Bilinear form

In mathematics, a **bilinear form** on a vector space *V* is a bilinear map *V* × *V* → *K*, where *K* is the field of 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 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* ≅ *Kn* be an *n*-dimensional vector space with basis {**e**1, ..., **e***n*}.

The *n* × *n* matrix *A*, defined by *Aij* = *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:

B
(

v

,

w

# )

x

T

A

y

=

∑

i
,
# j
1

n

x
i

a

i
j

y
j

.

{\displaystyle B(\mathbf{v}, \mathbf{w}) = \mathbf{x}^\text{T} A\mathbf{y} = \sum\_{i,j=1}^n x\_i a\_{ij} y\_j. }

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

f

j

=

∑

# i
1

n

S

i
,
j

e

i

,

{\displaystyle \mathbf{f}\_j=\sum\_{i=1}^n S\_{i,j}\mathbf{e}\_i,}

where the

S

i
,
j

{\displaystyle S\_{i,j}}

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

B
(
x
,
y
# )
0

{\displaystyle B(x,y)=0\,}

 for all

y
∈
V

{\displaystyle y \in V}

 implies that *x* = 0 and

B
(
x
,
y
# )
0

{\displaystyle B(x,y)=0\,}

 for all

x
∈
V

{\displaystyle x \in V}

 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; 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:

**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 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 (resp. skew-symmetric) if and only if its coordinate matrix (relative to any basis) is symmetric (resp. 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

B

=

1
2

(
B

t

B
)

B

−

=

1
2

(
B
−

t

B
)
,

{\displaystyle B^{+} = \tfrac{1}{2} (B + {}^{\text{t}}B) \qquad B^{-} = \tfrac{1}{2} (B - {}^{\text{t}}B) ,}

where 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 *B*** if *B*(**v**, **w**) = 0.
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 *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*

W

⊥

=
{

v

∣
B
(

v

,

w

# )
0

∀

w

∈
W
}

.

{\displaystyle W^{\perp}=\{\mathbf{v} \mid B(\mathbf{v}, \mathbf{w})=0\ \forall \mathbf{w}\in W\} \ . }

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". 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

∑

# k
1

p

x
k

y
k

−

∑

# k
p
1

n

x
k

y
k

{\displaystyle \sum\_{k=1}^p x\_k y\_k - \sum\_{k=p+1}^n x\_k y\_k }

is called the **real symmetric case** and labeled **R**(*p*, *q*), where *p* + *q* = *n*. Then he articulates the connection to traditional terminology:

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*.
## 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 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 (*V*, ‖·‖) is **bounded**, if there is a constant *C* such that for all **u**, **v** ∈ *V*,

B
(

u

,

v

)
≤
C

‖

u

‖

‖

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*,

B
(

u

,

u

)
≥
c

‖

u

‖

2

.

{\displaystyle B ( \mathbf{u} , \mathbf{u}) \ge c \left\| \mathbf{u} \right\| ^2 .}

## 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*∗, *x*, *y* ∈ *M*, *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*.

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*.