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onde ''a'',…,''f'' são '''coeficientes'''.<ref>Uma tradição remontando a [[Carl Friedrich Gauss|Gauss]] estabelece o uso de coeficientes pares associados a produtos de variáveis distintas, ou seja, 2''b'' ao invés de ''b'' em formas binárias, e 2''d'', 2''e'' e 2''f'' ao invés de ''d'', ''e'' e ''f'' em formas ternárias. As duas convenções são utilizadas na literastura</ref> Notar que [[Função quadrática|funções quadráticas]], tais como ''ax''<sup>2</sup>+''bx''+''c'' no caso de uma única variável, não são formas quadráticas, pois não são [[Polinômio homogêneo|homogêneas]] (a não ser que ''b'' e ''c'' sejam ambos 0).
{{em tradução|:en:Quadratic form}}
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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|complex]] numbers, [[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 '''Z''' or the [[p-adic integer|''p''-adic integers]] '''Z'''<sub>''p''</sub>.<ref>[[Localization of a ring#Terminology|away from 2]], i. e. 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 forms 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 ''n'' variables has important applications to [[algebraic topology]].
Using [[homogeneous coordinates]], a non-zero quadratic form in ''n'' variables defines an (''n''−2)-dimensional [[Quadric (projective geometry)|quadric]] in the (''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]].
A closely related notion with geometric overtones is a '''quadratic space''', which is a pair (''V'',''q''), with ''V'' a [[vector space]] over a field ''k'', and ''q'':''V'' → ''k'' a quadratic form on ''V''. An example is given by the three-dimensional [[Euclidean space]]
and the square of the [[Euclidean norm]] expressing the [[distance]] between a point with coordinates (''x'',''y'',''z'') and the origin:
: <math> q(x,y,z)=d((x,y,z),(0,0,0))^2=\|(x,y,z)\|^2=x^2+y^2+z^2. </math>
==History==
The study of particular quadratic forms, in particular the question of whether a given integer can be the value of a quadratic form over the integers, dates back many centuries. One such case is [[Fermat's theorem on sums of two squares]], which determines when an integer may be expressed in the form <math>x^2+y^2</math>, where <math>x,\ y</math> are integers. This problem is related to the problem of finding [[Pythagorean triple]]s, which appeared in the second millennium B.C.<ref>http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Babylonian_Pythagoras.html</ref>
In 628, the Indian mathematician [[Brahmagupta]] wrote ''[[Brahmasphutasiddhanta]]'' which includes, among many other things, a study of equations of the form <math>x^2-ny^2 = c</math>. In particular he considered what is now called [[Pell's equation]], <math>x^2-ny^2 = 1</math>, and found a method for its solution.<ref>http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Brahmagupta.html</ref> In Europe this problem was studied by [[William Brouncker, 2nd Viscount Brouncker|Brouncker]], [[Leonhard Euler|Euler]] and [[Joseph Louis Lagrange|Lagrange]].
In 1801 [[Carl Friedrich Gauss|Gauss]] published ''[[Disquisitiones Arithmeticae]],'' a major portion of which was devoted to a complete theory of [[binary quadratic form]]s over the [[integer]]s. Since then, the concept has been generalized, and the connections with [[quadratic number field]]s, the [[modular group]], and other areas of mathematics have been further elucidated.
== Real quadratic forms ==
{{see also|Sylvester's law of inertia|Definite form}}
Any ''n''×''n'' real [[symmetric matrix]] ''A'' determines a quadratic form ''q''<sub>''A''</sub> in ''n'' variables by the formula
: <math>q_A(x_1,\ldots,x_n) = \sum_{i,j=1}^{n}a_{ij}{x_i}{x_j}. </math>
Conversely, given a quadratic form in ''n'' variables, its coefficients can be arranged into an ''n''×''n'' symmetric matrix. One of the most important questions in the theory of quadratic forms is how much can one simplify a quadratic form ''q'' by a homogeneous linear change of variables. A fundamental theorem due to [[Carl Gustav Jacobi|Jacobi]] asserts that ''q'' can be brought to a '''diagonal form'''
: <math> \lambda_1 \tilde x_1^2 + \lambda_2 \tilde x_2^2 + \cdots + \lambda_n \tilde x_n^2, </math>
so that the corresponding symmetric matrix is [[diagonal matrix|diagonal]], and this is even possible to accomplish with a change of variables given by an [[orthogonal matrix]] – in this case the coefficients ''λ''<sub>1</sub>, ''λ''<sub>2</sub>, …, ''λ''<sub>''n''</sub> are in fact determined uniquely up to a permutation. If the change of variables is given by an invertible matrix, not necessarily orthogonal, then the coefficients ''λ''<sub>''i''</sub> can be made to be 0,1, and −1. [[Sylvester's law of inertia]] states that the numbers of 1 and −1 are [[invariant (mathematics)|invariants]] of the quadratic form, in the sense that any other diagonalization will contain them in the same quantities. The '''signature''' of the quadratic form is the triple (''n''<sub>0</sub>, ''n''<sub>+</sub>, ''n''<sub>−</sub>) where ''n''<sub>0</sub> is the number 0s and ''n''<sub>±</sub> 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''<sub>''i''</sub> 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 indefinite (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 <math>(p,q)</math> (''p'' 1s, ''q'' −1s) is often denoted as <math>\mathbf{R}^{p,q},</math> particularly in the physical theory of [[space-time]].
Below we reformulate these results in a different way.
Let ''q'' be a quadratic form defined on an ''n''-dimensional [[real number|real]] vector space. Let ''A'' be the matrix of the quadratic form ''q'' in a given basis. This means that ''A'' is a symmetric ''n''×''n'' matrix such that
: <math>q(v)=x^\mathrm{T} Ax,</math>
where ''x'' is the column vector of coordinates of ''v'' in the chosen basis. Under a change of basis, the column ''x'' is multiplied on the left by an ''n''×''n'' [[invertible matrix]] ''S'', and the symmetric square matrix ''A'' is transformed into another symmetric square matrix ''B'' of the same size according to the formula
: <math> A\to B=SAS^\mathrm{T}.</math>
Any symmetric matrix ''A'' can be transformed into a diagonal matrix
: <math>
B=\begin{pmatrix}
\lambda_1 & 0 & \cdots & 0\\
0 & \lambda_2 & \cdots & 0\\
\vdots & \vdots & \ddots & 0\\
0 & 0 & \cdots & \lambda_n
\end{pmatrix}
</math>
by a suitable choice of an orthogonal matrix ''S'', and the diagonal entries of ''B'' are uniquely determined — this is Jacobi's theorem. If ''S'' is allowed to be any invertible matrix then ''B'' can be made to have only 0,1, and −1 on the diagonal, and the number of the entries of each type (''n''<sub>0</sub> for 0, ''n''<sub>+</sub> for 1, and ''n''<sub>−</sub> for −1) depends only on ''A''. This is one of the formulations of Sylvester's law of inertia and the numbers ''n''<sub>+</sub> and ''n''<sub>−</sub> are called the '''positive''' and '''negative''' '''indices of inertia'''. Although their definition involved a choice of basis and consideration of the corresponding real symmetric matrix ''A'', Sylvester's law of inertia means that they are invariants of the quadratic form ''q''.
The quadratic form ''q'' is positive definite (resp., negative definite) if ''q''(''v'')>0 (resp., ''q''(''v'')<0) for every nonzero vector ''v''.<ref>If a non-strict inequality (with ≥ or ≤) holds then the quadratic form ''q'' is called semidefinite.</ref> When ''q''(''v'') assumes both positive and negative values, ''q'' is an '''indefinite''' quadratic form. The theorems of Jacobi and Sylvester show that any positive definite quadratic form in ''n'' variables can be brought to the sum of ''n'' squares by a suitable invertible linear transformation: geometrically, there is only ''one'' positive definite real quadratic form of every dimension. Its [[isometry group]] is a [[compact space|''compact'']] [[orthogonal group]] O(''n''). This stands in contrast with the case of indefinite forms, when the corresponding group, the [[indefinite orthogonal group]] O(''p'',''q''), is non-compact. Further, the isometry groups of ''Q'' and −''Q'' are the same (<math>O(p,q) \approx O(q,p)</math>), but the associated [[Clifford algebra]]s (and hence [[Pin group]]s) are different.
== Definitions ==
An ''n''-ary '''quadratic form''' over a field '''K''' is a [[homogeneous polynomial]] of degree 2 in ''n'' variables with coefficients in '''K''':
: <math>q(x_1,\ldots,x_n) = \sum_{i,j=1}^{n}a_{ij}{x_i}{x_j}, \quad a_{ij}\in K. </math>
This formula may be rewritten using matrices: let ''x'' be the [[column vector]] with components ''x''<sub>1</sub>, …, ''x''<sub>''n''</sub> and ''A'' = (''a''<sub>''ij''</sub>) be the ''n''×''n'' matrix over '''K''' whose entries are the coefficients of ''q''. Then
: <math> q(x)=x^\mathrm{T}Ax. </math>
Two ''n''-ary quadratic forms ''φ'' and ''ψ'' over '''K''' are '''equivalent''' if there exists a nonsingular linear transformation ''T'' ∈''GL''<sub>''n''</sub>('''K''') such that
: <math> \psi(x)=\varphi(Tx). </math>
''Let us assume that the characteristic of '''K''' is different from 2.''
(The theory of quadratic forms over a field of characteristic 2 has important differences and many definitions and theorems have to be modified.) The coefficient matrix ''A'' of ''q'' may be replaced by the [[symmetric matrix]] 1/2(''A'' + ''A''<sup>T</sup>) with the same quadratic form, so it may be assumed from the outset that ''A'' is symmetric. Moreover, a symmetric matrix ''A'' is uniquely determined by the corresponding quadratic form. Under an equivalence ''T'', the symmetric matrix ''A'' of ''φ'' and the symmetric matrix ''B'' of ''ψ'' are related as follows:
: <math> B=T^\mathrm{T}AT. </math>
The '''associated bilinear form''' of a quadratic form ''q'' is defined by
: <math> b_q(x,y)=\frac{1}{2}(q(x+y)-q(x)-q(y)) = x^\mathrm{T}Ay = y^\mathrm{T}Ax. </math>
Thus, ''b''<sub>''q''</sub> is a [[symmetric bilinear form]] over '''K''' with matrix ''A''. Conversely, any symmetric bilinear form ''b'' defines a quadratic form
: <math> q(x)=b(x,x) </math>
and these two processes are the inverses of one another. As a consequence, over a field of characteristic not equal to 2, the theories of symmetric bilinear forms and of quadratic forms in ''n'' variables are essentially the same.
=== Quadratic spaces ===
A quadratic form ''q'' in ''n'' variables over '''K''' induces a map from the ''n''-dimensional coordinate space '''K'''<sup>''n''</sup> into '''K''':
: <math> Q(v)=q(v), \quad v=[v_1,\ldots,v_n]^\mathrm{T}\in K^n. </math>
The map ''Q'' is a '''quadratic map''', which means that it has the properties:
* <math> Q(av) = a^2 Q(v)\quad \text{ for all } a\in K, v\in V. </math>
* The map ''B''<sub>''Q''</sub>: ''V''×''V'' → '''K''' defined below is bilinear over '''K''':
: <math> B_Q(v,w)=\frac{1}{2}(Q(v+w)-Q(v)-Q(w)). </math>
The pair (''V'',''Q'') consisting of a finite-dimensional vector space ''V'' over '''K''' and a quadratic map from ''V'' to '''K''' is called a '''quadratic space''' and ''B''<sub>''Q''</sub> is the associated bilinear form of ''Q''. The notion of a quadratic space is a coordinate-free version of the notion of quadratic form. Sometimes, ''Q'' is also called a quadratic form.
Two ''n''-dimensional quadratic spaces (''V'',''Q'') and (''V'' ′, ''Q'' ′) are '''isometric''' if there exists an invertible linear transformation ''T'': ''V'' →''V'' ′ ('''isometry''') such that
: <math> Q(v) = Q'(Tv) \text{ for all } v\in V.</math>
The isometry classes of ''n''-dimensional quadratic spaces over '''K''' correspond to the equivalence classes of ''n''-ary quadratic forms over '''K'''.
=== Further definitions ===
{{see also|Isotropic quadratic form}}
Two elements ''v'' and ''w'' of ''V'' are called '''[[orthogonal]]''' if ''B''(''v'', ''w'')=0. The '''kernel''' of a bilinear form ''B'' consists of the elements that are orthogonal to each elements of ''V''. ''Q'' is '''non-singular''' if the kernel of its associated bilinear form is 0. If there exists a non-zero ''v'' in ''V'' such that ''Q''(''v'') = 0, the quadratic form ''Q'' is [[Isotropic quadratic form|'''isotropic''']], otherwise it is '''anisotropic'''. This terminology also applies to vectors and subspaces of a quadratic space. If the restriction of ''Q'' to a subspace ''U'' of ''V'' is identically zero, ''U'' is '''totally singular'''.
The [[orthogonal group]] of a non-singular quadratic form ''Q'' is the group of the linear automorphisms of ''V'' that preserve ''Q'', i.e. the group of isometries of (''V'', ''Q'') into itself.
== Equivalence of forms ==
Every quadratic form ''q'' in ''n'' variables over a field of characteristic not equal to 2 is equivalent to a '''diagonal form'''
: <math>q(x)=a_1 x_1^2 + a_2 x_2^2+ \ldots +a_n x_n^2.</math>
Such a diagonal form is often denoted by <math>\langle a_1,\dots,a_n\rangle</math>.
Classification of all quadratic forms up to equivalence can thus be reduced to the case of diagonal forms.
==Geometric Meaning==
If we let the equation be <math>x^\mathrm{T}Ax+b^\mathrm{T}x=1</math> with symmetric matrix ''A'', then the geometric meaning is as follows.
If all eigenvalues of ''A'' are non-zero, then it is an [[ellipsoid]] or a [[hyperboloid]]. If all the eigenvalues are positive, then it is an ellipsoid; if all the eigenvalues are negative, it is an image ellipsoid; if some eigenvalues are positive and some are negative, then it is a hyperboloid.
If there exist one or more eigenvalues <math>\lambda_i=0</math>, then if the corresponding <math>b_i\ne0</math>, it is a [[paraboloid]] (either elliptic or hyperbolic); if the corresponding <math>b_i=0</math>, the dimension ''i'' degenerates and does not get into play, and the geometric meaning will be determined by other eigenvalues and other components of ''b''. When it is a paraboloid, whether it is elliptic or hyperbolic is determined by whether all other non-zero eigenvalues are of the same sign: if they are, then it is elliptic; otherwise, it is hyperbolic.
== Integral quadratic forms ==
Quadratic forms over the ring of integers are called '''integral quadratic forms''', whereas the corresponding modules are '''quadratic lattices''' (sometimes, simply [[lattice (group)|lattice]]s). They play an important role in [[number theory]] and [[topology]].
An integral quadratic form has integer coefficients, such as <math>x^2 + xy + y^2</math>; equivalently, given a lattice <math>\Lambda</math> in a vector space <math>V</math> (over a field with characteristic 0, such as <math>\mathbf{Q}</math> or <math>\mathbf{R}</math>), a quadratic form <math>Q</math> is integral ''with respect to'' <math>\Lambda</math> if and only if it is integer-valued on <math>\Lambda</math>, meaning <math>Q(x,y) \in \mathbf{Z}</math> if <math>x,y \in \Lambda</math>.
This is the current use of the term; in the past it was sometimes used differently, as detailed below.
=== Historical use ===
Historically there was some confusion and controversy over whether the notion of '''integral quadratic form''' should mean:
;''twos in'': the quadratic form associated to a symmetric matrix with integer coefficients
;''twos out'': a polynomial with integer coefficients (so the associated symmetric matrix may have half-integer coefficients off the diagonal)
This debate was due to the confusion of quadratic forms (represented by polynomials) and symmetric bilinear forms (represented by matrices), and "twos out" is now the accepted convention; "twos in" is instead the theory of integral symmetric bilinear forms (integral symmetric matrices).
In "twos in", binary quadratic forms are of the form <math>ax^2+2bxy+cy^2</math>, represented by the symmetric matrix <math>\begin{pmatrix}a & b\\ b&c\end{pmatrix}</math>; this is the convention [[Gauss]] uses in [[Disquisitiones Arithmeticae]].
In "twos out", binary quadratic forms are of the form <math>ax^2+bxy+cy^2</math>, represented by the symmetric matrix <math>\begin{pmatrix}a & b/2\\ b/2&c\end{pmatrix}</math>.
Several points of view mean that ''twos out'' has been adopted as the standard convention. Those include:
* better understanding of the 2-adic theory of quadratic forms, the 'local' source of the difficulty;
* the [[lattice (group)|lattice]] point of view, which was generally adopted by the experts in the arithmetic of quadratic forms during the 1950s;
* the actual needs for integral quadratic form theory in [[topology]] for [[intersection theory]];
* the [[Lie group]] and [[algebraic group]] aspects.
=== Universal quadratic forms ===
A quadratic form representing all of the positive integers is sometimes called ''universal''. [[Lagrange's four-square theorem]] shows that <math>w^2+x^2+y^2+z^2</math> is universal. [[Ramanujan]] generalized this to <math>aw^2+bx^2+cy^2+dz^2</math> and found 54 {''a,b,c,d''} such that it can generate all positive integers, namely,
:{1,1,1,d}; d = 1-7
:{1,1,2,d}; d = 2-14
:{1,1,3,d}; d = 3-6
:{1,2,2,d}; d = 2-7
:{1,2,3,d}; d = 3-10
:{1,2,4,d}; d = 4-14
:{1,2,5,d}; d = 6-10
There are also forms that can express nearly all positive integers except one, such as {1,2,5,5} which has 15 as the exception. Recently, the [[15 and 290 theorems]] have completely characterized universal integral quadratic forms: if all coefficients are integers, then it represents all positive integers if and only if it represents all integers up through 290; if it has an integral matrix, it represents all positive integers if and only if it represents all integers up through 15.
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História
O estudo de determinadas formas quadráticas , em particular a questão de saber se um determinado número inteiro pode ser o valor de uma forma quadrática sobre os inteiros , remonta muitos séculos. Um desses casos é o teorema de Fermat em somas de dois quadrados , que determina quando um inteiro pode ser expresso na forma x2 + y2 , onde x , y são inteiros . Este problema está relacionado com o problema de encontrar triplos de Pitágoras , que apareceu no segundo milênio antes de Cristo [3]
Em 628, o índio matemático Brahmagupta escreveu Brahmasphutasiddhanta que inclui, entre muitas outras coisas , um estudo de equações da forma x2 - = NY2 c . Em particular, ele considerou o que hoje é chamado de Pell equação x2 - ny2 = 1, e encontrou um método para a sua solução [4] Na Europa, este problema foi estudado por Brouncker , Euler e Lagrange .
Em 1801 Gauss publicou Disquisitiones Arithmeticae , a maior parte dos quais foi dedicada a uma teoria completa de formas quadráticas binárias sobre os inteiros . Desde
então, o conceito tem sido generalizada, e as conexões com campos quadrática número, o grupo modular , e outras áreas da matemática foram esclarecidas .
{{Referências}}
*{{Citation | last1=O'Meara | first1=T. | title=Introduction to Quadratic Forms | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-3-540-66564-9 | year=2000}}
*{{Citation|last1=Conway|first1=John Horton|author1-link=John Conway | last2=Fung | first2=Francis Y. C. | title=The Sensual (Quadratic) Form | publisher=The Mathematical Association of America | series=Carus Mathematical Monographs | isbn=978-0-88385-030-5 | year=1997}}
<!--=={{Ligações externas}}==
*{{eom|id=q/q076080|author=A.V.Malyshev|title=Quadratic form}}
*{{eom|id=b/b016370|author=A.V.Malyshev|title=Binary quadratic form}}-->
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