# Bilinear form

In mathematics, a **bilinear form** on a vector space *V* over a field *F* is a mapping *V* × *V* → *F* which is linear in both arguments. That is, *B* : *V* × *V* → *F* 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.

## 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 {*e*_{i}} 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 *u*^{i} and *v*^{j} 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 *B*_{2} is the transpose of the linear map *B*_{1} (if *V* is infinite-dimensional then *B*_{2} is the transpose of *B*^{1} 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 *B*_{1} is equal to the rank of *B*_{2}. If this number is equal to the dimension of *V* then *B*_{1} and *B*_{2} are linear isomorphisms from *V* to *V**. In this case *B* is said to be **nondegenerate**.

Given any linear map *A* : *V* → *V** 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* × *V* → *F* 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 *V* ⊗ *V* → *F*. If *B* is a bilinear form on *V* the corresponding linear map is given by

The set of all linear maps *V* ⊗ *V* → *F* is the dual space of *V* ⊗ *V*, so bilinear forms may be thought of as elements of

Likewise, symmetric bilinear forms may be thought of as elements of *S*^{2}*V** (the second symmetric power of *V**), and alternating bilinear forms as elements of Λ^{2}*V** (the second exterior power of *V**).