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
- <math>w \mapsto B(v, w)<math>
- <math>w \mapsto B(w, v)<math>
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 similiar 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 {ei} for V. The components of the matrix B are given by <math>B_{ij} = B(e_i,e_j)<math>. The action of the bilinear form on vectors u and v is then given by matrix multiplication:
- <math>B(u,v) = \mathbf{u}^T \mathbf{Bv} = \sum_{i,j=1}^{n}B_{ij}u^i v^j<math>
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 <math>B_1,B_2\colon V \to V^*<math> by
- <math>B_1(v)(w) = B(v,w)<math>
- <math>B_2(v)(w) = B(w,v)<math>
This is often denoted as
- <math>B_1(v) = B(v,{-})<math>
- <math>B_2(v) = B({-},v)<math>
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
- <math>B^*(v,w) = B(w,v).<math>
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 bilinear form B on V via
- <math>B(v,w) = A(v)(w)<math>
This form will be nondegenerate iff A is an isomorphism.
Symmetry
A bilinear form B : V × V → F is said to be:
- symmetric if <math>B(v,w)=B(w,v)<math> for all <math>v,w\in V<math>
- skew-symmetric if <math>B(v,w)=-B(w,v)<math> for all <math>v,w\in V<math>
- alternating if <math>B(v,v)=0<math> for all <math>v\in V<math>
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 <math>B_1,B_2\colon V \to V^*<math> 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
- <math>B^{\pm} = {1\over 2}(B \pm B^*)<math>
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
- <math>v\otimes w\mapsto B(v,w).<math>
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
- <math>(V\otimes V)^{*} \cong V^{*}\otimes V^{*}.<math>
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*).