View source for Bilinear form

{{Short description|Scalar-valued bilinear function}}
In [[mathematics]], a '''bilinear form''' is a [[bilinear map]] {{math|''V'' × ''V'' → ''K''}} on a [[vector space]] {{mvar|V}} (the elements of which are called ''[[Vector (mathematics)|vectors]]'') over a [[Field (mathematics)|field]] ''K'' (the elements of which are called ''[[scalar (mathematics)|scalar]]s''). In other words, a bilinear form is a function {{math|''B'' : ''V'' × ''V'' → ''K''}} that is [[linear map|linear]] in each argument separately:
* {{math|1=''B''('''u''' + '''v''', '''w''') = ''B''('''u''', '''w''') + ''B''('''v''', '''w''')}} {{spaces|3}} and {{spaces|3}} {{math|1=''B''(''λ'''''u''', '''v''') = ''λB''('''u''', '''v''')}}
* {{math|1=''B''('''u''', '''v''' + '''w''') = ''B''('''u''', '''v''') + ''B''('''u''', '''w''')}} {{spaces|3}} and {{spaces|3}} {{math|1=''B''('''u''', ''λ'''''v''') = ''λB''('''u''', '''v''')}}

The [[dot product]] on <math>\R^n</math> is an example of a bilinear form which is also an [[Inner product space|inner product]].<ref>{{Cite web| date=2021-01-16| title=Chapter 3. Bilinear forms — Lecture notes for MA1212| url=https://www.maths.tcd.ie/~pete/ma1212/chapter3.pdf}}</ref> An example of a bilinear form that is not an inner product would be the [[four-vector]] product.

The definition of a bilinear form can be extended to include [[module (mathematics)|modules]] over a [[Ring (mathematics)|ring]], with [[linear map]]s replaced by [[module homomorphism]]s.

When {{mvar|K}} is the field of [[complex number]]s {{math|'''C'''}}, one is often more interested in [[sesquilinear form]]s, which are similar to bilinear forms but are [[conjugate linear]] in one argument.

==Coordinate representation==
Let {{math|''V''}} be an {{mvar|n}}-[[dimension (vector space)|dimensional]] vector space with [[basis (linear algebra)|basis]] {{math|{'''e'''<sub>1</sub>, …, '''e'''<sub>''n''</sub>}<nowiki/>}}.

The {{math|''n'' × ''n''}} matrix ''A'', defined by {{math|1=''A<sub>ij</sub>'' = ''B''('''e'''<sub>''i''</sub>, '''e'''<sub>''j''</sub>)}} is called the ''matrix of the bilinear form'' on the basis {{math|{'''e'''<sub>1</sub>, …, '''e'''<sub>''n''</sub>}<nowiki/>}}.

If the {{math|''n'' × 1}} matrix {{math|''x''}} represents a vector {{math|'''x'''}} with respect to this basis, and similarly, the {{math|''n'' × 1}} matrix {{math|''y''}} represents another vector {{math|'''y'''}}, then:
<math display="block">B(\mathbf{x}, \mathbf{y}) = \mathbf{x}^\textsf{T} A\mathbf{y} = \sum_{i,j=1}^n x_i A_{ij} y_j. </math>

A bilinear form has different matrices on different bases. However, the matrices of a bilinear form on different bases are all [[congruent matrices|congruent]]. More precisely, if {{math|{'''f'''<sub>1</sub>, …, '''f'''<sub>''n''</sub>}<nowiki/>}} is another basis of {{mvar|V}}, then
<math display="block">\mathbf{f}_j=\sum_{i=1}^n S_{i,j}\mathbf{e}_i,</math>
where the <math>S_{i,j}</math> form an [[invertible matrix]] {{mvar|S}}. Then, the matrix of the bilinear form on the new basis is {{math|''S''<sup>T</sup>''AS''}}.

==Properties==
===Non-degenerate bilinear forms===
{{further|Degenerate bilinear form}}
Every bilinear form {{math|''B''}} on {{mvar|V}} defines a pair of linear maps from {{mvar|V}} to its [[dual space]] {{math|''V''<sup>∗</sup>}}. Define {{math|''B''<sub>1</sub>, ''B''<sub>2</sub>: ''V'' → ''V''<sup>∗</sup>}} by
{{block indent|left=1.6|text={{math|1=''B''<sub>1</sub>('''v''')('''w''') = ''B''('''v''', '''w''')}}}}
{{block indent|left=1.6|text={{math|1=''B''<sub>2</sub>('''v''')('''w''') = ''B''('''w''', '''v''')}}}}
This is often denoted as
{{block indent|left=1.6|text={{math|1=''B''<sub>1</sub>('''v''') = ''B''('''v''', ⋅)}}}}
{{block indent|left=1.6|text={{math|1=''B''<sub>2</sub>('''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 {{mvar|V}}, if either of {{math|''B''<sub>1</sub>}} or {{math|''B''<sub>2</sub>}} is an isomorphism, then both are, and the bilinear form {{math|''B''}} is said to be [[Degenerate form|nondegenerate]].  More concretely, for a finite-dimensional vector space, non-degenerate means that every non-zero element pairs non-trivially with some other element:
:<math>B(x,y)=0 </math> for all <math>y \in V</math> implies that {{math|1=''x'' = 0}} and
:<math>B(x,y)=0 </math> for all <math>x \in V</math> implies that {{math|1=''y'' = 0}}.

The corresponding notion for a module over a commutative ring is that a bilinear form is '''{{visible anchor|unimodular}}''' if {{math|''V'' → ''V''<sup>∗</sup>}} 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 {{math|1=''B''(''x'', ''y'') = 2''xy''}} is nondegenerate but not unimodular, as the induced map from {{math|1=''V'' = '''Z'''}} to {{math|1=''V''<sup>∗</sup> = '''Z'''}} is multiplication by 2.

If {{mvar|V}} is finite-dimensional then one can identify {{mvar|V}} with its double dual {{math|''V''<sup>∗∗</sup>}}. One can then show that {{math|''B''<sub>2</sub>}} is the [[Transpose of a linear map|transpose]] of the linear map {{math|''B''<sub>1</sub>}} (if {{mvar|V}} is infinite-dimensional then {{math|''B''<sub>2</sub>}} is the transpose of {{math|''B''<sub>1</sub>}} restricted to the image of {{mvar|V}} in {{math|1=''V''<sup>∗∗</sup>}}). Given {{math|''B''}} one can define the ''transpose'' of {{math|''B''}} to be the bilinear form given by
{{block indent|left=1.6|text=<sup>t</sup>''B''('''v''', '''w''') = ''B''('''w''', '''v''').}}

The '''left radical''' and '''right radical''' of the form {{math|''B''}} are the [[kernel (algebra)|kernel]]s of {{math|''B''<sub>1</sub>}} and {{math|''B''<sub>2</sub>}} 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 {{mvar|V}} is finite-dimensional then the [[rank (linear algebra)|rank]] of {{math|''B''<sub>1</sub>}} is equal to the rank of {{math|''B''<sub>2</sub>}}. If this number is equal to {{math|dim(''V'')}} then {{math|''B''<sub>1</sub>}} and {{math|''B''<sub>2</sub>}} are linear isomorphisms from {{mvar|V}} to {{math|''V''<sup>∗</sup>}}.  In this case {{math|''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:
{{block indent|left=1.6|text= '''Definition:''' ''B'' is '''nondegenerate''' if {{math|1=''B''('''v''', '''w''') = 0}} for all '''w''' implies {{math|1='''v''' = '''0'''}}.}}

Given any linear map {{math|1=''A'' : ''V'' → ''V''<sup>∗</sup>}} one can obtain a bilinear form ''B'' on ''V'' via
{{block indent|left=1.6|text=''B''('''v''', '''w''') = ''A''('''v''')('''w''').}}

This form will be nondegenerate if and only if {{math|''A''}} is an isomorphism.

If {{mvar|V}} is [[finite-dimensional]] then, relative to some [[basis (linear algebra)|basis]] for {{mvar|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 matrix|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 (ring theory)|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 {{math|1=''B''(''x'', ''y'') = 2''xy''}} over the integers.

===Symmetric, skew-symmetric, and alternating forms===
We define a bilinear form to be
* '''[[Symmetric bilinear form|symmetric]]''' if {{math|1=''B''('''v''', '''w''') = ''B''('''w''', '''v''')}} for all {{math|'''v'''}}, {{math|'''w'''}} in {{mvar|V}};
* '''[[Alternating form|alternating]]''' if {{math|1= ''B''('''v''', '''v''') = 0}} for all {{math|'''v'''}} in {{mvar|V}};
* '''{{visible anchor|skew-symmetric bilinear form|text=skew-symmetric}}''' or '''{{visible anchor|antisymmetric bilinear form|text=antisymmetric}}''' if {{math|1=''B''('''v''', '''w''') = −''B''('''w''', '''v''')}} for all {{math|'''v'''}}, {{math|'''w'''}} in {{mvar|V}};
*; Proposition: Every alternating form is skew-symmetric.
*; Proof: This can be seen by expanding {{math|''B''('''v''' + '''w''', '''v''' + '''w''')}}.

If the [[Characteristic (algebra)|characteristic]] of {{mvar|K}} is not 2 then the converse is also true: every skew-symmetric form is alternating. However, if {{math|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 (respectively skew-symmetric) [[if and only if]] its coordinate matrix (relative to any basis) is [[Symmetric matrix|symmetric]] (respectively [[Skew-symmetric matrix|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 {{math|char(''K'') ≠ 2}}).

A bilinear form is symmetric if and only if the maps {{math|''B''<sub>1</sub>, ''B''<sub>2</sub>: ''V'' → ''V''<sup>∗</sup>}} are equal, and skew-symmetric if and only if they are negatives of one another. If {{math|char(''K'') ≠ 2}} then one can decompose a bilinear form into a symmetric and a skew-symmetric part as follows
<math display="block">B^{+} = \tfrac{1}{2} (B + {}^{\text{t}}B)  \qquad  B^{-} = \tfrac{1}{2} (B - {}^{\text{t}}B) ,</math>
where {{math|<sup>t</sup>''B''}} is the transpose of {{math|''B''}} (defined above).

===Reflexive bilinear forms and orthogonal vectors===
{{block indent|left=1| '''Definition:''' A bilinear form {{math|''B'' : ''V'' × ''V'' → ''K''}} is called '''reflexive''' if {{math|1=''B''('''v''', '''w''') = 0}} implies {{math|1=''B''('''w''', '''v''') = 0}} for all '''v''', '''w''' in ''V''.}}
{{block indent|left=1| '''Definition:''' Let {{math|''B'' : ''V'' × ''V'' → ''K''}} be a reflexive bilinear form. '''v''', '''w''' in ''V'' are '''orthogonal with respect to ''B''''' if {{math|1=''B''('''v''', '''w''') = 0}}.}}

A bilinear form {{math|''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 {{math|'''v'''}}, with matrix representation {{math|''x''}}, is in the radical of a bilinear form with matrix representation {{math|''A''}}, if and only if {{math|1=''Ax'' = 0 ⇔ ''x''<sup>T</sup>''A'' = 0}}. The radical is always a subspace of {{math|''V''}}.  It is trivial if and only if the matrix {{math|''A''}} is nonsingular, and thus if and only if the bilinear form is nondegenerate.

Suppose {{mvar|W}} is a subspace.  Define the ''[[orthogonal complement]]''{{sfn|Adkins|Weintraub|1992|page=359}}
<math display="block"> W^{\perp} = \left\{\mathbf{v} \mid B(\mathbf{v}, \mathbf{w}) = 0 \text{ for all } \mathbf{w} \in W\right\} .</math>

For a non-degenerate form on a finite-dimensional space, the map {{math|''V/W'' → ''W''<sup>⊥</sup>}} is [[bijective]], and the dimension of {{math|''W''<sup>⊥</sup>}} is {{math|dim(''V'') − dim(''W'')}}.

===Bounded and elliptic bilinear forms===
'''Definition:''' A bilinear form on a [[normed vector space]]  {{math|(''V'', ‖⋅‖)}} is '''bounded''', if there is a constant {{math|''C''}} such that for all {{math|'''u''', '''v''' ∈ ''V''}},
<math display="block"> B ( \mathbf{u} , \mathbf{v}) \le C \left\| \mathbf{u} \right\| \left\|\mathbf{v} \right\| .</math>

'''Definition:''' A bilinear form on a normed vector space {{math|(''V'', ‖⋅‖)}} is '''elliptic''', or [[Coercive function#Coercive operators and forms|coercive]], if there is a constant {{math|''c'' > 0}} such that for all {{math|'''u''' ∈ ''V''}},
<math display="block"> B ( \mathbf{u} , \mathbf{u}) \ge c \left\| \mathbf{u} \right\| ^2 .</math>

==Associated quadratic form==
{{further|Quadratic form#Definitions}}
For any bilinear form {{math|''B'' : ''V'' × ''V'' → ''K''}}, there exists an associated [[quadratic form]] {{math|''Q'' : ''V'' → ''K''}} defined by {{math|''Q'' : ''V'' → ''K'' : '''v''' ↦ ''B''('''v''', '''v''')}}.

When {{math|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 {{math|1=char(''K'') = 2}} and {{math|dim ''V'' > 1}}, this correspondence between quadratic forms and symmetric bilinear forms breaks down.

==Relation to tensor products==
By the [[universal property]] of the [[tensor product]], there is a canonical correspondence between bilinear forms on {{mvar|V}} and linear maps {{math|''V'' ⊗ ''V'' → ''K''}}. If {{math|''B''}} is a bilinear form on {{mvar|V}} the corresponding linear map is given by
{{block indent|left=1.6|text= {{math|'''v''' ⊗ '''w''' ↦ ''B''('''v''', '''w''')}}}}
In the other direction, if {{math|''F'' : ''V'' ⊗ ''V'' → ''K''}} is a linear map the corresponding bilinear form is given by composing ''F'' with the bilinear map {{math|''V'' × ''V'' → ''V'' ⊗ ''V''}} that sends {{math|('''v''', '''w''')}} to {{math|'''v'''⊗'''w'''}}.

The set of all linear maps {{math|''V'' ⊗ ''V'' → ''K''}} is the [[dual space]] of {{math|''V'' ⊗ ''V''}}, so bilinear forms may be thought of as elements of {{math|(''V'' ⊗ ''V'')<sup>∗</sup>}} which (when {{mvar|V}} is finite-dimensional) is canonically isomorphic to {{math|''V''<sup>∗</sup> ⊗ ''V''<sup>∗</sup>}}.

Likewise, symmetric bilinear forms may be thought of as elements of {{math|(Sym<sup>2</sup>''V'')<sup>*</sup>}}  (dual of the second [[symmetric power]] of {{math|''V''}}) and alternating bilinear forms as elements of {{math|(Λ<sup>2</sup>''V'')<sup>∗</sup> ≃ Λ<sup>2</sup>''V''<sup>∗</sup>}} (the second [[exterior power]] of {{math|''V''<sup>∗</sup>}}). If {{math|char(''K'') ≠ 2}}, {{math|(Sym<sup>2</sup>''V'')<sup>*</sup> ≃ Sym<sup>2</sup>(''V''<sup>∗</sup>)}}.

==Generalizations==
===Pairs of distinct vector spaces===
Much of the theory is available for a [[bilinear mapping]] from two vector spaces over the same base field to that field
{{block indent|left=1.6|text={{math|''B'' : ''V'' × ''W'' → ''K''}}.}}

Here we still have induced linear mappings from {{mvar|V}} to {{math|''W''<sup>∗</sup>}}, and from {{mvar|W}} to {{math|''V''<sup>∗</sup>}}. 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 {{math|'''Z''' × '''Z''' → '''Z'''}} via {{math|(''x'', ''y'') ↦ 2''xy''}} is nondegenerate, but induces multiplication by 2 on the map {{math|'''Z''' → '''Z'''<sup>∗</sup>}}.

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 ''A<sub>ij</sub>'' having only +1 or −1 for non-zero elements. Some of the "inner products" are [[symplectic vector space|symplectic forms]] and some are [[sesquilinear form]]s or [[sesquilinear form#Hermitian form|Hermitian forms]]. Rather than a general field {{mvar|K}}, the instances with real numbers {{math|'''R'''}}, complex numbers {{math|'''C'''}}, and [[quaternions]] {{math|'''H'''}} are spelled out. The bilinear form
<math display="block">\sum_{k=1}^p x_k y_k - \sum_{k=p+1}^n  x_k y_k </math>
is called the '''real symmetric case''' and labeled {{math|'''R'''(''p'', ''q'')}}, where {{math|1=''p'' + ''q'' = ''n''}}.  Then he articulates the connection to traditional terminology:{{sfn|Harvey|1990|page=23}}
{{quote|
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''−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''.
}}

===General modules===
Given a [[Ring (mathematics)|ring]] {{mvar|R}} and a right [[Module (mathematics)|{{mvar|R}}-module]] {{math|''M''}} and its [[dual module]] {{math|''M''<sup>∗</sup>}}, a mapping {{math|''B'' : ''M''<sup>∗</sup> × ''M'' → ''R''}} is called a '''bilinear form''' if
{{block indent|left=1.6|text={{math|1=''B''(''u'' + ''v'', ''x'') = ''B''(''u'', ''x'') + ''B''(''v'', ''x'')}}}}
{{block indent|left=1.6|text={{math|1=''B''(''u'', ''x'' + ''y'') = ''B''(''u'', ''x'') + ''B''(''u'', ''y'')}}}}
{{block indent|left=1.6|text={{math|1=''B''(''αu'', ''xβ'') = ''αB''(''u'', ''x'')''β''}}}}
for all {{math|''u'', ''v'' ∈ ''M''<sup>∗</sup>}}, all {{math|''x'', ''y'' ∈ ''M''}} and all {{math|''α'', ''β'' ∈ ''R''}}.

The mapping {{math|⟨⋅,⋅⟩ : ''M''<sup>∗</sup> × ''M'' → ''R'' : (''u'', ''x'') ↦ ''u''(''x'')}} is known as the ''[[natural pairing]]'', also called the ''canonical bilinear form'' on {{math|''M''<sup>∗</sup> × ''M''}}.{{sfn|Bourbaki|1970|page=233}}

A linear map {{math|''S'' : ''M''<sup>∗</sup> → ''M''<sup>∗</sup> : ''u'' ↦ ''S''(''u'')}} induces the bilinear form {{math|''B'' : ''M''<sup>∗</sup> × ''M'' → ''R'' : (''u'', ''x'') ↦ ⟨''S''(''u''), ''x''⟩}}, and a linear map {{math|''T'' : ''M'' → ''M'' : ''x'' ↦ ''T''(''x'')}} induces the bilinear form {{math|''B'' : ''M''<sup>∗</sup> × ''M'' → ''R'' : (''u'', ''x'') ↦ ⟨''u'', ''T''(''x'')⟩}}.

Conversely, a bilinear form {{math|''B'' : ''M''<sup>∗</sup> × ''M'' → ''R''}} induces the ''R''-linear maps {{math|''S'' : ''M''<sup>∗</sup> → ''M''<sup>∗</sup> : ''u'' ↦ (''x'' ↦ ''B''(''u'', ''x''))}} and {{math|''T''′ : ''M'' → ''M''<sup>∗∗</sup> : ''x'' ↦ (''u'' ↦ ''B''(''u'', ''x''))}}.  Here, {{math|''M''<sup>∗∗</sup>}} denotes the [[double dual]] of {{math|''M''}}.

==See also==
{{cmn|colwidth=20em|
*[[Bilinear map]]
*[[:Category:Bilinear maps]]
*[[Inner product space]]
*[[Linear form]]
*[[Multilinear form]]
*[[Polar space]]
*[[Quadratic form]]
*[[Sesquilinear form]]
*[[System of bilinear equations]]
*[[Metric tensor]]
}}

==Citations==
{{reflist}}

==References==
{{refbegin}}
* {{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 |author-link=Nicolas Bourbaki}}
* {{citation | last=Cooperstein | first=Bruce | year=2010 | title=Advanced Linear Algebra | chapter=Ch 8: Bilinear Forms and Maps | pages=249–88 | 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=19–40 | publisher=[[Academic Press]] | isbn=0-12-329650-1 }}
* {{citation | last = Popov | first = V. L. | editor=Hazewinkel, M. | year=1987 | title = Bilinear form | url= https://www.encyclopediaofmath.org/index.php/Bilinear_form |encyclopedia=[[Encyclopedia of Mathematics]] | volume=1 | pages=390–392 | publisher=[[Kluwer Academic Publishers]]}}. Also: {{Google books|RVr5BwAAQBAJ|page=390}}
* {{citation | last=Jacobson | first=Nathan | year=2009 | title=Basic Algebra | volume=I | publisher=Courier Corporation | 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. | author-link=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. | author-link=Igor Shafarevich | author2=A. O. Remizov | year=2012 | title = Linear Algebra and Geometry | publisher=[[Springer Science+Business Media|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=Dmitriĭ 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 }}
{{refend}}

==External links==
{{Commonscat|Bilinear forms}}
*{{springer|title=Bilinear form|id=p/b016250}}
*{{planetmath reference|urlname=BilinearForm|title=Bilinear form}}

{{Functional Analysis}}
{{PlanetMath attribution|id=7553|title=Unimodular}}

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[[Category:Bilinear forms| ]]
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[[Category:Multilinear algebra]]