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| In [[mathematics]], a '''bilinear form''' on a [[vector space]] ''V'' is a [[bilinear map]]ping {{nowrap|''V'' × ''V'' → ''F''}}, where ''F'' is the [[field (mathematics)|field]] of [[scalar (mathematics)|scalar]]s. That is, a bilinear form is a function {{nowrap|''B'' : ''V'' × ''V'' → ''F''}} which is [[linear transformation|linear]] in each argument separately:
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| * ''B''('''u''' + '''v''', '''w''') = ''B''('''u''', '''w''') + ''B''('''v''', '''w''')
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| * ''B''('''u''', '''v''' + '''w''') = ''B''('''u''', '''v''') + ''B''('''u''', '''w''')
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| * ''B''(λ'''u''', '''v''') = ''B''('''u''', λ'''v''') = λ''B''('''u''', '''v''')
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| The definition of a bilinear form can be extended to include [[module (mathematics)|module]]s over a [[commutative ring]], with linear maps replaced by [[module homomorphism]]s. When ''F'' is the field of [[complex number]]s '''C''', one is often more interested in [[sesquilinear form]]s, which are similar to bilinear forms but are [[conjugate linear]] in one argument.
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| ==Coordinate representation==
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| Let ''V'' ≅ ''F<sup>n</sup>'' be an ''n''-dimensional vector space with basis {'''e'''<sub>1</sub>, ..., '''e'''<sub>''n''</sub>}. Define the ''n'' × ''n'' matrix ''A'' by ''A<sub>ij</sub>'' = ''B''('''e'''<sub>''i''</sub>, '''e'''<sub>''j''</sub>). If the ''n'' × 1 matrix ''x'' represents a vector '''v''' with respect to this basis, and analogously, ''y'' represents '''w''', then:
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| :<math>B(\mathbf{v}, \mathbf{w}) = x^\mathrm T Ay = \sum_{i,j=1}^n a_{ij} x_i y_j. </math>
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| Suppose {'''f'''<sub>1</sub>, ..., '''f'''<sub>''n''</sub>} is another basis for ''V'', such that:
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| : ['''f'''<sub>1</sub>, ..., '''f'''<sub>''n''</sub>] = ['''e'''<sub>1</sub>, ..., '''e'''<sub>''n''</sub>]''S''
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| where ''S'' ∈ GL(''n'', ''F''). Now the new matrix representation for the bilinear form is given by: ''S''<sup>T</sup>''AS''.
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| ==Maps to the dual space==
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| Every bilinear form ''B'' on ''V'' defines a pair of linear maps from ''V'' to its [[dual space]] ''V*''. Define ''B''<sub>1</sub>, ''B''<sub>2</sub>: ''V'' → ''V*'' by
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| :''B''<sub>1</sub>('''v''')('''w''') = ''B''('''v''', '''w''')
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| :''B''<sub>2</sub>('''v''')('''w''') = ''B''('''w''', '''v''')
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| This is often denoted as
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| :''B''<sub>1</sub>('''v''') = ''B''('''v''', ⋅)
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| :''B''<sub>2</sub>('''v''') = ''B''(⋅, '''v''')
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| where the ( ⋅ ) indicates the slot into which the argument for the resulting [[linear functional]] is to be placed.
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| For a finite-dimensional vector space ''V'', if either of ''B''<sub>1</sub> or ''B''<sub>2</sub> is an isomorphism, then both are, and the bilinear form ''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 ''x'' = 0 and
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| :<math>B(x,y)=0\,</math> for all <math>x \in V</math> implies that ''y'' = 0.
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| The corresponding notion for a module over a ring is that a bilinear form is '''{{visible anchor|unimodular}}''' if <math>V \to V^*</math> is an isomorphism. Given a finite dimensional 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>B(x,y) = 2xy</math> is nondegenerate but not unimodular, as the induced map from ''V'' = '''Z''' to ''V*'' = '''Z''' is multiplication by 2.
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| If ''V'' is finite-dimensional then one can identify ''V'' with its double dual ''V**''. One can then show that ''B''<sub>2</sub> is the [[transpose]] of the linear map ''B''<sub>1</sub> (if ''V'' is infinite-dimensional then ''B''<sub>2</sub> is the transpose of ''B''<sub>1</sub> restricted to the image of ''V'' in ''V**''). Given ''B'' one can define the ''transpose'' of ''B'' to be the bilinear form given by
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| :''B*''('''v''', '''w''') = ''B''('''w''', '''v''').
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| The ''left radical'' and ''right radical'' of the form ''B'' are the [[kernel (algebra)|kernel]]s of ''B''<sub>1</sub> and ''B''<sub>2</sub> respectively;<ref>{{harvnb|Jacobson|2009}} p.346</ref> they are the vectors orthogonal to the whole space on the left and on the right.<ref>{{cite book | title=Principal Structures and Methods of Representation Theory | series=Translations of Mathematical Monographs | first=Dmitriĭ Petrovich | last=Zhelobenko | publisher=[[American Mathematical Society]] | year=2006 | isbn=0-8218-3731-1 | page=11 }}</ref>
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| If ''V'' is finite-dimensional then the [[rank (linear algebra)|rank]] of ''B''<sub>1</sub> is equal to the rank of ''B''<sub>2</sub>. If this number is equal to dim(''V'') then ''B''<sub>1</sub> and ''B''<sub>2</sub> 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. In fact, for finite dimensional spaces, this is often taken as the ''definition'' of nondegeneracy:
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| <blockquote>'''Definition:''' ''B'' is nondegenerate if and only if ''B''('''v''', '''w''') = 0 for all '''w''' implies '''v''' = '''0'''.</blockquote>
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| Given any linear map ''A'' : ''V'' → ''V*'' one can obtain a bilinear form ''B'' on ''V'' via
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| :''B''('''v''', '''w''') = ''A''('''v''')('''w''').
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| This form will be nondegenerate if and only if ''A'' is an isomorphism.
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| If ''V'' is [[finite-dimensional]] then, relative to some [[basis (linear algebra)|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 matrix|non-singular]]). These statements are independent of the chosen basis. For a module over a 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 <math>B(x,y) = 2xy</math> over the integers.
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| ==Symmetric, skew-symmetric and alternating forms==
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| We define a form to be
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| *'''[[Symmetric bilinear form|symmetric]]''' if ''B''('''v''', '''w''') = ''B''('''w''', '''v''') for all '''v''', '''w''' in ''V'';
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| *'''[[Alternating form|alternating]]''' if ''B''('''v''', '''v''') = 0 for all '''v''' in ''V'';
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| *'''skew-symmetric''' if ''B''('''v''', '''w''') = −''B''('''w''', '''v''') for all '''v''', '''w''' in ''V'';
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| <blockquote>'''Proposition:''' Every alternating form is skew-symmetric. </blockquote>
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| <blockquote> '''Proof:''' This can be seen by expanding ''B''('''v'''+'''w''', '''v'''+'''w'''). </blockquote>
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| If the [[characteristic (algebra)|characteristic]] of ''F'' is not 2 then the converse is also true: every skew-symmetric form is alternating. If, however, char(''F'') = 2 then a skew-symmetric form is the same as a symmetric form and there exist symmetric/skew-symmetric forms which are not alternating.
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| A bilinear form is symmetric (resp. skew-symmetric) [[if and only if]] its coordinate matrix (relative to any basis) is [[Symmetric matrix|symmetric]] (resp. [[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 char(''F'') ≠ 2).
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| A bilinear form is symmetric if and only if the maps ''B''<sub>1</sub>, ''B''<sub>2</sub>: ''V'' → ''V*'' are equal, and skew-symmetric if and only if they are negatives of one another. If char(''F'') ≠ 2 then one can decompose a bilinear form into a symmetric and a skew-symmetric part as follows
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| :<math>B^{\pm} = \frac{1}{2} (B \pm B^*)</math>
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| where ''B*'' is the transpose of ''B'' (defined above).
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| Also if char(''F'') ≠ 2 then one can define a [[quadratic form]] in terms of its associated symmetric form. One can likewise define quadratic forms corresponding to skew-symmetric forms, [[Hermitian form]]s, and [[skew-Hermitian form]]s; the general concept is [[ε-quadratic form]].
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| ==Reflexivity and orthogonality==
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| <blockquote>'''Definition:''' A bilinear form ''B'' : ''V'' × ''V'' → ''F'' is called '''reflexive''' if ''B''('''v''', '''w''') = 0 implies ''B''('''w''', '''v''') = 0 for all '''v''', '''w''' in ''V''.</blockquote>
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| <blockquote>'''Definition:''' Let ''B'' : ''V'' × ''V'' → ''F'' be a reflexive bilinear form. '''v''', '''w''' in ''V'' are '''orthogonal with respect to ''B''''' if and only if ''B''('''v''', '''w''') = 0 or ''B''('''w''', '''v''') = 0.</blockquote>
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| A form ''B'' is reflexive if and only if it is either symmetric or alternating.<ref>{{harvnb|Grove|1997}}</ref> 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''<sup>T</sup>''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.
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| Suppose ''W'' is a subspace. Define the ''[[orthogonal complement]]''<ref>Adkins & Weintraub (1992) p.359</ref>
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| :<math>W^{\perp}=\{\mathbf{v}| B(\mathbf{v}, \mathbf{w})=0\ \forall \mathbf{w}\in W\} \ . </math>
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| For a non-degenerate form on a finite dimensional space, the map ''W'' ↔ ''W''<sup>⊥</sup> is bijective, and the dimension of ''W''<sup>⊥</sup> is dim(''V'') − dim(''W'').
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| ==Different spaces==
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| Much of the theory is available for a [[bilinear mapping]] to the base field
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| :''B'' : ''V'' × ''W'' → ''F''.
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| In this situation 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'''.
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| 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''' × '''Z''' → '''Z'''}} via {{nowrap|(''x'',''y'') ↦ 2''xy''}} is nondegenerate, but induces multiplication by 2 on the map {{nowrap|'''Z''' → '''Z'''*}}.
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| Terminology varies in coverage of bilinear forms. For example, F. Reese Harvey discusses "eight types of inner product".<ref>Harvey p. 22</ref> 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 ''F'', the instances with real numbers '''R''', complex numbers '''C''', and [[quaternions]] '''H''' are spelled out. The bilinear form
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| :<math>\sum_{k=1}^p x_k y_k - \sum_{k=p+1}^n x_k y_k </math>
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| is called the '''real symmetric case''' and labeled R(''p'', ''q''), where ''p'' + ''q'' = ''n''. Then he articulates the connection to traditional terminology:
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| :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''.<ref>Harvey p 23</ref>
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| ==Relation to tensor products==
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| By the [[universal property]] of the [[tensor product]], bilinear forms on ''V'' are in 1-to-1 correspondence with linear maps ''V'' ⊗ ''V'' → ''F''. If ''B'' is a bilinear form on ''V'' the corresponding linear map is given by
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| :'''v''' ⊗ '''w''' ↦ ''B''('''v''', '''w''')
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| The set of all linear maps ''V'' ⊗ ''V'' → ''F'' is the [[dual space]] of ''V'' ⊗ ''V'', so bilinear forms may be thought of as elements of
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| :(''V'' ⊗ ''V'')* ≅ ''V*'' ⊗ ''V*''
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| Likewise, symmetric bilinear forms may be thought of as elements of Sym<sup>2</sup>(''V*'') (the second [[symmetric power]] of ''V*''), and alternating bilinear forms as elements of Λ<sup>2</sup>''V*'' (the second [[exterior power]] of ''V*'').
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| ==On normed vector spaces==
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| <blockquote>'''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''
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| :<math>B(\mathbf{u}, \mathbf{v}) \le C \|\mathbf{u}\| \|\mathbf{v}\|.</math></blockquote> | |
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| <blockquote>'''Definition:''' A bilinear form on a normed vector space (''V'', ‖·‖ ) is '''elliptic''', or [[Coercive_function#Coercive_operators_and_forms|coercive]], if there is a constant ''c'' > 0 such that for all '''u''' ∈ ''V''
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| :<math>B(\mathbf{u}, \mathbf{u}) \ge c \|\mathbf{u}\|^2.</math></blockquote>
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| ==See also==
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| *[[Bilinear operator]]
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| *[[Multilinear form]]
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| *[[Quadratic form]]
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| *[[Inner product space]]
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| *[[positive semi-definite|Positive semi definite]]
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| *[[Sesquilinear form]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * {{cite book | last=Jacobson | first=Nathan | title=Basic Algebra | volume=I | edition=2nd | year=2009 | isbn=978-0-486-47189-1 }}
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| * {{cite book | last1=Adkins | first1=William A. | last2=Weintraub | first2=Steven H. | title=Algebra: An Approach via Module Theory | series=[[Graduate Texts in Mathematics]] | volume=136 | publisher=[[Springer-Verlag]] | year=1992 | isbn=3-540-97839-9 | zbl=0768.00003 }}
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| * {{cite book | first=Bruce | last=Cooperstein | 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 }}
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| * {{cite book | last=Grove | first=Larry C. | title=Groups and characters | year=1997 | publisher=Wiley-Interscience | isbn=978-0-471-16340-4}}
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| * {{cite book | last1=Halmos | first1=Paul R. | author1-link=Paul R. Halmos | title=Finite-dimensional vector spaces | series=Undergraduate Texts in Mathematics | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-90093-3 | year=1974 | zbl=0288.15002 }}
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| * Harvey, F. Reese (1990) ''Spinors and calibrations'', Ch 2:The Eight Types of Inner Product Spaces, pp 19–40, [[Academic Press]], ISBN 0-12-329650-1 .
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| * M. Hazewinkel ed. (1988) [[Encyclopedia of Mathematics]], v.1, p. 390, [[Kluwer Academic Publishers]]
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| * {{cite book | first1=J. | last1=Milnor | author1-link=John Milnor| first2=D. | last2=Husemoller | title=Symmetric Bilinear Forms | series=[[Ergebnisse der Mathematik und ihrer Grenzgebiete]] | volume=73 | publisher=[[Springer-Verlag]] | year=1973 | isbn=3-540-06009-X | zbl=0292.10016 }}
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| * {{cite book | last=Shilov | first=Georgi E. | title=Linear Algebra | editor-last=Silverman | editor-first=Richard A. | year=1977 | publisher=Dover | isbn=0-486-63518-X}}
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| * {{cite book | last = Shafarevich | first = I. R. | authorlink = Igor Shafarevich | coauthors = A. O. Remizov | title = Linear Algebra and Geometry | publisher = [[Springer Science+Business Media|Springer]] | year = 2012 | url = http://www.springer.com/mathematics/algebra/book/978-3-642-30993-9 | isbn = 978-3-642-30993-9}}
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| ==External links==
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| * {{springer|title=Bilinear form|id=p/b016250}}
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| *{{planetmath reference|id=1612|title=Bilinear form}}
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| {{Functional Analysis}}
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| {{PlanetMath attribution|id=7553|title=Unimodular}}
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| [[Category:Bilinear forms]]
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