In mathematics, a **bilinear form** is a bilinear map *V* × *V* → *K* on a vector space V (the elements of which are called *vectors*) over a field *K* (the elements of which are called *scalars*). In other words, a bilinear form is a function *B* : *V* × *V* → *K* that is linear in each argument separately:

*B*(**u**+**v**,**w**) =*B*(**u**,**w**) +*B*(**v**,**w**) and*B*(*λ***u**,**v**) =*λB*(**u**,**v**)*B*(**u**,**v**+**w**) =*B*(**u**,**v**) +*B*(**u**,**w**) and*B*(**u**,*λ***v**) =*λB*(**u**,**v**)

The dot product on is an example of a bilinear form.^{[1]}

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.

Let *V* be an n-dimensional vector space with basis {**e**_{1}, …, **e**_{n}}.

The *n* × *n* matrix *A*, defined by *A _{ij}* =

If the *n* × 1 matrix *x* represents a vector **x** with respect to this basis, and similarly, the *n* × 1 matrix *y* represents another vector **y**, then:

A bilinear form has different matrices on different bases. However, the matrices of a bilinear form on different bases are all congruent. More precisely, if {**f**_{1}, …, **f**_{n}} is another basis of V, then

where the form an invertible matrix S. Then, the matrix of the bilinear form on the new basis is

Every bilinear form *B* on V defines a pair of linear maps from V to its dual space *V*^{∗}. Define *B*_{1}, *B*_{2}: *V* → *V*^{∗} 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:

- for all implies that
*x*= 0 and - for all implies that
*y*= 0.

The corresponding notion for a module over a commutative ring is that a bilinear form is **unimodular** if *V* → *V*^{∗} 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 *B*(*x*, *y*) = 2*xy* is nondegenerate but not unimodular, as the induced map from *V* = **Z** to *V*^{∗} = **Z** 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;^{[2]} they are the vectors orthogonal to the whole space on the left and on the right.^{[3]}

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 *A* : *V* → *V*^{∗} 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 *B*(*x*, *y*) = 2*xy* over the integers.

We define a bilinear form to be

**symmetric**if*B*(**v**,**w**) =*B*(**w**,**v**) for all**v**,**w**in V;**alternating**if*B*(**v**,**v**) = 0 for all**v**in V;**skew-symmetric**or**antisymmetric**if*B*(**v**,**w**) = −*B*(**w**,**v**) for all**v**,**w**in V;- Proposition
- Every alternating form is skew-symmetric.
- Proof
- This can be seen by expanding
*B*(**v**+**w**,**v**+**w**).

If the characteristic of K is not 2 then the converse is also true: every skew-symmetric form is alternating. However, if char(*K*) = 2 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 char(*K*) ≠ 2).

A bilinear form is symmetric if and only if the maps *B*_{1}, *B*_{2}: *V* → *V*^{∗} are equal, and skew-symmetric if and only if they are negatives of one another. If char(*K*) ≠ 2 then one can decompose a bilinear form into a symmetric and a skew-symmetric part as follows

where

For any bilinear form *B* : *V* × *V* → *K*, there exists an associated quadratic form *Q* : *V* → *K* defined by *Q* : *V* → *K* : **v** ↦ *B*(**v**, **v**).

When char(*K*) ≠ 2, 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 char(*K*) = 2 and dim *V* > 1, this correspondence between quadratic forms and symmetric bilinear forms breaks down.

A bilinear form *B* is reflexive if and only if it is either symmetric or alternating.^{[4]} 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 *Ax* = 0 ⇔ *x*^{T}*A* = 0. 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*^{[5]}

For a non-degenerate form on a finite-dimensional space, the map *V/W* → *W*^{⊥} is bijective, and the dimension of *W*^{⊥} is dim(*V*) − dim(*W*).

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 **Z** × **Z** → **Z** via (*x*, *y*) ↦ 2*xy* is nondegenerate, but induces multiplication by 2 on the map **Z** → **Z**^{∗}.

Terminology varies in coverage of bilinear forms. For example, F. Reese Harvey discusses "eight types of inner product".^{[6]} To define them he uses diagonal matrices *A _{ij}* 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

is called the

Some of the real symmetric cases are very important. The positive definite case

R(n, 0) is calledEuclidean space, while the case of a single minus,R(n−1, 1) is calledLorentzian space. Ifn= 4, then Lorentzian space is also calledMinkowski spaceorMinkowski spacetime. The special caseR(p,p) will be referred to as thesplit-case.

By the universal property of the tensor product, there is a canonical correspondence between bilinear forms on V and linear maps *V* ⊗ *V* → *K*. If *B* is a bilinear form on V the corresponding linear map is given by

In the other direction, if *F* : *V* ⊗ *V* → *K* is a linear map the corresponding bilinear form is given by composing *F* with the bilinear map *V* × *V* → *V* ⊗ *V* that sends (**v**, **w**) to **v**⊗**w**.

The set of all linear maps *V* ⊗ *V* → *K* is the dual space of *V* ⊗ *V*, so bilinear forms may be thought of as elements of (*V* ⊗ *V*)^{∗} which (when V is finite-dimensional) is canonically isomorphic to *V*^{∗} ⊗ *V*^{∗}.

Likewise, symmetric bilinear forms may be thought of as elements of (Sym^{2}*V*)^{*} (dual of the second symmetric power of *V*) and alternating bilinear forms as elements of (Λ^{2}*V*)^{∗} ≃ Λ^{2}*V*^{∗} (the second exterior power of *V*^{∗}). If char*K* ≠ 2, (Sym^{2}*V*)^{*} ≃ Sym^{2}(*V*^{∗}).

**Definition:** A bilinear form on a normed vector space (*V*, ‖⋅‖) is **bounded**, if there is a constant *C* such that for all **u**, **v** ∈ *V*,

**Definition:** A bilinear form on a normed vector space (*V*, ‖⋅‖) is **elliptic**, or coercive, if there is a constant *c* > 0 such that for all **u** ∈ *V*,

Given a ring R and a right R-module *M* and its dual module *M*^{∗}, a mapping *B* : *M*^{∗} × *M* → *R* is called a **bilinear form** if

for all *u*, *v* ∈ *M*^{∗}, all *x*, *y* ∈ *M* and all *α*, *β* ∈ *R*.

The mapping ⟨⋅,⋅⟩ : *M*^{∗} × *M* → *R* : (*u*, *x*) ↦ *u*(*x*) is known as the *natural pairing*, also called the *canonical bilinear form* on *M*^{∗} × *M*.^{[8]}

A linear map *S* : *M*^{∗} → *M*^{∗} : *u* ↦ *S*(*u*) induces the bilinear form *B* : *M*^{∗} × *M* → *R* : (*u*, *x*) ↦ ⟨*S*(*u*), *x*⟩, and a linear map *T* : *M* → *M* : *x* ↦ *T*(*x*) induces the bilinear form *B* : *M*^{∗} × *M* → *R* : (*u*, *x*) ↦ ⟨*u*, *T*(*x*)⟩.

Conversely, a bilinear form *B* : *M*^{∗} × *M* → *R* induces the *R*-linear maps *S* : *M*^{∗} → *M*^{∗} : *u* ↦ (*x* ↦ *B*(*u*, *x*)) and *T*′ : *M* → *M*^{∗∗} : *x* ↦ (*u* ↦ *B*(*u*, *x*)). Here, *M*^{∗∗} denotes the double dual of *M*.