TheInfoList

OR:

In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a quadratic form is a
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exam ...
with terms all of degree two (" form" is another name for a
homogeneous polynomial In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; ...
). For example, :$4x^2 + 2xy - 3y^2$ is a quadratic form in the variables and . The coefficients usually belong to a fixed field , such as the real 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 $K=\mathbb R$, and the quadratic form takes zero only when all variables are simultaneously zero, then it is a definite quadratic form, otherwise it is an isotropic quadratic form. Quadratic forms occupy a central place in various branches of mathematics, including
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathe ...
, 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 In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...
),
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multil ...
( 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 In algebra, a quadratic equation () is any equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where represents an unknown value, and , , and represent known numbers, where . (If and then the equation is linear, not quad ...
, 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: :$\begin q\left(x\right) &= ax^2&&\textrm \\ q\left(x,y\right) &= ax^2 + bxy + cy^2&&\textrm \\ q\left(x,y,z\right) &= ax^2 + bxy + cy^2 + dyz + ez^2 + fxz&&\textrm \end$ where ''a'', …, ''f'' are the coefficients. The notation $\langle a_1, \ldots, a_n\rangle$ is often used for the quadratic form : $q\left(x\right) = a_1 x_1^2 + a_2 x_2^2 + \cdots + a_n x_n^2.$ 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 or
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s,
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rati ...
s, or
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language o ...
s. 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 engine ...
, 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. 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 forms have been extensively studied in
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathe ...
, 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 u ...
. Using homogeneous coordinates, a non-zero quadratic form in ''n'' variables defines an (''n''−2)-dimensional quadric in the (''n''−1)-dimensional
projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
. This is a basic construction in
projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, pro ...
. In this way one may visualize 3-dimensional real quadratic forms as conic sections. An example is given by the three-dimensional
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
and the
square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length ad ...
of the
Euclidean norm Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean s ...
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: : $q\left(x,y,z\right) = d\left(\left(x,y,z\right), \left(0,0,0\right)\right)^2 = \, \left(x,y,z\right)\, ^2 = x^2 + y^2 + z^2.$ A closely related notion with geometric overtones is a quadratic space, which is a pair , with ''V'' a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
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 trea ...
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 Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form x^2 - ny^2 = 1, where ''n'' is a given positive nonsquare integer, and integer solutions are sought for ''x'' and ''y''. In Cartesian coordinates ...
, , and found a method for its solution. In Europe this problem was studied by Brouncker,
Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in m ...
and Lagrange. In 1801 Gauss published '' Disquisitiones Arithmeticae,'' a major portion of which was devoted to a complete theory of binary quadratic forms over the
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language o ...
s. Since then, the concept has been generalized, and the connections with quadratic number fields, the modular group, and other areas of mathematics have been further elucidated.

# Associated symmetric matrix

Any matrix determines a quadratic form in variables by : $q_A\left(x_1,\ldots,x_n\right) = \sum_^\sum_^a_ = \mathbf x^\mathrm A \mathbf x,$ where $A = \left(a_\right)$.

## Example

Consider the case of quadratic forms in three variables $x, y, z.$ The matrix has the form :$A=\begin a&b&c\\d&e&f\\g&h&k \end.$ The above formula gives :$q_A\left(x,y,z\right)=ax^2 + ey^2 +kz^2 + \left(b+d\right)xy + \left(c+g\right)xz + \left(f+h\right)yz.$ 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 $b+d, c+g$ and $f+h.$ In particular, the quadratic form $q_A$ is defined by a unique symmetric matrix :$A=\begin a&\frac2&\frac2\\\frac2&e&\frac2\\\frac2&\frac2&k \end.$ This generalizes to any number of variables as follows.

## General case

Given a quadratic form $q_A,$ defined by the matrix $A=\left\left(a_\right\right),$ the matrix $B = \left(\frac 2\right)$ is symmetric, defines the same quadratic form as , and is the unique symmetric matrix that defines $q_A.$ So, over the real numbers (and, more generally, over a field 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" : $\lambda_1 \tilde x_1^2 + \lambda_2 \tilde x_2^2 + \cdots + \lambda_n \tilde x_n^2,$ 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 multiplicat ...
, 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 In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
. 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, $\left(-1\right)^.$ These results are reformulated in a different way below. Let ''q'' be a quadratic form defined on an ''n''-dimensional real 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 : $q\left(v\right)=x^\mathrm Ax,$ 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 multiplicat ...
''S'', and the symmetric square matrix ''A'' is transformed into another symmetric square matrix ''B'' of the same size according to the formula : $A\to B=S^\mathrmAS.$ Any symmetric matrix ''A'' can be transformed into a diagonal matrix : $B=\begin \lambda_1 & 0 & \cdots & 0\\ 0 & \lambda_2 & \cdots & 0\\ \vdots & \vdots & \ddots & 0\\ 0 & 0 & \cdots & \lambda_n \end$ 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 is a ''
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Briti ...
'' 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 algebras (and hence pin groups) are different.

# Definitions

A quadratic form over a field ''K'' is a map $q: V \to K$ from a finite-dimensional ''K''-vector space to ''K'' such that $q\left(av\right) = a^2q\left(v\right)$ for all $a \in K, v \in V$ and the function $q\left(u+v\right) - q\left(u\right) - q\left(v\right)$ is bilinear. More concretely, an ''n''-ary quadratic form over a field ''K'' is a
homogeneous polynomial In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; ...
of degree 2 in ''n'' variables with coefficients in ''K'': : $q\left(x_1,\ldots,x_n\right) = \sum_^\sum_^a_, \quad a_\in K.$ This formula may be rewritten using matrices: let ''x'' be the column vector with components ''x''1, ..., ''x''''n'' and be the ''n''×''n'' matrix over ''K'' whose entries are the coefficients of ''q''. Then : $q\left(x\right)=x^\mathrmAx.$ A vector $v = \left(x_1,\ldots,x_n\right)$ is a
null vector In mathematics, given a vector space ''X'' with an associated quadratic form ''q'', written , a null vector or isotropic vector is a non-zero element ''x'' of ''X'' for which . In the theory of real bilinear forms, definite quadratic forms and ...
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 : $\psi\left(x\right)=\varphi\left(Cx\right).$ 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: : $B=C^\mathrmAC.$ The associated bilinear form of a quadratic form ''q'' is defined by : $b_q\left(x,y\right)=\tfrac\left(q\left(x+y\right)-q\left(x\right)-q\left(y\right)\right) = x^\mathrmAy = y^\mathrmAx.$ Thus, ''b''''q'' is a symmetric bilinear form over ''K'' with matrix ''A''. Conversely, any symmetric bilinear form ''b'' defines a quadratic form : $q\left(x\right)=b\left(x,x\right),$ 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.

Given an ''n''-dimensional
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
''V'' over a field ''K'', a ''quadratic form'' on ''V'' is a function $Q:V\to K$ that has the following property: for some basis, the function ''q'' that maps the coordinates of $v\in V$ to $Q\left(v\right)$ is a quadratic form. In particular, if $V=K^n$ 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 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 In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the ''d ...
of degree 2, which means that it has the property that, for all ''a'' in ''K'' and ''v'' in ''V'': :$Q\left(av\right) = a^2 Q\left(v\right).$ When the characteristic of ''K'' is not 2, the bilinear map over ''K'' is defined: :$B\left(v,w\right)= \tfrac\left(Q\left(v+w\right)-Q\left(v\right)-Q\left(w\right)\right).$ 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 – . 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 : $Q\left(v\right) = Q\text{'}\left(Tv\right) \text v\in V.$ 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, 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 :$\forall x, y \isin A \quad Q\left(x y\right) = Q\left(x\right) Q\left(y\right) ,$ then it is a composition algebra.

# 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 : $q\left(x\right)=a_1 x_1^2 + a_2 x_2^2+ \cdots +a_n x_n^2.$ Such a diagonal form is often denoted by $\langle a_1,\ldots,a_n\rangle.$ 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 $\mathbf = \left(x,y,z\right)^\text$, and let $A$ be a symmetric 3-by-3 matrix. Then the geometric nature of the solution set of the equation $\mathbf^\textA\mathbf+\mathbf^\text\mathbf=1$ depends on the eigenvalues of the matrix $A$. 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 $A$ are non-zero, then the solution set is an
ellipsoid An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface;  that is, a surface that may be defined as the ...
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 $\lambda_i = 0$, then the shape depends on the corresponding $b_i$. If the corresponding $b_i \neq 0$, 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 pl ...
(either elliptic or hyperbolic); if the corresponding $b_i = 0$, then the dimension $i$ degenerates and does not come into play, and the geometric meaning will be determined by other eigenvalues and other components of $\mathbf$. 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 Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathe ...
and
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
. 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 $ax^2+2bxy+cy^2$, represented by the symmetric matrix :$\begina & b\\ b&c\end$ this is the convention Gauss uses in '' Disquisitiones Arithmeticae''. In "twos out", binary quadratic forms are of the form $ax^2+bxy+cy^2$, represented by the symmetric matrix :$\begina & b/2\\ b/2&c\end.$ 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 In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
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 additi ...
and algebraic group 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 Lagrange's four-square theorem, also known as Bachet's conjecture, states that every natural number can be represented as the sum of four integer squares. That is, the squares form an additive basis of order four. p = a_0^2 + a_1^2 + a_2^2 + a_ ...
shows that $w^2+x^2+y^2+z^2$ is universal. Ramanujan generalized this to $aw^2+bx^2+cy^2+dz^2$ 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 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.

* ''ε''-quadratic form * Cubic form * Discriminant of a quadratic form * Hasse–Minkowski theorem * Quadric * Ramanujan's ternary quadratic form * Square class * Witt group * Witt's theorem

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