Quadratic form
From Wikipedia, the free encyclopedia
|
In mathematics, a quadratic form is a homogeneous polynomial of 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, which leads to the use of Hilbert spaces.
DefinitionQuadratic forms in one, two, and three variables are given by:
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 k, and q:V → k is a quadratic form on V. For example, the distance between two points in 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 fields. Note that general quadratic functions and quadratic equations are not examples of quadratic forms, as they are not always homogeneous: quadratic functions are functions on affine space, while quadratic forms are "functions" on projective space (properly, sections of Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{O}(2) , the square of the twisting sheaf). Any non-zero quadratic form in n variables defines an (n-2)-dimensional quadric in projective space. In this way one may visualize 3-dimensional quadratic forms as conic sections. Symmetric formsWhen working over a ring where 2 is invertible (for instance, over a field of 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 Failed to parse (Missing texvc executable; please see math/README to configure.): xy , while a quadratic form generalizes Failed to parse (Missing texvc executable; please see math/README to configure.): x^2 , and one can pass between these via the polarization identities. Given a quadratic form Failed to parse (Missing texvc executable; please see math/README to configure.): Q , one obtains a symmetric form Failed to parse (Missing texvc executable; please see math/README to configure.): B , called the associated symmetric form or associated bilinear form, via:
This corresponds to:
(which need not be symmetric), one obtains a quadratic form via:
This corresponds to:
one takes
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 Failed to parse (Missing texvc executable; please see math/README to configure.): B(u,u) , for example, over the integers, Failed to parse (Missing texvc executable; please see math/README to configure.): Q(u)=x^2+xy+y^2 , or more simply Failed to parse (Missing texvc executable; please see math/README to configure.): Q(u)=xy . DetailsLet us describe this equivalence in the 2 dimensional case. Any 2 dimensional quadratic form may be written as
. 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 2×2 matrix:
Where the superscript vT 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 2×2 symmetric matrices M, which correspond to symmetric forms. This observation generalises quickly to forms in n variables and n×n symmetric matrices. For example, in the case of real-valued quadratic forms, the characteristic of the real numbers is 0, so real quadratic forms and real symmetric bilinear forms are the same objects, from different points of view. If V is free of rank n we write the bilinear form B as a symmetric matrix B relative to some basis {ei} for V. The components of B are given by Failed to parse (Missing texvc executable; please see math/README to configure.): B_{ij} = B(e_i,e_j) . If 2 is invertible the quadratic form Q is then given by
where ui are the components of u in this basis. Abstract definition
Let V be a module over a commutative ring R; often R is a field, such as the real numbers, in which case V is a vector space. A quadratic form is an element of the symmetric square of the dual space,
This is precisely the coordinate-free formulation of "homogeneous degree 2 polynomial", as the symmetric algebra of Failed to parse (Missing texvc executable; please see math/README to configure.): V^* corresponds to polynomials on Failed to parse (Missing texvc executable; please see math/README to configure.): V . Bilinear forms are the full tensor product Failed to parse (Missing texvc executable; please see math/README to configure.): V^* \otimes V^* , and symmetric forms are the subspace of symmetric tensors. 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 Failed to parse (Missing texvc executable; please see math/README to configure.): V to be 2-dimensional):
correspond to bilinear forms
correspond to symmetric forms
yields the quadratic form Failed to parse (Missing texvc executable; please see math/README to configure.): ax^2 + bxy+cyx + dy^2 = ax^2 + (b+c)xy + dy^2 , which is a quotient map with kernel Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{pmatrix}0 & b\\-b & 0\end{pmatrix} . One can likewise define quadratic forms corresponding to skew-symmetric forms, Hermitian forms, and skew-Hermitian forms; the general concept is ε-quadratic form. Away from 2When 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 Failed to parse (Missing texvc executable; please see math/README to configure.): Q\colon V \to R is called a quadratic form on V if
and Failed to parse (Missing texvc executable; please see math/README to configure.): v \in V , and
Here B is called the associated symmetric form; it is a symmetric bilinear form. Further definitionsTwo 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 when there is a non-zero v in V such that Failed to parse (Missing texvc executable; please see math/README to configure.): Q(v) = 0 . Otherwise it is called anisotropic. A vector or a subspace of a quadratic space may also be referred to as isotropic. If Failed to parse (Missing texvc executable; please see math/README to configure.): Q(V) = 0 then Failed to parse (Missing texvc executable; please see math/README to configure.): Q is called totally singular. PropertiesSome other properties of quadratic forms:
Integral quadratic formQuadratic forms over the ring of integers are called integral quadratic forms or integral lattices. They are important in number theory and topology. An integral quadratic form is one with integer coefficients, such as Failed to parse (Missing texvc executable; please see math/README to configure.): x^2 + xy + y^2
in a vector space Failed to parse (Missing texvc executable; please see math/README to configure.): V
(over a field with characteristic 0, such as Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{Q}
or Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{R}
), a quadratic form Failed to parse (Missing texvc executable; please see math/README to configure.): Q is integral with respect to Failed to parse (Missing texvc executable; please see math/README to configure.): \Lambda if and only if it is integer-valued on Failed to parse (Missing texvc executable; please see math/README to configure.): \Lambda , meaning Failed to parse (Missing texvc executable; please see math/README to configure.): Q(x,y) \in \mathbf{Z} if Failed to parse (Missing texvc executable; please see math/README to configure.): x,y \in \Lambda . This is the current use of the term; in the past it was sometimes used differently, as detailed below. Historical useHistorically there was some confusion and controversy over whether the notion of integral quadratic form should mean:
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 Failed to parse (Missing texvc executable; please see math/README to configure.): ax^2+2bxy+cy^2 , represented by the symmetric matrix Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{pmatrix}a & b\\ b&c\end{pmatrix}
In "twos out", binary quadratic forms are of the form Failed to parse (Missing texvc executable; please see math/README to configure.): ax^2+bxy+cy^2 , represented by the symmetric matrix Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{pmatrix}a & b/2\\ b/2&c\end{pmatrix} . Several points of view mean that twos out has been adopted as the standard convention. Those include:
Universal quadratic formsA quadratic form representing all positive integers is sometimes called universal. Lagrange's four-square theorem shows that Failed to parse (Missing texvc executable; please see math/README to configure.): w^2+x^2+y^2+z^2 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 formsAssume Failed to parse (Missing texvc executable; please see math/README to configure.): Q is a quadratic form defined on a real vector space.
(resp. Failed to parse (Missing texvc executable; please see math/README to configure.): Q(v)<0 ) for every vector Failed to parse (Missing texvc executable; please see math/README to configure.): v\ne 0.
is said to be semidefinite.
for some Failed to parse (Missing texvc executable; please see math/README to configure.): v and Failed to parse (Missing texvc executable; please see math/README to configure.): Q(v)>0 for some other Failed to parse (Missing texvc executable; please see math/README to configure.): v , Failed to parse (Missing texvc executable; please see math/README to configure.): Q is said to be indefinite. Let Failed to parse (Missing texvc executable; please see math/README to configure.): A be the real symmetric matrix associated with Failed to parse (Missing texvc executable; please see math/README to configure.): Q as described above, so for any column vector Failed to parse (Missing texvc executable; please see math/README to configure.): v it holds that
is positive (semi)definite, negative (semi)definite, indefinite, if and only if the matrix Failed to parse (Missing texvc executable; please see math/README to configure.): A has the same properties (see positive-definite matrix). Ultimately, these properties can be characterized in terms of the eigenvalues of Failed to parse (Missing texvc executable; please see math/README to configure.): A. See alsoReferences
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 ur:چکوری ہئیت |
