In
mathematics, a quadratic form is a
polynomial with terms all of
degree two ("
form
Form is the shape, visual appearance, or configuration of an object. In a wider sense, the form is the way something happens.
Form also refers to:
*Form (document), a document (printed or electronic) with spaces in which to write or enter data
* ...
" is another name for a
homogeneous polynomial). For example,
:
is a quadratic form in the variables and . The coefficients usually belong to a fixed
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
, such as the
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
or
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
numbers, and one speaks of a quadratic form over . If
, and the quadratic form takes zero only when all variables are simultaneously zero, then it is a
definite quadratic form
In linguistics, definiteness is a semantic feature of noun phrases, distinguishing between referents or senses that are identifiable in a given context (definite noun phrases) and those which are not (indefinite noun phrases). The prototypical ...
, otherwise it is an
isotropic quadratic form.
Quadratic forms occupy a central place in various branches of mathematics, including
number theory,
linear algebra,
group theory
In abstract algebra, group theory studies the algebraic structures known as groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as ...
(
orthogonal group),
differential geometry (
Riemannian metric,
second fundamental form),
differential topology
In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...
(
intersection forms of
four-manifolds), and
Lie theory (the
Killing form
In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras. Cartan's criteria (criterion of solvability and criterion of semisimplicity) show ...
).
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
homogeneous polynomials.
Introduction
Quadratic forms are homogeneous quadratic polynomials in ''n'' variables. In the cases of one, two, and three variables they are called unary,
binary
Binary may refer to:
Science and technology Mathematics
* Binary number, a representation of numbers using only two digits (0 and 1)
* Binary function, a function that takes two arguments
* Binary operation, a mathematical operation that ta ...
, and ternary and have the following explicit form:
:
where ''a'', …, ''f'' are the coefficients.
The notation
is often used for the quadratic form
:
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
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
or
complex numbers,
rational numbers, or
integers. In
linear algebra,
analytic geometry
In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.
Analytic geometry is used in physics and engineer ...
, 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
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
. In the arithmetic theory of quadratic forms, the coefficients belong to a fixed
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
, frequently the integers Z or the
''p''-adic integers Z
''p''.
Binary quadratic form
In mathematics, a binary quadratic form is a quadratic homogeneous polynomial in two variables
: q(x,y)=ax^2+bxy+cy^2, \,
where ''a'', ''b'', ''c'' are the coefficients. When the coefficients can be arbitrary complex numbers, most results a ...
s have been extensively studied in
number theory, in particular, in the theory of
quadratic fields,
continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer ...
s, and
modular forms. The theory of integral quadratic forms in ''n'' variables has important applications to
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
.
Using
homogeneous coordinates, a non-zero quadratic form in ''n'' variables defines an (''n''−2)-dimensional
quadric
In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension ''D'') in a -dimensional space, and it is d ...
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
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a speci ...
.
An example is given by the three-dimensional
Euclidean space and the
square of the
Euclidean norm expressing the
distance
Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
between a point with coordinates and the origin:
:
A closely related notion with geometric overtones is a quadratic space, which is a pair , with ''V'' a
vector space over a field ''K'', and a quadratic form on ''V''. See below for the definition of a quadratic form on a vector space.
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 , where ''x'', ''y'' are integers. This problem is related to the problem of finding
Pythagorean triple
A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A primitive Pythagorean triple is ...
s, which appeared in the second millennium B.C.
In 628, the Indian mathematician
Brahmagupta
Brahmagupta ( – ) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical tr ...
wrote ''
Brāhmasphuṭasiddhānta
The ''Brāhmasphuṭasiddhānta'' ("Correctly Established Doctrine of Brahma", abbreviated BSS)
is the main work of Brahmagupta, written c. 628. This text of mathematical astronomy contains significant mathematical content, including a good underst ...
'', which includes, among many other things, a study of equations of the form . In particular he considered what is now called
Pell's equation, , and found a method for its solution. In Europe this problem was studied by
Brouncker,
Euler and
Lagrange.
In 1801
Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
published ''
Disquisitiones Arithmeticae,'' a major portion of which was devoted to a complete theory of
binary quadratic form
In mathematics, a binary quadratic form is a quadratic homogeneous polynomial in two variables
: q(x,y)=ax^2+bxy+cy^2, \,
where ''a'', ''b'', ''c'' are the coefficients. When the coefficients can be arbitrary complex numbers, most results a ...
s over the
integers. Since then, the concept has been generalized, and the connections with
quadratic number field
In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf, the rational numbers.
Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 a ...
s, the
modular group, and other areas of mathematics have been further elucidated.
Associated symmetric matrix
Any matrix determines a quadratic form in variables by
:
where
.
Example
Consider the case of quadratic forms in three variables
The matrix has the form
:
The above formula gives
:
So, two different matrices define the same quadratic form if and only if they have the same elements on the diagonal and the same values for the sums
and
In particular, the quadratic form
is defined by a unique
symmetric matrix
:
This generalizes to any number of variables as follows.
General case
Given a quadratic form
defined by the matrix
the matrix
is
symmetric
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
, defines the same quadratic form as , and is the unique symmetric matrix that defines
So, over the real numbers (and, more generally, over a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
of
characteristic different from two), there is a
one-to-one correspondence between quadratic forms and
symmetric matrices that determine them.
Real quadratic forms
A fundamental question is the classification of the real quadratic form under
linear change of variables.
Jacobi proved that, for every real quadratic form, there is an
orthogonal diagonalization, that is an
orthogonal change of variables that puts the quadratic form in a "diagonal form"
:
where the associated symmetric matrix is
diagonal. Moreover, the coefficients are determined uniquely up to a permutation.
If the change of variables is given by an
invertible matrix
In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that
:\mathbf = \mathbf = \mathbf_n \
where denotes the -by- identity matrix and the multiplicati ...
, that is not necessarily orthogonal, one can suppose that all coefficients are 0, 1, or −1.
Sylvester's law of inertia states that the numbers of each 1 and −1 are
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 , 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 (all 1) or negative definite (all −1). If none of the terms are 0, then the form is called ; 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 ''bilinear'' form. A real vector space with an indefinite nondegenerate quadratic form of index (denoting ''p'' 1s and ''q'' −1s) is often denoted as R
''p'',''q'' particularly in the physical theory of
spacetime.
The
discriminant of a quadratic form, concretely the class of the determinant of a representing matrix in ''K''/(''K''
×)
2 (up to non-zero squares) can also be defined, and for a real quadratic form is a cruder invariant than signature, taking values of only “positive, zero, or negative”. Zero corresponds to degenerate, while for a non-degenerate form it is the parity of the number of negative coefficients,
These results are reformulated in a different way below.
Let ''q'' be a quadratic form defined on an ''n''-dimensional
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
vector space. Let ''A'' be the matrix of the quadratic form ''q'' in a given basis. This means that ''A'' is a symmetric matrix such that
:
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
invertible matrix
In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that
:\mathbf = \mathbf = \mathbf_n \
where denotes the -by- identity matrix and the multiplicati ...
''S'', and the symmetric square matrix ''A'' is transformed into another symmetric square matrix ''B'' of the same size according to the formula
:
Any symmetric matrix ''A'' can be transformed into a diagonal matrix
:
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''
0 for 0, ''n''
+ for 1, and ''n''
− for −1) depends only on ''A''. This is one of the formulations of Sylvester's law of inertia and the numbers ''n''
+ and ''n''
− 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 (resp., ) for every nonzero vector ''v''. 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 In mathematics, the isometry group of a metric space is the set of all bijective isometries (i.e. bijective, distance-preserving maps) from the metric space onto itself, with the function composition as group operation. Its identity element is the ...
is a ''
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 (, but the associated
Clifford algebra
In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercom ...
s (and hence
pin group
The PIN Group was a German courier and postal services company. It belonged to PIN Group S.A., a Luxembourg-based corporate affiliation made up of several German postal companies.
History and shareholding
The PIN Group originally traded under ...
s) are different.
Definitions
A quadratic form over a field ''K'' is a map
from a finite-dimensional ''K''-vector space to ''K'' such that
for all
and the function
is bilinear.
More concretely, an ''n''-ary quadratic form over a field ''K'' is a
homogeneous polynomial of degree 2 in ''n'' variables with coefficients in ''K'':
:
This formula may be rewritten using matrices: let ''x'' be the
column vector
In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example,
\boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end.
Similarly, a row vector is a 1 \times n matrix for some n, c ...
with components ''x''
1, ..., ''x''
''n'' and be the ''n''×''n'' matrix over ''K'' whose entries are the coefficients of ''q''. Then
:
A vector
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 such that
:
Let the
characteristic of ''K'' be different from 2. The coefficient matrix ''A'' of ''q'' may be replaced by the
symmetric matrix 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 ''C'', the symmetric matrix ''A'' of ''φ'' and the symmetric matrix ''B'' of ''ψ'' are related as follows:
:
The associated bilinear form of a quadratic form ''q'' is defined by
:
Thus, ''b''
''q'' is a
symmetric bilinear form over ''K'' with matrix ''A''. Conversely, any symmetric bilinear form ''b'' defines a quadratic form
:
and these two processes are the inverses of each other. 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 space
Given an ''n''-dimensional
vector space ''V'' over a field ''K'', a ''quadratic form'' on ''V'' is a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
that has the following property: for some basis, the function ''q'' that maps the coordinates of
to
is a quadratic form. In particular, if
with its
standard basis
In mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as \mathbb^n or \mathbb^n) is the set of vectors whose components are all zero, except one that equals 1. For example, in the ...
, one has
:
The
change of basis
In mathematics, an ordered basis of a vector space of finite dimension allows representing uniquely any element of the vector space by a coordinate vector, which is a sequence of scalars called coordinates. If two different bases are consi ...
formulas show that the property of being a quadratic form does not depend on the choice of a specific basis in ''V'', although the quadratic form ''q'' depends on the choice of the basis.
A finite-dimensional vector space with a quadratic form is called a quadratic space.
The map ''Q'' is a
homogeneous function of degree 2, which means that it has the property that, for all ''a'' in ''K'' and ''v'' in ''V'':
:
When the characteristic of ''K'' is not 2, the bilinear map over ''K'' is defined:
:
This bilinear form ''B'' is symmetric. That is, for all ''x'', ''y'' in ''V'', and it determines ''Q'': for all ''x'' in ''V''.
When the characteristic of ''K'' is 2, so that 2 is not a
unit, it is still possible to use a quadratic form to define a symmetric bilinear form . However, ''Q''(''x'') can no longer be recovered from this ''B''′ in the same way, since for all ''x'' (and is thus alternating).
[This alternating form associated with a quadratic form in characteristic 2 is of interest related to the ]Arf invariant
In mathematics, the Arf invariant of a nonsingular quadratic form over a field of characteristic 2 was defined by Turkish mathematician when he started the systematic study of quadratic forms over arbitrary fields of characteristic 2. The Arf ...
– . Alternatively, there always exists a bilinear form ''B''″ (not in general either unique or symmetric) such that .
The pair consisting of a finite-dimensional vector space ''V'' over ''K'' and a quadratic map ''Q'' from ''V'' to ''K'' is called a quadratic space, and ''B'' as defined here is the associated symmetric 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 and are isometric if there exists an invertible linear transformation (isometry) such that
:
The isometry classes of ''n''-dimensional quadratic spaces over ''K'' correspond to the equivalence classes of ''n''-ary quadratic forms over ''K''.
Generalization
Let ''R'' be a
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
, ''M'' be an ''R''-
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Modul ...
, and be an ''R''-bilinear form. A mapping is the ''associated quadratic form'' of ''b'', and is the ''polar form'' of ''q''.
A quadratic form may be characterized in the following equivalent ways:
*There exists an ''R''-bilinear form such that ''q''(''v'') is the associated quadratic form.
* for all and , and the polar form of ''q'' is ''R''-bilinear.
Related concepts
Two elements ''v'' and ''w'' of ''V'' are called
orthogonal
In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''.
By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
if . The kernel of a bilinear form ''B'' consists of the elements that are orthogonal to every element of ''V''. ''Q'' is non-singular if the kernel of its associated bilinear form is . If there exists a non-zero ''v'' in ''V'' such that , the quadratic form ''Q'' is
isotropic
Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...
, 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, then ''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'': that is, the group of isometries of into itself.
If a quadratic space has a product so that ''A'' is an
algebra over a field
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition ...
, and satisfies
:
then it is a
composition algebra
In mathematics, a composition algebra over a field is a not necessarily associative algebra over together with a nondegenerate quadratic form that satisfies
:N(xy) = N(x)N(y)
for all and in .
A composition algebra includes an involution ...
.
Equivalence of forms
Every quadratic form ''q'' in ''n'' variables over a field of characteristic not equal to 2 is
equivalent
Equivalence or Equivalent may refer to:
Arts and entertainment
*Album-equivalent unit, a measurement unit in the music industry
*Equivalence class (music)
*''Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre
*'' Equival ...
to a diagonal form
:
Such a diagonal form is often denoted by
Classification of all quadratic forms up to equivalence can thus be reduced to the case of diagonal forms.
Geometric meaning
Using
Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
in three dimensions, let
, and let
be a
symmetric
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
3-by-3 matrix. Then the geometric nature of the
solution set
In mathematics, a solution set is the set of values that satisfy a given set of equations or inequalities.
For example, for a set of polynomials over a ring ,
the solution set is the subset of on which the polynomials all vanish (evaluate t ...
of the equation
depends on the eigenvalues of the matrix
.
If all
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
s of
are non-zero, then the solution set is an
ellipsoid or a
hyperboloid. If all the eigenvalues are positive, then it is an ellipsoid; if all the eigenvalues are negative, then it is an ''imaginary ellipsoid'' (we get the equation of an ellipsoid but with imaginary radii); if some eigenvalues are positive and some are negative, then it is a hyperboloid.
If there exist one or more eigenvalues
, then the shape depends on the corresponding
. If the corresponding
, then the solution set is a
paraboloid
In geometry, a paraboloid is a quadric surface that has exactly one axis of symmetry and no center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar property of symmetry.
Every plane ...
(either elliptic or hyperbolic); if the corresponding
, then the dimension
degenerates and does not come into play, and the geometric meaning will be determined by other eigenvalues and other components of
. When the solution set 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
lattices). They play an important role in
number theory and
topology.
An integral quadratic form has integer coefficients, such as ; equivalently, given a lattice Λ in a vector space ''V'' (over a field with characteristic 0, such as Q or R), a quadratic form ''Q'' is integral ''with respect to'' Λ if and only if it is integer-valued on Λ, meaning if .
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
, represented by the symmetric matrix
:
this is the convention
Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
uses in ''
Disquisitiones Arithmeticae''.
In "twos out", binary quadratic forms are of the form
, represented by the symmetric matrix
:
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 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
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the ad ...
and
algebraic group
In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory.
...
aspects.
Universal quadratic forms
An integral quadratic form whose image consists of all the positive integers is sometimes called ''universal''.
Lagrange's four-square theorem shows that
is universal.
Ramanujan generalized this to
and found 54 multisets that can each generate all positive integers, namely,
:, 1 ≤ ''d'' ≤ 7
:, 2 ≤ ''d'' ≤ 14
:, 3 ≤ ''d'' ≤ 6
:, 2 ≤ ''d'' ≤ 7
:, 3 ≤ ''d'' ≤ 10
:, 4 ≤ ''d'' ≤ 14
:, 6 ≤ ''d'' ≤ 10
There are also forms whose image consists of all but one of the positive integers. For example, has 15 as the exception. Recently, the
15 and 290 theorems
In mathematics, the 15 theorem or Conway–Schneeberger Fifteen Theorem, proved by John H. Conway and W. A. Schneeberger in 1993, states that if a positive definite quadratic form with integer matrix represents all positive integers up to 15, the ...
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.
See also
*
''ε''-quadratic form
*
Cubic form
*
Discriminant of a quadratic form
*
Hasse–Minkowski theorem
*
Quadric
In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension ''D'') in a -dimensional space, and it is d ...
*
Ramanujan's ternary quadratic form
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Square class
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Witt group
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Witt's theorem
Notes
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Further reading
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{{Authority control
Linear algebra
Real algebraic geometry
Squares in number theory