Bilinear form

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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

are linear for each v in V. This definition applies equally well to modules over a commutative ring with linear maps being module homomorphisms.

Note that a bilinear form is a special case of a bilinear operator.

When F is the field of complex numbers C one is often more interested in sesquilinear forms. These are similar to bilinear forms but are conjugate linear in one argument instead of linear.

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 . The action of the bilinear form on vectors u and v is then given by matrix multiplication:

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

This is often denoted as

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

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.

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

This form will be nondegenerate iff A is an isomorphism.

Symmetry

A bilinear form B : V × VF is said to be:

  • symmetric if for all
  • skew-symmetric if for all (this is called skew-symmetric by mathematicians and antisymmetric by physicists)
  • alternating if 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.

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).

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

The set of all linear maps VVF is the dual space of VV, so bilinear forms may be thought of as elements of

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*).

See also

External links

zh:雙線性形