Editing Quadratic form (section)

== Introduction ==
Quadratic forms are homogeneous quadratic polynomials in {{math|''n''}} variables. In the cases of one, two, and three variables they are called '''unary''', '''[[binary quadratic form|binary]]''', and '''ternary''' and have the following explicit form:
<math display="block">\begin{align}
      q(x) &= ax^2&&\textrm{(unary)} \\
    q(x,y) &= ax^2 + bxy + cy^2&&\textrm{(binary)} \\
  q(x,y,z) &= ax^2 + bxy + cy^2 + dyz + ez^2 + fxz&&\textrm{(ternary)}
\end{align}</math>

where {{math|''a''}}, ..., {{math|''f''}} are the '''coefficients'''.<ref>A tradition going back to [[Gauss]] dictates the use of manifestly even coefficients for the products of distinct variables, that is, {{math|2''b''}} in place of {{math|''b''}} in binary forms and {{math|2''b''}}, {{math|2''d''}}, {{math|2''f''}} in place of {{math|''b''}}, {{math|''d''}}, {{math|''f''}} in ternary forms. Both conventions occur in the literature.</ref>

The theory of quadratic forms and methods used in their study depend in a large measure on the nature of the coefficients, which may be [[real number|real]] or [[complex number]]s, [[rational number]]s, or [[integer]]s. In [[linear algebra]], [[analytic geometry]], and in the majority of applications of quadratic forms, the coefficients are real or complex numbers. In the algebraic theory of quadratic forms, the coefficients are elements of a certain [[field (algebra)|field]]. In the arithmetic theory of quadratic forms, the coefficients belong to a fixed [[commutative ring]], frequently the integers {{math|'''Z'''}} or the [[p-adic integer|{{math|''p''}}-adic integers]] {{math|'''Z'''<sub>''p''</sub>}}.<ref>[[Localization of a ring#Terminology|away from 2]], that is, if 2 is invertible in the ring, quadratic forms are equivalent to [[symmetric bilinear form]]s (by the [[polarization identities]]), but at 2 they are different concepts; this distinction is particularly important for quadratic forms over the integers.</ref> [[Binary quadratic form]]s have been extensively studied in [[number theory]], in particular, in the theory of [[quadratic field]]s, [[continued fraction]]s, and [[modular forms]]. The theory of integral quadratic forms in {{math|''n''}} variables has important applications to [[algebraic topology]].

Using [[homogeneous coordinates]], a non-zero quadratic form in {{math|''n''}} variables defines an {{math|(''n'' − 2)}}-dimensional [[Quadric (projective geometry)|quadric]] in the {{math|(''n'' − 1)}}-dimensional [[projective space]]. This is a basic construction in [[projective geometry]]. In this way one may visualize 3-dimensional real quadratic forms as [[conic sections]].
An example is given by the three-dimensional [[Euclidean space]] and the [[Square (algebra)|square]] of the [[Euclidean norm]] expressing the [[distance]] between a point with coordinates {{math|(''x'', ''y'', ''z'')}} and the origin:
<math display="block">q(x,y,z) = d((x,y,z), (0,0,0))^2 = \left\|(x,y,z)\right\|^2 = x^2 + y^2 + z^2.</math>

A closely related notion with geometric overtones is a '''quadratic space''', which is a pair {{math|(''V'', ''q'')}}, with {{math|''V''}} a [[vector space]] over a field {{math|''K''}}, and {{math|''q'' : ''V'' → ''K''}} a quadratic form on ''V''. See ''{{section link|#Definitions}}'' below for the definition of a quadratic form on a vector space.