Bilinear form

From Exampleproblems

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

$w \mapsto B(v, w)$

$w \mapsto B(w, v)$

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.

1 Coordinate representation
2 Maps to the dual space
3 Symmetry
4 Relation to tensor products
5 See also
6 External links

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 $B i j = B (e i, e j)$ . The action of the bilinear form on vectors u and v is then given by matrix multiplication:

$B(u,v) = \mathbf{u}^T \mathbf{Bv} = \sum_{i,j=1}^{n}B_{ij}u^i v^j$

where uⁱ 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 $B_1,B_2\colon V \to V^*$ by

B 1 (v)(w) = B (v, w)

B 2 (v)(w) = B (w, v)

This is often denoted as

B 1 (v) = B (v, - )

B 2 (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

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.

Given any linear map A : V → V* one can obtain a 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 × V → F is said to be:

symmetric if $B (v, w) = B (w, v)$ for all $v,w\in V$
skew-symmetric if $B (v, w) = - B (w, v)$ for all $v,w\in V$ (this is called skew-symmetric by mathematicians and antisymmetric by physicists)
alternating if $B (v, v) = 0$ for all $v\in V$

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 $B_1,B_2\colon V \to V^*$ 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

$B^{\pm} = {1\over 2}(B \pm B^*)$

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

$v\otimes w\mapsto B(v,w).$

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

$(V\otimes V)^{*} \cong V^{*}\otimes V^{*}.$

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*).