Bilinear form

In mathematics, a bilinear form on a vector space "V" is a bilinear map , where "K" is the field of scalars. In other words, a bilinear form is a function that is linear in each argument separately:
The dot product on formula_1 is an example of a bilinear form.
The definition of a bilinear form can be extended to include modules over a ring, with linear maps replaced by module homomorphisms.
When "K" is the field of complex numbers C, one is often more interested in sesquilinear forms, which are similar to bilinear forms but are conjugate linear in one argument.
Coordinate representation.
Let be an -dimensional vector space with basis 
The matrix "A", defined by is called the "matrix of the bilinear form" on the basis 
If the matrix represents a vector with respect to this basis, and analogously, represents another vector , then:
A bilinear form has different matrices on different bases. However, the matrices of a bilinear on different bases are all congruent. More precisely, if is another basis of , then 
where the formula_4 form an invertible matrix . Then, the matrix of the bilinear form on the new basis is .
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 dot ( ⋅ ) indicates the slot into which the argument for the resulting linear functional is to be placed (see Currying).
For a finite-dimensional vector space "V", if either of "B"1 or "B"2 is an isomorphism, then both are, and the bilinear form "B" is said to be nondegenerate. More concretely, for a finite-dimensional vector space, non-degenerate means that every non-zero element pairs non-trivially with some other element:
The corresponding notion for a module over a commutative ring is that a bilinear form is if is an isomorphism. Given a finitely generated module over a commutative ring, the pairing may be injective (hence "nondegenerate" in the above sense) but not unimodular. For example, over the integers, the pairing is nondegenerate but not unimodular, as the induced map from to is multiplication by 2.
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
The left radical and right radical of the form "B" are the kernels of "B"1 and "B"2 respectively; they are the vectors orthogonal to the whole space on the left and on the right.
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 dim("V") then "B"1 and "B"2 are linear isomorphisms from "V" to "V"∗. In this case "B" is nondegenerate. By the rank–nullity theorem, this is equivalent to the condition that the left and equivalently right radicals be trivial. For finite-dimensional spaces, this is often taken as the "definition" of nondegeneracy:
Given any linear map one can obtain a bilinear form "B" on "V" via
This form will be nondegenerate if and only if "A" is an isomorphism.
If "V" is finite-dimensional then, relative to some basis for "V", a bilinear form is degenerate if and only if the determinant of the associated matrix is zero. Likewise, a nondegenerate form is one for which the determinant of the associated matrix is non-zero (the matrix is non-singular). These statements are independent of the chosen basis. For a module over a commutative ring, a unimodular form is one for which the determinant of the associate matrix is a unit (for example 1), hence the term; note that a form whose matrix determinant is non-zero but not a unit will be nondegenerate but not unimodular, for example over the integers.
Symmetric, skew-symmetric and alternating forms.
We define a bilinear form to be
If the characteristic of "K" is not 2 then the converse is also true: every skew-symmetric form is alternating. If, however, then a skew-symmetric form is the same as a symmetric form and there exist symmetric/skew-symmetric forms that are not alternating.
A bilinear form is symmetric (respectively skew-symmetric) if and only if its coordinate matrix (relative to any basis) is symmetric (respectively skew-symmetric). A bilinear form is alternating if and only if its coordinate matrix is skew-symmetric and the diagonal entries are all zero (which follows from skew-symmetry when ).
A bilinear form is symmetric if and only if the maps are equal, and skew-symmetric if and only if they are negatives of one another. If then one can decompose a bilinear form into a symmetric and a skew-symmetric part as follows
where t"B" is the transpose of "B" (defined above).
Derived quadratic form.
For any bilinear form , there exists an associated quadratic form defined by .
When , the quadratic form "Q" is determined by the symmetric part of the bilinear form "B" and is independent of the antisymmetric part. In this case there is a one-to-one correspondence between the symmetric part of the bilinear form and the quadratic form, and it makes sense to speak of the symmetric bilinear form associated with a quadratic form.
When and , this correspondence between quadratic forms and symmetric bilinear forms breaks down.
Reflexivity and orthogonality.
A bilinear form "B" is reflexive if and only if it is either symmetric or alternating. In the absence of reflexivity we have to distinguish left and right orthogonality. In a reflexive space the left and right radicals agree and are termed the "kernel" or the "radical" of the bilinear form: the subspace of all vectors orthogonal with every other vector. A vector v, with matrix representation "x", is in the radical of a bilinear form with matrix representation "A", if and only if . The radical is always a subspace of "V". It is trivial if and only if the matrix "A" is nonsingular, and thus if and only if the bilinear form is nondegenerate.
Suppose "W" is a subspace. Define the "orthogonal complement"
For a non-degenerate form on a finite-dimensional space, the map is bijective, and the dimension of "W"⊥ is .
Different spaces.
Much of the theory is available for a bilinear mapping from two vector spaces over the same base field to that field
Here we still have induced linear mappings from "V" to "W"∗, and from "W" to "V"∗. It may happen that these mappings are isomorphisms; assuming finite dimensions, if one is an isomorphism, the other must be. When this occurs, "B" is said to be a perfect pairing.
In finite dimensions, this is equivalent to the pairing being nondegenerate (the spaces necessarily having the same dimensions). For modules (instead of vector spaces), just as how a nondegenerate form is weaker than a unimodular form, a nondegenerate pairing is a weaker notion than a perfect pairing. A pairing can be nondegenerate without being a perfect pairing, for instance via is nondegenerate, but induces multiplication by 2 on the map .
Terminology varies in coverage of bilinear forms. For example, F. Reese Harvey discusses "eight types of inner product". To define them he uses diagonal matrices "Aij" having only +1 or −1 for non-zero elements. Some of the "inner products" are symplectic forms and some are sesquilinear forms or Hermitian forms. Rather than a general field "K", the instances with real numbers R, complex numbers C, and quaternions H are spelled out. The bilinear form
is called the real symmetric case and labeled , where . Then he articulates the connection to traditional terminology:
Relation to tensor products.
By the universal property of the tensor product, there is a canonical correspondence between bilinear forms on "V" and linear maps . If "B" is a bilinear form on "V" the corresponding linear map is given by
In the other direction, if is a linear map the corresponding bilinear form is given by composing "F" with the bilinear map that sends to .
The set of all linear maps is the dual space of , so bilinear forms may be thought of as elements of which (when "V" is finite-dimensional) is canonically isomorphic to .
Likewise, symmetric bilinear forms may be thought of as elements of Sym2("V"∗) (the second symmetric power of "V"∗), and alternating bilinear forms as elements of Λ2"V"∗ (the second exterior power of "V"∗).
On normed vector spaces.
Definition: A bilinear form on a normed vector space is bounded, if there is a constant "C" such that for all ,
Definition: A bilinear form on a normed vector space is elliptic, or coercive, if there is a constant such that for all ,
Generalization to modules.
Given a ring "R" and a right "R"-module "M" and its dual module "M"∗, a mapping is called a bilinear form if
for all , all and all .
The mapping is known as the "natural pairing", also called the "canonical bilinear form" on .
A linear map induces the bilinear form , and a linear map induces the bilinear form .
Conversely, a bilinear form induces the "R"-linear maps and . Here, "M"∗∗ denotes the double dual of "M".