# Bilinear form

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

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.

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

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

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 **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

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.

If the characteristic of K is not 2 then the converse is also true: every skew-symmetric form is alternating. If, however, 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

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.

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

**R**, complex numbers

**C**, and quaternions

**H**are spelled out. The bilinear form

Some of the real symmetric cases are very important. The positive definite case **R**(*n*, 0) is called *Euclidean space*, while the case of a single minus, **R**(*n*−1, 1) is called *Lorentzian space*. If *n* = 4, then Lorentzian space is also called *Minkowski space* or *Minkowski spacetime*. The special case **R**(*p*, *p*) will be referred to as the *split-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*^{∗}) (the second symmetric power of *V*^{∗}), and alternating bilinear forms as elements of Λ^{2}*V*^{∗} (the second exterior power of *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

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*.

*This article incorporates material from Unimodular on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.*