynomials of degree
. In the case
(
) one obtains quadratic (cubic) forms on
(cf. Quadratic form; Cubic form). The form
completely determines the symmetrization
of a form
:
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In particular, for ,
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The mappings and
define a homomorphism of the algebra
on the algebra of all polynomial functions (cf. Polynomial function)
, which is an isomorphism if
is a finitely-generated free module over an infinite integral domain
.
References
[1] | N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , 1 , Addison-Wesley (1974) pp. Chapt.1;2 (Translated from French) |
[2] | N. Bourbaki, "Elements of mathematics. Algebra: Modules. Rings. Forms" , 2 , Addison-Wesley (1975) pp. Chapt.4;5;6 (Translated from French) |
[3] | S. Lang, "Algebra" , Addison-Wesley (1984) |
Multilinear form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Multilinear_form&oldid=16677