Quadratic form 251478 221467425 2008-06-24T17:02:42Z AndersBot 6986250 robot Adding: [[uk:Квадратична форма]] In [[mathematics]], a '''quadratic form''' is a [[homogeneous polynomial]] of [[Degree_(mathematics)|degree]] two in a number of variables. Quadratic forms are central objects in mathematics, occurring for instance in [[number theory]], [[Riemannian geometry]] (as [[curvature]]), and [[Lie theory]] (via the [[Killing form]]). They are also ubiquitous in physics and chemistry, as the [[energy]] of a system, particularly in relation to the [[L2_norm|L<sup>2</sup> norm]], which leads to the use of [[Hilbert space]]s. == Definition == Quadratic forms in one, two, and three variables are given by: :<math>F(x) = ax^2</math> :<math>F(x,y) = ax^2 + by^2 + cxy</math> :<math>F(x,y,z) = ax^2 + by^2 + cz^2 + dxy + exz + fyz</math> [[Localization of a ring#Terminology|Away from 2]], 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. The term quadratic form is also often used to refer to a '''quadratic space''', which is a pair (''V'',''q'') where ''V'' is a [[vector space]] over a [[field (mathematics)|field]] ''k'', and ''q'':''V'' &rarr; ''k'' is a quadratic form on ''V''. For example, the [[distance formula|distance]] between two points in [[Three-dimensional space|three-dimensional]] [[Euclidean space]] is found by taking the square root of a quadratic form involving six variables, the three coordinates of each of the two points. A quadratic form in 2 variables is called a '''binary quadratic form''', and these are extensively studied in [[number theory]] (particularly in the theory of [[modular forms]]), together with their associated [[quadratic field]]s. Note that general [[quadratic function]]s and [[quadratic equation]]s are not examples of quadratic forms, as they are not always [[homogeneous polynomial|homogeneous]]: quadratic functions are functions on affine space, while quadratic forms are "functions" on projective space (properly, sections of <math>\mathcal{O}(2)</math>, the square of the [[twisting sheaf]]). Any non-zero quadratic form in ''n'' variables defines an (n-2)-dimensional [[Quadric (projective geometry)|quadric]] in [[projective space]]. In this way one may visualize 3-dimensional quadratic forms as [[conic sections]]. ==Symmetric forms== When working over a ring where 2 is invertible (for instance, over a [[field (mathematics)|field]] of [[characteristic (field)|characteristic]] not equal to 2), a quadratic form is equivalent to a [[symmetric bilinear form]], in this context often called simply a ''symmetric form''. They are thus frequently confused, as in integral quadratic forms (below), or in higher [[Witt groups]]. However, they are distinct concepts, and the distinction is frequently important. Intuitively, a symmetric form generalizes <math>xy</math>, while a quadratic form generalizes <math>x^2</math>, and one can pass between these via the [[polarization identities]]. Given a quadratic form <math>Q</math>, one obtains a symmetric form <math>B</math>, called the '''associated symmetric form''' or '''associated bilinear form''', via: :<math>B(u,v) = Q(u+v) - Q(u) - Q(v)</math> This corresponds to: :<math>2xy = (x+y)^2 - x^2 - y^2</math> Conversely, given a [[bilinear form]] <math>B</math> (which need not be symmetric), one obtains a quadratic form via: :<math>Q(u) = B(u,u)</math> This corresponds to: :<math>x^2 = x\cdot x</math> If one composes these two operations, one gets multiplication by 2 (if one starts with either a quadratic form or a ''symmetric'' bilinear form); thus if 2 is invertible, these operations are invertible (the [[polarization identities]]); by analogy with :<math>xy = \frac{1}{2}\left((x+y)^2 - x^2 - y^2\right)</math> one takes :<math>B(u,v) = \frac{1}{2}\left(Q(u+v) - Q(u) - Q(v)\right)</math> which gives a 1-1 correspondence between quadratic forms on ''V'' and symmetric forms on ''V''. But if 2 is not invertible, symmetric forms and quadratic forms are different: some quadratic forms cannot be written in the form <math>B(u,u)</math>, for example, over the integers, <math>Q(u)=x^2+xy+y^2</math>, or more simply <math>Q(u)=xy</math>. ===Details=== Let us describe this equivalence in the 2 dimensional case. Any 2 dimensional quadratic form may be written as :<math>F(x,y) = ax^2 + bxy + cy^2</math>. Let us write '''v''' = (''x'',''y'') for any vector in the vector space. The quadratic form ''F'' can be expressed in terms of matrices if we let ''M'' be the 2&times;2 matrix: : <math> M= \begin{bmatrix} a & b/2 \\ b/2 & c \end{bmatrix}. </math> Then [[matrix multiplication]] gives us the following equality: : ''F''('''v''')='''v'''<sup>T</sup>&middot;''M''&middot;'''v''' Where the superscript '''v'''<sup>T</sup> denotes the [[transpose of a matrix]]. Notice we have used that the characteristic is not 2, since we divided by 2 to define ''M''. So we see the correspondence between 2 dimensional quadratic forms ''F'' and 2&times;2 [[symmetric matrix|symmetric matrices]] ''M'', which correspond to symmetric forms. This observation generalises quickly to forms in ''n'' variables and ''n''&times;''n'' symmetric matrices. For example, in the case of [[real number|real]]-valued quadratic forms, the characteristic of the real numbers is 0, so real quadratic forms and real [[symmetric bilinear form]]s are the same objects, from different points of view. If ''V'' is free of [[Hamel dimension|rank]] ''n'' we write the bilinear form ''B'' as a [[symmetric matrix]] '''B''' relative to some [[basis (linear algebra)|basis]] {''e''<sub>''i''</sub>} for ''V''. The components of '''B''' are given by <math>B_{ij} = B(e_i,e_j)</math>. If 2 is invertible the quadratic form ''Q'' is then given by :<math>2 Q(u) = \mathbf{u}^T \mathbf{Bu} = \sum_{i,j=1}^{n}B_{ij}u^i u^j</math> where ''u''<sup>''i''</sup> are the components of ''u'' in this basis. ==Abstract definition== {{details|ε-quadratic form}} Let ''V'' be a [[module (mathematics) | module]] over a [[commutative ring]] ''R''; often ''R'' is a [[field (mathematics)|field]], such as the [[real number]]s, in which case ''V'' is a [[vector space]]. A quadratic form is an element of the [[symmetric algebra|symmetric square]] of the [[dual space]], :<math>\mbox{Sym}^2\left(V^*\right) := V^* \otimes V^* / \langle v\otimes w - w\otimes v\rangle.</math> This is precisely the coordinate-free formulation of "homogeneous degree 2 polynomial", as the symmetric algebra of <math>V^*</math> corresponds to polynomials on <math>V</math>. Bilinear forms are the full tensor product <math>V^* \otimes V^*</math>, and symmetric forms are the subspace of [[symmetric tensor]]s. Note that the space of quadratic forms is a ''quotient'' of the space of bilinear forms, while symmetric forms are a ''subspace''. In terms of matrices, (we take <math>V</math> to be 2-dimensional): * matrices <math>\begin{pmatrix}a & b\\c & d\end{pmatrix}</math> correspond to bilinear forms * the subspace of symmetric matrices <math>\begin{pmatrix}a & b\\b & c\end{pmatrix}</math> correspond to symmetric forms * the bilinear form <math>\begin{pmatrix}a & b\\c & d\end{pmatrix}</math> yields the quadratic form <math>ax^2 + bxy+cyx + dy^2 = ax^2 + (b+c)xy + dy^2 </math>, which is a quotient map with kernel <math>\begin{pmatrix}0 & b\\-b & 0\end{pmatrix}</math>. One can likewise define quadratic forms corresponding to [[Alternating bilinear form|skew-symmetric form]]s, [[Hermitian form]]s, and [[skew-Hermitian form]]s; the general concept is [[ε-quadratic form]]. ===Away from 2=== When 2 is invertible in the ring ''R'', one can define a quadratic form in terms of its associated symmetric form in the following way. A map <math>Q\colon V \to R</math> is called a '''quadratic form''' on ''V'' if *''Q''(''av'') = ''a''<sup>2</sup> ''Q''(''v'') for all <math>a \in R</math> and <math>v \in V</math>, and *''B''(''u'',''v'') = ''Q''(''u''+''v'') &minus; ''Q''(''u'') &minus; ''Q''(''v'') is a [[bilinear form]] on ''V''. Here ''B'' is called the '''associated symmetric form'''; it is a [[symmetric bilinear form]]. ===Further definitions=== Two elements ''u'' and ''v'' of ''V'' are called '''[[orthogonal]]''' if ''B''(''u'', ''v'')=0. The '''kernel''' of the bilinear form ''B'' consists of the elements that are orthogonal to all elements of ''V'', and the '''kernel''' of the quadratic form ''Q'' consists of all elements ''u'' of the kernel of ''B'' with ''Q''(''u'')=0. If 2 is invertible then ''Q'' and its associated bilinear form ''B'' have the same kernel. The bilinear form ''B'' is called '''non-singular''' if its kernel is 0, and the quadratic form ''Q'' is called '''non-singular''' if its kernel is 0. The [[orthogonal group]] of a non-singular quadratic form ''Q'' is the group of automorphisms of ''V'' that preserve the quadratic form ''Q''. A quadratic form ''Q'' is called ''[[Isotropic quadratic form|isotropic]]'' when there is a non-zero ''v'' in ''V'' such that <math>Q(v) = 0 </math>. Otherwise it is called ''[[Isotropic quadratic form|anisotropic]]''. A vector or a subspace of a quadratic space may also be referred to as ''isotropic''. If <math>Q(V) = 0 </math> then <math>Q</math> is called [[totally singular]]. == Properties == Some other properties of quadratic forms: *''Q'' obeys the [[parallelogram law]]: ::<math>Q(u+v) + Q(u-v) = 2Q(u) + 2Q(v)</math> *The vectors ''u'' and ''v'' are orthogonal with respect to ''B'' if and only if ::<math>Q(u+v) = Q(u) + Q(v)</math> == Integral quadratic form == Quadratic forms over the ring of integers are called '''''integral quadratic forms''''' or '''integral [[lattice (group)|lattice]]s'''. They are important in [[number theory]] and [[topology]]. An integral quadratic form is one with 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 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. 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. == Real quadratic forms == Assume <math>Q</math> is a quadratic form defined on a [[real number|real]] vector space. * It is said to be ''[[definite bilinear form|positive definite]]'' (resp. ''negative definite'') if <math>Q(v)>0</math> (resp. <math>Q(v)<0</math>) for every vector <math>v\ne 0.</math> * If we loosen the strict inequality to &ge; or &le;, the form <math>Q</math> is said to be ''[[semidefinite]]''. * If <math>Q(v)<0</math> for some <math>v</math> and <math>Q(v)>0</math> for some other <math>v</math>, <math>Q</math> is said to be ''[[definite bilinear form|indefinite]]''. Let <math>A</math> be the real symmetric matrix associated with <math>Q</math> as described above, so for any column vector <math>v</math> it holds that : <math>Q(v)=v^T Av. </math> Then, <math>Q</math> is positive (semi)definite, negative (semi)definite, indefinite, if and only if the matrix <math>A</math> has the same properties (see [[positive-definite matrix]]). Ultimately, these properties can be characterized in terms of the [[eigenvalue]]s of <math>A.</math> == See also == *[[Quadratic form (statistics)]] *[[Discriminant#Discriminant of a quadratic form]] ==References== <references/> * {{Citation | last1=O'Meara | first1=T. | title=Introduction to Quadratic Forms | publisher=[[Springer-Verlag|Springer-Verlag]] | location=Berlin, New York | isbn=978-3-540-66564-9 | year=2000}} * {{Citation | last1=Conway | first1=John Horton | author1-link= John Horton 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}} [[Category:Quadratic forms]] [[ca:Forma quadràtica]] [[de:Quadratische Form]] [[fr:Forme quadratique]] [[ko:2차형식]] [[it:Forma quadratica]] [[nl:Kwadratische vorm]] [[ja:二次形式]] [[pl:Forma kwadratowa]] [[ru:Квадратичная форма]] [[uk:Квадратична форма]] [[ur:چکوری ہئیت]] [[zh:二次型]]