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:

*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.

## Coordinate representation

Let *V* ≅ *K ^{n}* be an n-dimensional vector space with basis {

**e**

_{1}, …,

**e**

_{n}}.

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

*B*(

**e**

_{i},

**e**

_{j}) is called the

*matrix of the bilinear form*on the basis {

**e**

_{1}, …,

**e**

_{n}}.

If the *n* × 1 matrix *x* represents a vector **v** with respect to this basis, and analogously, *y* represents another vector **w**, 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 {**f**_{1}, …, **f**_{n}} is another basis of V, then

*S*

^{T}

*AS*.

## 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}: *V* → *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 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

^{t}

*B*(

**v**,

**w**) =

*B*(

**w**,

**v**).

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:

**Definition:**

*B*is

**nondegenerate**if

*B*(

**v**,

**w**) = 0 for all

**w**implies

**v**=

**0**.

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 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.

## Symmetric, skew-symmetric and alternating forms

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**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. 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

^{t}

*B*is the transpose of

*B*(defined above).

## Derived quadratic form

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.

## Reflexivity and orthogonality

**Definition:**A bilinear form

*B*:

*V*×

*V*→

*K*is called

**reflexive**if

*B*(

**v**,

**w**) = 0 implies

*B*(

**w**,

**v**) = 0 for all

**v**,

**w**in

*V*.

**Definition:**Let

*B*:

*V*×

*V*→

*K*be a reflexive bilinear form.

**v**,

**w**in

*V*are

**orthogonal with respect to**if

*B**B*(

**v**,

**w**) = 0.

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*).

## Different spaces

Much of the theory is available for a bilinear mapping from two vector spaces over the same base field to that field

*B*:

*V*×

*W*→

*K*.

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

**real symmetric case**and labeled

**R**(

*p*,

*q*), where

*p*+

*q*=

*n*. Then he articulates the connection to traditional terminology:

^{[7]}

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.

## 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 *V* ⊗ *V* → *K*. If *B* is a bilinear form on V the corresponding linear map is given by

**v**⊗

**w**↦

*B*(

**v**,

**w**)

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*^{∗}).

## On normed vector spaces

**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*,

## Generalization to modules

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

*B*(

*u*+

*v*,

*x*) =

*B*(

*u*,

*x*) +

*B*(

*v*,

*x*)

*B*(

*u*,

*x*+

*y*) =

*B*(

*u*,

*x*) +

*B*(

*u*,

*y*)

*B*(

*αu*,

*xβ*) =

*αB*(

*u*,

*x*)

*β*

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*.

## See also

- Bilinear map
- Bilinear operator
- Inner product space
- Linear form
- Multilinear form
- Quadratic form
- Sesquilinear form
- Polar space

## Citations

**^**"Chapter 3. Bilinear forms — Lecture notes for MA1212" (PDF). 2021-01-16.**^**Jacobson 2009, p. 346.**^**Zhelobenko 2006, p. 11.**^**Grove 1997.**^**Adkins & Weintraub 1992, p. 359.**^**Harvey 1990, p. 22.**^**Harvey 1990, p. 23.**^**Bourbaki 1970, p. 233.

## References

- Adkins, William A.; Weintraub, Steven H. (1992),
*Algebra: An Approach via Module Theory*, Graduate Texts in Mathematics,**136**, Springer-Verlag, ISBN 3-540-97839-9, Zbl 0768.00003 - Bourbaki, N. (1970),
*Algebra*, Springer - Cooperstein, Bruce (2010), "Ch 8: Bilinear Forms and Maps",
*Advanced Linear Algebra*, CRC Press, pp. 249–88, ISBN 978-1-4398-2966-0 - Grove, Larry C. (1997),
*Groups and characters*, Wiley-Interscience, ISBN 978-0-471-16340-4 - Halmos, Paul R. (1974),
*Finite-dimensional vector spaces*, Undergraduate Texts in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90093-3, Zbl 0288.15002 - Harvey, F. Reese (1990), "Chapter 2: The Eight Types of Inner Product Spaces",
*Spinors and calibrations*, Academic Press, pp. 19–40, ISBN 0-12-329650-1 - Popov, V. L. (1987), "Bilinear form", in Hazewinkel, M. (ed.),
*Encyclopedia of Mathematics*,**1**, Kluwer Academic Publishers, pp. 390–392. Also:*Bilinear form*, p. 390, at Google Books - Jacobson, Nathan (2009),
*Basic Algebra*,**I**(2nd ed.), ISBN 978-0-486-47189-1 - Milnor, J.; Husemoller, D. (1973),
*Symmetric Bilinear Forms*, Ergebnisse der Mathematik und ihrer Grenzgebiete,**73**, Springer-Verlag, ISBN 3-540-06009-X, Zbl 0292.10016 - Porteous, Ian R. (1995),
*Clifford Algebras and the Classical Groups*, Cambridge Studies in Advanced Mathematics,**50**, Cambridge University Press, ISBN 978-0-521-55177-9 - Shafarevich, I. R.; A. O. Remizov (2012),
*Linear Algebra and Geometry*, Springer, ISBN 978-3-642-30993-9 - Shilov, Georgi E. (1977), Silverman, Richard A. (ed.),
*Linear Algebra*, Dover, ISBN 0-486-63518-X - Zhelobenko, Dmitriĭ Petrovich (2006),
*Principal Structures and Methods of Representation Theory*, Translations of Mathematical Monographs, American Mathematical Society, ISBN 0-8218-3731-1

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