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Encyclopedia > Symmetric bilinear form

In mathematics, a bilinear form on a vector space V over a field F is a mapping V × VF which is linear in both arguments. That is, B : V × VF is bilinear if the maps History Main article: History of mathematics In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. ... The fundamental concept in linear algebra is that of a vector space or linear space. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed and the associative, commutative, and distributive rules hold, which are familiar from the arithmetic of ordinary numbers. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...

are linear for each v in V. This definition applies equal well to modules over a commutative ring with linear maps being module homomorphisms. In abstract algebra, a module is a generalization of a vector space. ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... In abstract algebra, a module is a generalization of a vector space. ...


Note that a bilinear form is a special case of a bilinear operator. In mathematics, a bilinear operator is a generalized multiplication which satisfies the distributive law. ...


When F is the field of complex numbers C one is often more interested in sesquilinear forms. These are similiar to bilinear forms but are conjugate linear in one argument instead of linear. The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ... In mathematics, a sesquilinear form on a complex vector space V is a map V × V → C that is linear in one argument and antilinear in the other. ... In mathematics, a real linear transformation f from a complex vector space V to another is said to be antilinear (or conjugate-linear or semilinear) if for all c, d in C and all x, y in V. See also: complex conjugate, sesquilinear form ...

Contents

Coordinate representation

If V is finite-dimensional with dimension n then any bilinear form B on V can represented in coordinates by a matrix B relative to some ordered basis {ei} for V. The components of the matrix B are given by Bij = B(ei,ej). The action of the bilinear form on vectors u and v is then given by matrix multiplication: In mathematics, the dimension of a vector space V is the cardinality (i. ... For the square matrix section, see square matrix. ... In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V...

where ui and vj are the components of u and v in this basis.


Maps to the dual space

Every bilinear form B on V defines a pair of linear maps from V to its dual space V*. Define by In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). ...

B1(v)(w) = B(v,w)
B2(v)(w) = B(w,v)

This is often denoted as

B1(v) = B(v, − )
B2(v) = B( − ,v)

where the (–) indicates the slot into which the argument is to be placed.


If V is finite-dimensional then one can identify V with its double dual V**. One can then show that B2 is the transpose of the linear map B1 (if V is infinite-dimensional then B2 is the transpose of B1 restricted to the image of V in V**). Given B one can define the transpose of B to be the bilinear form given by See transposition for meanings of this term in telecommunication and music. ...

B * (v,w) = B(w,v).

If V is finite-dimensional then the rank of B1 is equal to the rank of B2. If this number is equal to the dimension of V then B1 and B2 are linear isomorphisms from V to V*. In this case B is said to be nondegenerate. In linear algebra, the column rank (row rank respectively) of a matrix A with entries in some field is defined to be the maximal number of columns (rows respectively) of A which are linearly independent. ... In mathematics, a degenerate bilinear form f(x,y) on a vector space V is one such that for some non-zero x in V for all y ∈ V. A nondegenerate form is one that is not degenerate. ...


Given any linear map A : VV* one can bilinear form B on V via

B(v,w) = A(v)(w)

This form will be nondegenerate iff A is an isomorphism.


Symmetry

A bilinear form B : V × VF is said to

  • symmetric if B(v,w) = B(w,v) for all
  • skew-symmetric if B(v,w) = − B(w,v) for all
  • alternating if B(v,v) = 0 for all

Every alternating form is skew-symmetric; this may be seen by expanding

B(v+w,v+w).

If the characteristic of F is not 2 then the converse is also true (every skew-symmetric form is alternating). If, however, char(F) = 2 then a skew-symmetric form is the same thing as a symmetric form and not all of these are alternating. In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0 (where n1R is defined as 1R + ... + 1R with n summands). ...


A bilinear form is symmetric (resp. skew-symmetric) iff its coordinate matrix (relative to any basis) is symmetric (resp. skew-symmetric). A bilinear form is alternating iff its coordinate matrix is skew-symmetric and the diagonal entries are all zero (which follows from skew-symmetry when char(F) ≠ 2). ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Although P iff Q is most standard, common alternative phrases include P is necessary and sufficient for Q and P...


A bilinear form is symmetric iff the maps are equal, and skew-symmetric iff they are negatives of one another. If char(F) ≠ 2 then one can always decompose a bilinear form into a symmetric and an skew-symmetric part as follows

where B* is the transpose of B (defined above).


Relation to tensor products

By the universal property of the tensor product, bilinear forms on V are in 1-to-1 correspondence with linear maps VVF. If B is a bilinear form on V the corresponding linear map is given by In category theory, abstract algebra and other fields of mathematics, frequently constructions are defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. ... In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras and modules. ...

The set of all linear maps VVF is the dual space of VV, so bilinear forms may be thought of as elements of In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). ...

Likewise, symmetric bilinear forms may be thought of as elements of S2V* (the second symmetric power of V*), and alternating bilinear forms as elements of Λ2V* (the second exterior power of V*). In mathematics, the symmetric algebra S(V) on a vector space V over a field K is a certain commutative unital associative K-algebra containing V. In fact, it is the most general such algebra, which can be expressed by a universal property. ... In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V is a certain unital associative algebra which contains V as a subspace. ...


See also

In mathematics, a bilinear operator is a generalized multiplication which satisfies the distributive law. ... In multilinear algebra, a multilinear form is a map of the type , where V is a vector space over the field K, that is separately linear in each its N variables. ... In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ... In mathematics, a sesquilinear form on a complex vector space V is a map V × V → C that is linear in one argument and antilinear in the other. ...

External links

  • Bilinear form (http://planetmath.org/?op=getobj&from=objects&id=1612) on PlanetMath.

 
 

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