Editing Quadratic form (section)

== Definitions ==
A '''quadratic form''' over a field {{math|''K''}} is a map {{math|''q'' : ''V'' → ''K''}} from a finite-dimensional {{math|''K''}}-vector space to {{math|''K''}} such that {{math|1=''q''(''av'') = ''a''<sup>2</sup>''q''(''v'')}} for all {{math|''a'' ∈ ''K''}}, {{math|''v'' ∈ ''V''}} and the function {{math|''q''(''u'' + ''v'') − ''q''(''u'') − ''q''(''v'')}} is bilinear.

More concretely, an {{math|''n''}}-ary '''quadratic form''' over a field {{math|''K''}} is a [[homogeneous polynomial]] of degree 2 in {{math|''n''}} variables with coefficients in {{math|''K''}}:
<math display="block">q(x_1,\ldots,x_n) = \sum_{i=1}^{n}\sum_{j=1}^{n}a_{ij}{x_i}{x_j}, \quad a_{ij}\in K. </math>

This formula may be rewritten using matrices: let {{math|''x''}} be the [[column vector]] with components {{math|''x''<sub>1</sub>}}, ..., {{math|''x''<sub>''n''</sub>}} and {{math|1=''A'' = (''a''<sub>''ij''</sub>)}} be the {{math|''n'' × ''n''}} matrix over {{math|''K''}} whose entries are the coefficients of {{math|''q''}}. Then
<math display="block"> q(x) = x^\mathsf{T} A x. </math>

A vector {{math|1=''v'' = (''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>)}} is a [[null vector]] if {{math|1=''q''(''v'') = 0}}.

Two {{math|''n''}}-ary quadratic forms {{math|''φ''}} and {{math|''ψ''}} over {{math|''K''}} are '''equivalent''' if there exists a nonsingular linear transformation {{math|''C'' ∈ [[General linear group|GL]](''n'', ''K'')}} such that
<math display="block"> \psi(x) = \varphi(Cx). </math>

Let the [[characteristic (field)|characteristic]] of {{math|''K''}} be different from 2.{{refn|The theory of quadratic forms over a field of characteristic 2 has important differences and many definitions and theorems must be modified.}} The coefficient matrix {{math|''A''}} of {{math|''q''}} may be replaced by the [[symmetric matrix]] {{math|(''A'' + ''A''<sup>T</sup>)/2}} with the same quadratic form, so it may be assumed from the outset that {{math|''A''}} is symmetric. Moreover, a symmetric matrix {{math|''A''}} is uniquely determined by the corresponding quadratic form. Under an equivalence {{math|''C''}}, the symmetric matrix {{math|''A''}} of {{math|''φ''}} and the symmetric matrix {{math|''B''}} of {{math|''ψ''}} are related as follows:
<math display="block"> B = C^\mathsf{T} A C. </math>

The '''associated bilinear form''' of a quadratic form {{math|''q''}} is defined by
<math display="block"> b_q(x,y)=\tfrac{1}{2}(q(x+y)-q(x)-q(y)) = x^\mathsf{T}Ay = y^\mathsf{T}Ax. </math>

Thus, {{math|''b''<sub>''q''</sub>}} is a [[symmetric bilinear form]] over {{math|''K''}} with matrix {{math|''A''}}. Conversely, any symmetric bilinear form {{math|''b''}} defines a quadratic form
<math display="block"> q(x)=b(x,x),</math>
and these two processes are the inverses of each other. As a consequence, over a field of characteristic not equal to 2, the theories of symmetric bilinear forms and of quadratic forms in {{math|''n''}} variables are essentially the same.

=== Quadratic space ===
{{see also|Bilinear form#Associated quadratic form}}
Given an {{math|''n''}}-dimensional [[vector space]] {{math|''V''}} over a field {{math|''K''}}, a ''quadratic form'' on {{math|''V''}} is a [[function (mathematics)|function]] {{math|''Q'' : ''V'' → ''K''}} that has the following property: for some basis, the function {{math|''q''}} that maps the coordinates of {{math|''v'' ∈ ''V''}} to {{math|''Q''(''v'')}} is a quadratic form. In particular, if {{math|1=''V'' = ''K''<sup>''n''</sup>}} with its [[standard basis]], one has 
<math display="block"> q(v_1,\ldots, v_n)= Q([v_1,\ldots,v_n])\quad \text{for} \quad [v_1,\ldots,v_n] \in K^n. </math>

The [[change of basis]] formulas show that the property of being a quadratic form does not depend on the choice of a specific basis in {{math|''V''}}, although the quadratic form {{math|''q''}} depends on the choice of the basis.

A finite-dimensional vector space with a quadratic form is called a '''quadratic space'''.

The map {{math|''Q''}} is a [[homogeneous function]] of degree 2, which means that it has the property that, for all {{math|''a''}} in {{math|''K''}} and {{math|''v''}} in {{math|''V''}}:
<math display="block"> Q(av) = a^2 Q(v). </math>

When the characteristic of {{math|''K''}} is not 2, the bilinear map {{math|''B'' : ''V'' × ''V'' → ''K''}} over {{math|''K''}} is defined:
<math display="block"> B(v,w)= \tfrac{1}{2}(Q(v+w)-Q(v)-Q(w)).</math>
This bilinear form {{math|''B''}} is symmetric. That is, {{math|1=''B''(''x'', ''y'') = ''B''(''y'', ''x'')}} for all {{math|''x''}}, {{math|''y''}} in {{math|''V''}}, and it determines {{math|''Q''}}: {{math|1=''Q''(''x'') = ''B''(''x'', ''x'')}} for all {{math|''x''}} in {{math|''V''}}.

When the characteristic of {{math|''K''}} is 2, so that 2 is not a [[Unit (ring theory)|unit]], it is still possible to use a quadratic form to define a symmetric bilinear form {{math|1=''B''′(''x'', ''y'') = ''Q''(''x'' + ''y'') − ''Q''(''x'') − ''Q''(''y'')}}. However, {{math|''Q''(''x'')}} can no longer be recovered from this {{math|''B''′}} in the same way, since {{math|1=''B''′(''x'', ''x'') = 0}} for all {{math|''x''}} (and is thus alternating).<ref>This alternating form associated with a quadratic form in characteristic 2 is of interest related to the [[Arf invariant]] – {{citation|author=Irving Kaplansky | year=1974 | title=Linear Algebra and Geometry|page=27}}.</ref> Alternatively, there always exists a bilinear form {{math|''B''″}} (not in general either unique or symmetric) such that {{math|1=''B''″(''x'', ''x'') = ''Q''(''x'')}}.

The pair {{math|(''V'', ''Q'')}} consisting of a finite-dimensional vector space {{math|''V''}} over {{math|''K''}} and a quadratic map {{math|''Q''}} from {{math|''V''}} to {{math|''K''}} is called a '''quadratic space''', and {{math|''B''}} as defined here is the associated symmetric bilinear form of {{math|''Q''}}. The notion of a quadratic space is a coordinate-free version of the notion of quadratic form. Sometimes, {{math|''Q''}} is also called a quadratic form.

{{anchor|isometry}}Two {{math|''n''}}-dimensional quadratic spaces {{math|(''V'', ''Q'')}} and {{math|(''V''′, ''Q''′)}} are '''isometric''' if there exists an invertible linear transformation {{math|''T'' : ''V'' → ''V''′}} ('''isometry''') such that
<math display="block"> Q(v) = Q'(Tv) \text{ for all } v\in V.</math>

The isometry classes of {{math|''n''}}-dimensional quadratic spaces over {{math|''K''}} correspond to the equivalence classes of {{math|''n''}}-ary quadratic forms over {{math|''K''}}.

=== Generalization ===
Let {{math|''R''}} be a [[commutative ring]], {{math|''M''}} be an {{math|''R''}}-[[Module (mathematics)|module]], and {{math|''b'' : ''M'' × ''M'' → ''R''}} be an {{math|''R''}}-bilinear form.{{refn|The bilinear form to which a quadratic form is associated is not restricted to being symmetric, which is of significance when 2 is not a unit in {{math|''R''}}.}} A mapping {{math|''q'' : ''M'' → ''R'' : ''v'' ↦ ''b''(''v'', ''v'')}} is the ''associated quadratic form'' of {{math|''b''}}, and {{math|''B'' : ''M'' × ''M'' → ''R'' : (''u'', ''v'') ↦ ''q''(''u'' + ''v'') − ''q''(''u'') − ''q''(''v'')}} is the ''polar form'' of {{math|''q''}}.

A quadratic form {{math|''q'' : ''M'' → ''R''}} may be characterized in the following equivalent ways:
* There exists an {{math|''R''}}-bilinear form {{math|''b'' : ''M'' × ''M'' → ''R''}} such that {{math|''q''(''v'')}} is the associated quadratic form.
* {{math|1=''q''(''av'') = ''a''<sup>2</sup>''q''(''v'')}} for all {{math|''a'' ∈ ''R''}} and {{math|''v'' ∈ ''M''}}, and the polar form of {{math|''q''}} is {{math|''R''}}-bilinear.

=== Related concepts ===
{{see also|Isotropic quadratic form}}
Two elements {{math|''v''}} and {{math|''w''}} of {{math|''V''}} are called '''[[orthogonal]]''' if {{math|1=''B''(''v'', ''w'') = 0}}. The '''kernel''' of a bilinear form {{math|''B''}} consists of the elements that are orthogonal to every element of {{math|''V''}}. {{math|''Q''}} is '''non-singular''' if the kernel of its associated bilinear form is {{math|{{mset|0}}}}. If there exists a non-zero {{math|''v''}} in {{math|''V''}} such that {{math|1=''Q''(''v'') = 0}}, the quadratic form {{math|''Q''}} is '''[[Isotropic quadratic form|isotropic]]''', otherwise it is '''[[definite quadratic form|definite]]'''. This terminology also applies to vectors and subspaces of a quadratic space. If the restriction of {{math|''Q''}} to a subspace {{math|''U''}} of {{math|''V''}} is identically zero, then {{math|''U''}} is '''totally singular'''.

The orthogonal group of a non-singular quadratic form {{math|''Q''}} is the group of the linear automorphisms of {{math|''V''}} that preserve {{math|''Q''}}:  that is, the group of isometries of {{math|(''V'', ''Q'')}} into itself.

If a quadratic space {{math|(''A'', ''Q'')}} has a product so that {{math|''A''}} is an [[algebra over a field]], and satisfies
<math display="block">\forall x, y \isin A \quad Q(x y) = Q(x) Q(y) ,</math> then it is a [[composition algebra]].