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bilinear form

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**bilinear form**on a vector space

*V*is a bilinear map {{nowrap|

*V*Ã—

*V*â†’

*K*}}, where

*K*is the field of scalars. In other words, a bilinear form is a function {{nowrap|

*B*:

*V*Ã—

*V*â†’

*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 *B*(**u**+**v**,**w**) =*B*(**u**,**w**) +*B*(**v**,**w**) and*B*(*Î»***u**,**v**) =*Î»B*(**u**,**v**) **B*(**u**,**v**+**w**) =*B*(**u**,**v**) +*B*(**u**,**w**) and*B*(**u**,*Î»***v**) =*Î»B*(**u**,**v**)*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 {{nowrap|*V*â‰…

*Kn*}} be an

*n*-dimensional vector space with basis {{nowrap|{

**e**1, ...,

**e**

*n*}.}} Define the {{nowrap|

*n*â€‰Ã—â€‰

*n*}} matrix

*A*by {{nowrap|1=

*Aij*=

*B*(

**e**

*i**,**e**j*)}}. If the

*n*â€‰Ã—â€‰1 matrix

*x*represents a vector

**v**with respect to this basis, and analogously,

*y*represents

**w**, then:

B(mathbf{v}, mathbf{w}) = mathbf{x}^mathrm T Amathbf{y} = sum_{i,j=1}^n x_i a_{ij} y_j.

Suppose {{nowrap|{**f**1, ...,

**f**

*n*}{{void}}}} is another basis for

*V*, such that:

[

where {{nowrap|**f**1, ...,**f***n**] = [**e***1, ...,**e*n*]*S**S*âˆˆ GL(

*n*,

*K*)}}. Now the new matrix representation for the bilinear form is given by:

*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*âˆ—. Define {{nowrap|

*B*1,

*B*2:

*V*â†’

*V*âˆ—}} by

*B*1(

**v**)(

**w**) =

*B*(

**v**,

**w**)

*B*2(

**v**)(

**w**) =

*B*(

**w**,

**v**)

*B*1(

**v**) =

*B*(

**v**, â‹…)

*B*2(

**v**) =

*B*(â‹…,

**v**)

*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, for all y in V implies that {{nowrap|1=

The corresponding notion for a module over a commutative ring is that a bilinear form is *x*= 0}} and B(x,y)=0, for all x in V implies that {{nowrap|1=*y*= 0}}.**{{visible anchor|unimodular}}**if {{nowrap|

*V*â†’

*V*âˆ—}} 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 {{nowrap|1=

*B*(

*x*,

*y*) = 2

*xy*}} is nondegenerate but not unimodular, as the induced map from {{nowrap|1=

*V*=

**Z**}} to {{nowrap|1=

*V*âˆ— =

**Z**}} is multiplication by 2.If

*V*is finite-dimensional then one can identify

*V*with its double dual

*V*âˆ—âˆ—. 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*âˆ—âˆ—). Given

*B*one can define the

*transpose*of

*B*to be the bilinear form given by

t

The *B*(**v**,**w**) =*B*(**w**,**v**).**left radical**and

**right radical**of the form

*B*are the kernels of

*B*1 and

*B*2 respectively;{{sfn|Jacobson|2009|page=346}} they are the vectors orthogonal to the whole space on the left and on the right.{{sfn|Zhelobenko|2006|page=11}}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*âˆ—. In this case

*B*is nondegenerate. By the rankâ€“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 {{nowrap|1=

*B*(

**v**,

**w**) = 0}} for all

**w**implies {{nowrap|1=

**v**=

**0**}}.Given any linear map {{nowrap|1=

*A*:

*V*â†’

*V*âˆ—}} one can obtain a bilinear form

*B*on

*V*via

*B*(

**v**,

**w**) =

*A*(

**v**)(

**w**).

*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 {{nowrap|1=

*B*(

*x*,

*y*) = 2

*xy*}} over the integers.

## Symmetric, skew-symmetric and alternating forms

We define a bilinear form to be**symmetric**if {{nowrap|1=*B*(**v**,**w**) =*B*(**w**,**v**)}} for all**v**,**w**in*V*;**alternating**if {{nowrap|1=*B*(**v**,**v**) = 0}} for all**v**in*V*;**skew-symmetric**if {{nowrap|1=*B*(**v**,**w**) = âˆ’*B*(**w**,**v**)}} for all**v**,**w**in*V*;- :
**Proposition:**Every alternating form is skew-symmetric. - :
**Proof:**This can be seen by expanding {{nowrap|*B*(**v**+**w**,**v**+**w**)}}.

*K*is not 2 then the converse is also true: every skew-symmetric form is alternating. If, however, {{nowrap|1=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 {{nowrap|char(

*K*) â‰ 2}}).A bilinear form is symmetric if and only if the maps {{nowrap|

*B*1,

*B*2:

*V*â†’

*V*âˆ—}} are equal, and skew-symmetric if and only if they are negatives of one another. If {{nowrap|char(

*K*) â‰ 2}} then one can decompose a bilinear form into a symmetric and a skew-symmetric part as follows

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 {{nowrap|*B*:

*V*Ã—

*V*â†’

*K*}}, there exists an associated quadratic form {{nowrap|

*Q*:

*V*â†’

*K*}} defined by {{nowrap|

*Q*:

*V*â†’

*K*:

**v**â†¦

*B*(

**v**,

**v**)}}.When {{nowrap|char(

*K*) â‰ 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 {{nowrap|1=char(

*K*) = 2}} and {{nowrap|dim

*V*> 1}}, this correspondence between quadratic forms and symmetric bilinear forms breaks down.

## Reflexivity and orthogonality

**Definition:**A bilinear form {{nowrap|

*B*:

*V*Ã—

*V*â†’

*K*}} is called

**reflexive**if {{nowrap|1=

*B*(

**v**,

**w**) = 0}} implies {{nowrap|1=

*B*(

**w**,

**v**) = 0}} for all

**v**,

**w**in

*V*.

**Definition:**Let {{nowrap|

*B*:

*V*Ã—

*V*â†’

*K*}} be a reflexive bilinear form.

**v**,

**w**in

*V*are

**orthogonal with respect to**if {{nowrap|1=

*B**B*(

**v**,

**w**) = 0}}.A bilinear form

*B*is reflexive if and only if it is either symmetric or alternating.{{sfn|Grove|1997}} 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 {{nowrap|1=

*Ax*= 0 â‡”

*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*{{sfn|Adkins|Weintraub|1992|page=359}}

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 {{nowrap|*V/W*â†’

*W*âŠ¥}} is bijective, and the dimension of

*W*âŠ¥ is {{nowrap|dim(

*V*) âˆ’ 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*Ã—

*W*â†’

*K*.

*V*to

*W*âˆ—, and from

*W*to

*V*âˆ—. 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 {{nowrap|

**Z**Ã—

**Z**â†’

**Z**}} via {{nowrap|(

*x*,

*y*) â†¦ 2

*xy*}} is nondegenerate, but induces multiplication by 2 on the map {{nowrap|

**Z**â†’

**Z**âˆ—}}.Terminology varies in coverage of bilinear forms. For example, F. Reese Harvey discusses "eight types of inner product".{{sfn|Harvey|1990|page=22}} To define them he uses diagonal matrices

*Aij*having only +1 or âˆ’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

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 {{nowrap|

**R**(

*p*,

*q*)}}, where {{nowrap|1=

*p*+

*q*=

*n*}}. Then he articulates the connection to traditional terminology:{{sfn|Harvey|1990|page=23}}

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

**R**(*n*, 0)}} is called*Euclidean space*, while the case of a single minus, {{nowrap|**R**(*n*âˆ’1, 1)}} is called*Lorentzian space*. If {{nowrap|1=*n*= 4}}, then Lorentzian space is also called*Minkowski space*or*Minkowski spacetime*. The special case {{nowrap|**R**(*p*,*p*)}} will be referred to as the*split-case*.## Relation to tensor products

By the universal property of the tensor product, bilinear forms on*V*are in 1-to-1 correspondence with linear maps {{nowrap|

*V*âŠ—

*V*â†’

*K*}}. If

*B*is a bilinear form on

*V*a corresponding linear map is given by

**v**âŠ—

**w**â†¦

*B*(

**v**,

**w**)

*V*âŠ—

*V*â†’

*K*}} is the dual space of {{nowrap|

*V*âŠ—

*V*}}, so bilinear forms may be thought of as elements of

(

Likewise, symmetric bilinear forms may be thought of as elements of Sym2(*V*âŠ—*V*)âˆ— â‰…*V*âˆ— âŠ—*V*âˆ—*V*âˆ—) (the second symmetric power of

*V*âˆ—), and alternating bilinear forms as elements of Î›2

*V*âˆ— (the second exterior power of

*V*âˆ—).

## On normed vector spaces

**Definition:**A bilinear form on a normed vector space {{nowrap|(

*V*, â€–Â·â€–)}} is

**bounded**, if there is a constant

*C*such that for all {{nowrap|

**u**,

**v**âˆˆ

*V*}},

B ( mathbf{u} , mathbf{v}) le C left| mathbf{u} right| left|mathbf{v} right| .

**Definition:**A bilinear form on a normed vector space {{nowrap|(

*V*, â€–Â·â€–)}} is

**elliptic**, or coercive, if there is a constant {{nowrap|

*c*> 0}} such that for all {{nowrap|

**u**âˆˆ

*V*}},

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*âˆ—, a mapping {{nowrap|

*B*:

*M*âˆ— Ã—

*M*â†’

*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*(

*Î±u*,

*xÎ²*) =

*Î±B*(

*u*,

*x*)

*Î²*

*u*,

*v*âˆˆ

*M*âˆ—}}, {{nowrap|

*x*,

*y*âˆˆ

*M*}}, {{nowrap|

*Î±*,

*Î²*âˆˆ

*R*}}.The mapping {{nowrap|⋅,⋅{{rangle}} :

*M*âˆ— Ã—

*M*â†’

*R*: (

*u*,

*x*) â†¦

*u*(

*x*)}} is known as the

*natural pairing*, also called the

*canonical bilinear form*on {{nowrap|

*M*âˆ— Ã—

*M*}}.{{sfn|Bourbaki|1970|page=233}}A linear map {{nowrap|

*S*:

*M*âˆ— â†’

*M*âˆ— :

*u*â†¦

*S*(

*u*)}} induces the bilinear form {{nowrap|

*B*:

*M*âˆ— Ã—

*M*â†’

*R*: (

*u*,

*x*) â†¦

*S*(

*u*),

*x*{{rangle}}}}, and a linear map {{nowrap|

*T*:

*M*â†’

*M*:

*x*â†¦

*T*(

*x*)}} induces the bilinear form {{nowrap|

*B*:

*M*âˆ— Ã—

*M*â†’

*R*: (

*u*,

*x*) â†¦

*u*,

*T*(

*x*)){{rangle}}}}.Conversely, a bilinear form {{nowrap|

*B*:

*M*âˆ— Ã—

*M*â†’

*R*}} induces the

*R*-linear maps {{nowrap|

*S*:

*M*âˆ— â†’

*M*âˆ— :

*u*â†¦ (

*x*â†¦

*B*(

*u*,

*x*))}} and {{nowrap|

*T*â€² :

*M*â†’

*M*âˆ—âˆ— :

*x*â†¦ (

*u*â†¦

*B*(

*u*,

*x*))}}. Here,

*M*âˆ—âˆ— 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

{{reflist}}## References

- {{citation | last1=Adkins | first1=William A. | last2=Weintraub | first2=Steven H. | year=1992 | title=Algebra: An Approach via Module Theory | series=Graduate Texts in Mathematics | volume=136 | publisher=Springer-Verlag | isbn=3-540-97839-9 | zbl=0768.00003 }}
- {{citation | last=Bourbaki | first=N. | year=1970 | title=Algebra | publisher=Springer }}
- {{citation | last=Cooperstein | first=Bruce | year=2010 | title=Advanced Linear Algebra | chapter=Ch 8: Bilinear Forms and Maps | pages=249â€“88 | publisher=CRC Press | isbn=978-1-4398-2966-0 }}
- {{citation | last=Grove | first=Larry C. | year=1997 | title=Groups and characters | publisher=Wiley-Interscience | isbn=978-0-471-16340-4}}
- {{citation | last1=Halmos | first1=Paul R. | author1-link=Paul R. Halmos | year=1974 | title=Finite-dimensional vector spaces | series=Undergraduate Texts in Mathematics | publisher=Springer-Verlag | location=Berlin, New York | isbn=978-0-387-90093-3 | zbl=0288.15002 }}
- {{citation | last1=Harvey | first1=F. Reese | year=1990 | title=Spinors and calibrations | chapter=Chapter 2: The Eight Types of Inner Product Spaces | pages=19â€“40 | publisher=Academic Press | isbn=0-12-329650-1 }}
- {{citation | editor=Hazewinkel, M. | year=1988 | journal=Encyclopedia of Mathematics | volume=1 | page=390 | publisher=Kluwer Academic Publishers }}
- {{citation | last=Jacobson | first=Nathan | year=2009 | title=Basic Algebra | volume=I | edition=2nd | isbn=978-0-486-47189-1 }}
- {{citation | last1=Milnor | first1=J. | author1-link=John Milnor| first2=D. | last2=Husemoller | year=1973 | title=Symmetric Bilinear Forms | series=Ergebnisse der Mathematik und ihrer Grenzgebiete | volume=73 | publisher=Springer-Verlag | isbn=3-540-06009-X | zbl=0292.10016 }}
- {{citation | last=Porteous | first=Ian R. | authorlink=Ian R. Porteous | year=1995 | title=Clifford Algebras and the Classical Groups | series=Cambridge Studies in Advanced Mathematics | volume=50 | publisher=Cambridge University Press | ISBN=978-0-521-55177-9 }}
- {{citation | last=Shafarevich | first=I. R. | authorlink=Igor Shafarevich | author2=A. O. Remizov | year=2012 | title = Linear Algebra and Geometry | publisher=Springer | url=https://www.springer.com/mathematics/algebra/book/978-3-642-30993-9 | isbn=978-3-642-30993-9}}
- {{citation | last=Shilov | first=Georgi E. | title=Linear Algebra | editor-last=Silverman | editor-first=Richard A. | year=1977 | publisher=Dover | isbn=0-486-63518-X}}
- {{citation | last=Zhelobenko | first=DmitriÄ Petrovich | year=2006 | title=Principal Structures and Methods of Representation Theory | series=Translations of Mathematical Monographs | publisher=American Mathematical Society | isbn=0-8218-3731-1 }}

## External links

- {{springer|title=Bilinear form|id=p/b016250}}
- {{planetmath reference|id=1612|title=Bilinear form}}

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