Quadratic form
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,
 4x^2 + 2xy  3y^2
is a quadratic form in the variables x and y.
Quadratic forms occupy a central place in various branches of mathematics, including number theory, linear algebra, group theory (orthogonal group), differential geometry (Riemannian metric), differential topology (intersection forms of fourmanifolds), and Lie theory (the Killing form).
Contents
 Introduction 1
 History 2
 Real quadratic forms 3

Definitions 4
 Quadratic spaces 4.1
 Further definitions 4.2
 Equivalence of forms 5
 Geometric meaning 6

Integral quadratic forms 7
 Historical use 7.1
 Universal quadratic forms 7.2
 See also 8
 Notes 9
 References 10
 Further reading 11
 External links 12
Introduction
Quadratic forms are homogeneous quadratic polynomials in n variables. In the cases of one, two, and three variables they are called unary, binary, and ternary and have the following explicit form:
 q(x) = ax^2\quad \textrm{(unary)}
 q(x,y) = ax^2 + bxy + cy^2\quad \textrm{(binary)}
 q(x,y,z) = ax^2 + by^2 + cz^2 + dxy + exz + fyz \quad\textrm{(ternary)}
where a, ..., f are the coefficients.^{[1]} Note that quadratic functions, such as ax^{2} + bx + c in the one variable case, are not quadratic forms, as they are typically not homogeneous (unless b and c are both 0).
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 numbers, rational numbers, or integers. In linear algebra, analytic geometry, and in the majority of applications of quadratic forms, the coefficients are real or complex numbers. In the algebraic theory of quadratic forms, the coefficients are elements of a certain field. In the arithmetic theory of quadratic forms, the coefficients belong to a fixed commutative ring, frequently the integers Z or the padic integers Z_{p}.^{[2]} Binary quadratic forms have been extensively studied in number theory, in particular, in the theory of quadratic fields, continued fractions, and modular forms. The theory of integral quadratic forms in n variables has important applications to algebraic topology.
Using homogeneous coordinates, a nonzero quadratic form in n variables defines an (n−2)dimensional quadric in the (n−1)dimensional projective space. This is a basic construction in projective geometry. In this way one may visualize 3dimensional real quadratic forms as conic sections.
A closely related notion with geometric overtones is a quadratic space, which is a pair (V,q), with V a vector space over a field K, and q: V → K a quadratic form on V. An example is given by the threedimensional Euclidean space and the square of the Euclidean norm expressing the distance between a point with coordinates (x,y,z) and the origin:
 q(x,y,z)=d((x,y,z),(0,0,0))^2=\(x,y,z)\^2=x^2+y^2+z^2.
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 x^{2} + y^{2}, where x, y are integers. This problem is related to the problem of finding Pythagorean triples, which appeared in the second millennium B.C.^{[3]}
In 628, the Indian mathematician Brahmagupta wrote Brāhmasphuṭasiddhānta which includes, among many other things, a study of equations of the form x^{2} − ny^{2} = c. In particular he considered what is now called Pell's equation, x^{2} − ny^{2} = 1, and found a method for its solution.^{[4]} In Europe this problem was studied by Brouncker, Euler 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 integers. 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.
Real quadratic forms
Any n×n real symmetric matrix A determines a quadratic form q_{A} in n variables by the formula
 q_A(x_1,\ldots,x_n) = \sum_{i=1}^{n}\sum_{j=1}^{n}a_{ij}{x_i}{x_j} = \mathbf x^\mathrm{T} A \mathbf x.
Conversely, given a quadratic form in n variables, its coefficients can be arranged into an n×n symmetric matrix. One of the most important questions in the theory of quadratic forms is how much can one simplify a quadratic form q by a homogeneous linear change of variables. A fundamental theorem due to Jacobi asserts that q can be brought to a diagonal form
 \lambda_1 \tilde x_1^2 + \lambda_2 \tilde x_2^2 + \cdots + \lambda_n \tilde x_n^2,
so that the corresponding symmetric matrix is diagonal, and this is even possible to accomplish with a change of variables given by an orthogonal matrix – in this case the coefficients λ_{1}, λ_{2}, ..., λ_{n} are in fact determined uniquely up to a permutation. If the change of variables is given by an invertible matrix, not necessarily orthogonal, then the coefficients λ_{i} can be made to be 0,1, and −1. Sylvester's law of inertia states that the numbers of 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 (n_{0}, n_{+}, n_{−}) where n_{0} is the number 0s and n_{±} is the number of ±1s. Sylvester's law of inertia shows that this is a welldefined 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 nondegenerate; this includes positive definite, negative definite, and indefinite (a mix of 1 and −1); equivalently, a nondegenerate quadratic form is one whose associated symmetric form is a nondegenerate bilinear form. A real vector space with an indefinite nondegenerate quadratic form of index (p, q) (p 1s, 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 nonzero 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 nondegenerate form it is the parity of the number of negative coefficients, (1)^{n_}.
These results are reformulated in a different way below.
Let q be a quadratic form defined on an ndimensional real vector space. Let A be the matrix of the quadratic form q in a given basis. This means that A is a symmetric n×n matrix such that
 q(v)=x^\mathrm{T} 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 n×n invertible matrix S, and the symmetric square matrix A is transformed into another symmetric square matrix B of the same size according to the formula
 A\to B=SAS^\mathrm{T}.
Any symmetric matrix A can be transformed into a diagonal matrix
 B=\begin{pmatrix} \lambda_1 & 0 & \cdots & 0\\ 0 & \lambda_2 & \cdots & 0\\ \vdots & \vdots & \ddots & 0\\ 0 & 0 & \cdots & \lambda_n \end{pmatrix}
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 q(v) > 0 (resp., q(v) < 0) for every nonzero vector v.^{[5]} 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 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 noncompact. Further, the isometry groups of Q and −Q are the same (O(p, q) ≈ O(q, p)), but the associated Clifford algebras (and hence Pin groups) are different.
Definitions
An nary quadratic form over a field K is a homogeneous polynomial of degree 2 in n variables with coefficients in K:
 q(x_1,\ldots,x_n) = \sum_{i=1}^{n}\sum_{j=1}^{n}a_{ij}{x_i}{x_j}, \quad a_{ij}\in K.
This formula may be rewritten using matrices: let x be the column vector with components x_{1}, ..., x_{n} and A = (a_{ij}) be the n×n matrix over K whose entries are the coefficients of q. Then
 q(x)=x^\mathrm{T}Ax.
Two nary quadratic forms φ and ψ over K are equivalent if there exists a nonsingular linear transformation C ∈ GL(n, K) such that
 \psi(x)=\varphi(Cx).
Let us assume that the characteristic of K is different from 2. (The theory of quadratic forms over a field of characteristic 2 has important differences and many definitions and theorems have to be modified.) The coefficient matrix A of q may be replaced by the symmetric matrix (A + A^{T})/2 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^\mathrm{T}AC.
The associated bilinear form of a quadratic form q is defined by
 b_q(x,y)=\tfrac{1}{2}(q(x+y)q(x)q(y)) = x^\mathrm{T}Ay = y^\mathrm{T}Ax.
Thus, b_{q} is a symmetric bilinear form over K with matrix A. Conversely, any symmetric bilinear form b defines a quadratic form
 q(x)=b(x,x)
and these two processes are the inverses of one another. As a consequence, over a field of characteristic not equal to 2, the theories of symmetric bilinear forms and of quadratic forms in n variables are essentially the same.
Quadratic spaces
A quadratic form q in n variables over K induces a map from the ndimensional coordinate space K^{n} into K:
 Q(v)=q(v), \quad v=[v_1,\ldots,v_n]^\mathrm{T}\in K^n.
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:
 Q(av) = a^2 Q(v).
When the characteristic of K is not 2, the map B : V × V → K defined below is bilinear over K:
 B(v,w)= \tfrac{1}{2}(Q(v+w)Q(v)Q(w)).
This bilinear form B is symmetric, i.e. B(x, y) = B(y, x) for all x, y in V, and it determines to Q: Q(x) = B(x, x) 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 B′(x, y) = Q(x + y) − Q(x) − Q(y). However, Q(x) can no longer be recovered from this B′ in the same way, since B′(x, x) = 0 for all x (and is thus alternating^{[6]}). Alternately, there always exists a bilinear form B″ (not in general either unique or symmetric) such that B″(x, x) = Q(x).
The pair (V, Q) consisting of a finitedimensional vector space V over K and a quadratic map 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 coordinatefree version of the notion of quadratic form. Sometimes, Q is also called a quadratic form.
Two ndimensional quadratic spaces (V, Q) and (V′, Q′) are isometric if there exists an invertible linear transformation T : V → V′ (isometry) such that
 Q(v) = Q'(Tv) \text{ for all } v\in V.
The isometry classes of ndimensional quadratic spaces over K correspond to the equivalence classes of nary quadratic forms over K.
Further definitions
Two elements v and w of V are called orthogonal if B(v,w) = 0. The kernel of a bilinear form B consists of the elements that are orthogonal to every element of V. Q is nonsingular if the kernel of its associated bilinear form is 0. If there exists a nonzero v in V such that Q(v) = 0, the quadratic form Q is isotropic, otherwise it is anisotropic. This terminology also applies to vectors and subspaces of a quadratic space. If the restriction of Q to a subspace U of V is identically zero, U is totally singular.
The orthogonal group of a nonsingular quadratic form Q is the group of the linear automorphisms of V that preserve Q, i.e. the group of isometries of (V, Q) into itself.
Equivalence of forms
Every quadratic form q in n variables over a field of characteristic not equal to 2 is equivalent to a diagonal form
 q(x)=a_1 x_1^2 + a_2 x_2^2+ \ldots +a_n x_n^2.
Such a diagonal form is often denoted by
 \langle a_1,\dots,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 in three dimensions, let \mathbf{x} = (x,y,z)^\text{T}, and let A be a symmetric 3by3 matrix. Then the geometric nature of the solution set of the equation \mathbf{x}^\text{T}A\mathbf{x}+\mathbf{b}^\text{T}\mathbf{x}=1 depends on the eigenvalues of the matrix A.
If all eigenvalues of A are nonzero, 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 \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 (either elliptic or hyperbolic); if the corresponding b_i = 0, then the dimension i degenerates and does not get into play, and the geometric meaning will be determined by other eigenvalues and other components of \mathbf{b}. When the solution set is a paraboloid, whether it is elliptic or hyperbolic is determined by whether all other nonzero 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 x^{2} + xy + y^{2}; 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 integervalued on Λ, meaning Q(x,y) ∈ Z if x,y ∈ Λ.
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 halfinteger 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
 \begin{pmatrix}a & b\\ b&c\end{pmatrix}
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
 \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:
 better understanding of the 2adic 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 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 foursquare theorem 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 {a,b,c,d} that can each generate all positive integers, namely,
 {1,1,1,d}, 1 ≤ d ≤ 7
 {1,1,2,d}, 2 ≤ d ≤ 14
 {1,1,3,d}, 3 ≤ d ≤ 6
 {1,2,2,d}, 2 ≤ d ≤ 7
 {1,2,3,d}, 3 ≤ d ≤ 10
 {1,2,4,d}, 4 ≤ d ≤ 14
 {1,2,5,d}, 6 ≤ d ≤ 10
There are also forms whose image consists of all but one of the positive integers. For example, {1,2,5,5} 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.
See also
 εquadratic form
 Quadratic form (statistics)
 Quadric
 Discriminant of a quadratic form
 Cubic form
 Witt group
 Witt's theorem
 Hasse–Minkowski theorem
 Orthogonal group
 Square class
 Ramanujan's ternary quadratic form
Notes
 ^ A tradition going back to Gauss dictates the use of manifestly even coefficients for the products of distinct variables, i.e. 2b in place of b in binary forms and 2d, 2e, 2f in place of d, e, f in ternary forms. Both conventions occur in the literature
 ^ away from 2, i. e. if 2 is invertible in the ring, quadratic forms are equivalent to symmetric bilinear forms (by the polarization identities), but at 2 they are different concepts; this distinction is particularly important for quadratic forms over the integers.
 ^ Babylonian Pythagoras
 ^ Brahmagupta biography
 ^ If a nonstrict inequality (with ≥ or ≤) holds then the quadratic form q is called semidefinite.
 ^ This alternating form associated with a quadratic form in characteristic 2 is of interest related to the .
References
Further reading
 Kitaoka, Yoshiyuki (1993). Arithmetic of quadratic forms. Cambridge Tracts in Mathematics 106. Cambridge University Press.
External links
 A.V.Malyshev (2001), "Quadratic form", in Hazewinkel, Michiel,
 A.V.Malyshev (2001), "Binary quadratic form", in Hazewinkel, Michiel,