Quadratic form: Difference between revisions

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Quadratic forms occupy a central place in various branches of mathematics, including [[number theory]], [[linear algebra]], [[group theory]] ([[orthogonal group]]), [[differential geometry]] ([[Riemannian metric]], [[second fundamental form]]), [[differential topology]] ([[intersection form (4-manifold)|intersection forms]] of [[four-manifold]]s), and [[Lie theory]] (the [[Killing form]]).

Quadratic forms are not to be confused with a [[quadratic equation]], which has only one variable and includes terms of degree two or less. A quadratic form is one case of the more general concept of [[Homogeneoushomogeneous polynomial|homogeneous polynomials]]s.

== Introduction ==

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\end{align}</math>

where ''a'', …..., ''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, 2''b'' in place of ''b'' in binary forms and 2''b'', 2''d'', 2''f'' in place of ''b'', ''d'', ''f'' in ternary forms. Both conventions occur in the literature.</ref>

The notation <math>\langle a_1, \ldots, a_n\rangle</math> is often used{{cncitation needed|date=December 2020}} for the quadratic form

: <math>q(x) = a_1 x_1^2 + a_2 x_2^2 + \cdots + a_n x_n^2.</math>

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: <math>q(x,y,z) = d((x,y,z), (0,0,0))^2 = \|(x,y,z)\|^2 = x^2 + y^2 + z^2.</math>

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

== History ==

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Any {{math|''n''×''n''}} matrix {{math|''A''}} determines a quadratic form {{math|''q''_''A''}} in {{math|''n''}} variables by

: <math>q_A(x_1,\ldots,x_n) = \sum_{i=1}^{n}\sum_{j=1}^{n}a_{ij}{x_i}{x_j} = \mathbf x^\mathrm{T} A \mathbf x, </math>

where <math>A = (a_{ij})</math>.

===Example===

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===General case===

Given a quadratic form <math>q_A,</math> defined by the matrix <math>A=\left(a_{ij}\right),</math>

the matrix <math display = block>B = \left(\frac{a_{ij}+a_{ji}} 2\right)</math> is [[symmetric matrix|symmetric]], defines the same quadratic form as {{mvar|A}}, and is the unique symmetric matrix that defines <math>q_A.</math>

So, over the real numbers (and, more generally, over a [[field (mathematics)|field]] of [[characteristic (algebra)|characteristic]] different from two), there is a [[one-to-one correspondence]] between quadratic forms and [[symmetric matrices]] that determine them.

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== Real quadratic forms ==

{{see also|Sylvester's law of inertia|Definite quadratic form|Isotropic quadratic form}}

A fundamental problem is the classification of real quadratic forms under a [[linear transformation|linear change of variables]].

[[Carl Gustav Jacobi|Jacobi]] proved that, for every real quadratic form, there is an [[orthogonal diagonalization]]; that is, an [[orthogonal transformation|orthogonal change of variables]] that puts the quadratic form in a "[[diagonal form]]"

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where the associated symmetric matrix is [[diagonal matrix|diagonal]]. Moreover, the coefficients {{math|''λ''₁, ''λ''₂, ..., ''λ''_''n''}} are determined uniquely [[up to]] a [[permutation]].<ref>[[Maxime Bôcher]] (with E.P.R. DuVal)(1907) ''Introduction to Higher Algebra'', [https://babel.hathitrust.org/cgi/pt?id=uc1.b4248862;view=1up;seq=147 § 45 Reduction of a quadratic form to a sum of squares] via [[HathiTrust]]</ref>

If the change of variables is given by an [[invertible matrix]] that is not necessarily orthogonal, one can suppose that all coefficients {{math|''λ''_''i''}} are 0, 1, or −1. [[Sylvester's law of inertia]] states that the numbers of each 1 and −1 are [[invariant (mathematics)|invariants]] of the quadratic form, in the sense that any other diagonalization will contain the same number of each. The '''signature''' of the quadratic form is the triple {{nowrap|(''n''₀, ''n''₊, ''n''₋)}}, where ''n''₀ is the number of 0s and ''n''_± is the number of ±1s. Sylvester's law of inertia shows that this is a well-defined quantity attached to the quadratic form.

The case when all ''λ''_''i'' have the same sign is especially important: in this case the quadratic form is called '''[[positive definite form|positive definite]]''' (all 1) or '''negative definite''' (all −1). If none of the terms are 0, then the form is called '''{{visible anchor|nondegenerate}}'''; this includes positive definite, negative definite, and [[isotropic quadratic form]] (a mix of 1 and −1); equivalently, a nondegenerate quadratic form is one whose associated symmetric form is a [[nondegenerate form|nondegenerate ''bilinear'' form]]. A real vector space with an indefinite nondegenerate quadratic form of index {{nowrap|(''p'', ''q'')}} (denoting ''p'' 1s and ''q'' −1s) is often denoted as '''R'''^''p'',''q'' particularly in the physical theory of [[spacetime]].

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: <math> q(x)=x^\mathrm{T}Ax. </math>

A vector <math>v = (x_1,\ldots,x_n)</math> is a [[null vector]] if ''q''(''v'') = 0.

Two ''n''-ary quadratic forms ''φ'' and ''ψ'' over ''K'' are '''equivalent''' if there exists a nonsingular linear transformation {{nowrap|''C'' ∈ [[General linear group|GL]](''n'', ''K'')}} such that

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==External links==

{{CommonscatCommons category|Quadratic forms}}

*{{eom|id=q/q076080|author=A.V.Malyshev|title=Quadratic form}}

*{{eom|id=b/b016370|author=A.V.Malyshev|title=Binary quadratic form}}