Quadratic form

In, a quadratic form is a with terms all of  two. For example,
 * $$4x^2 + 2xy - 3y^2$$

is a quadratic form in the variables x and y. The coefficients usually belong to a fixed field K, such as the real or complex numbers, and we speak of a quadratic form over K.

Quadratic forms occupy a central place in various branches of mathematics, including, , ,  ,  ( of s), and  (the ).

Quadratic forms are not to be confused with a 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.

Introduction
Quadratic forms are homogeneous quadratic polynomials in n variables. In the cases of one, two, and three variables they are called unary, , and ternary and have the following explicit form:


 * $$\begin{align}

q(x) &= ax^2&\textrm{(unary)} \\ q(x,y) &= ax^2 + bxy + cy^2&\textrm{(binary)} \\ q(x,y,z) &= ax^2 + bxy + cy^2 + dyz + ez^2 + fxz&\textrm{(ternary)} \end{align}$$

where a, …, f are the coefficients.

The notation $$\langle a_1, \dots, a_n\rangle$$ is often used for the quadratic form
 * $$q(x) = a_1 x_1^2 + a_2 x_2^2 + \ldots + a_n x_n^2.$$

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 or s, s, or s. In, , 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. In the arithmetic theory of quadratic forms, the coefficients belong to a fixed, frequently the integers Z or the Zp. Binary quadratic forms have been extensively studied in , in particular, in the theory of s, s, and. The theory of integral quadratic forms in n variables has important applications to.

Using, a non-zero quadratic form in n variables defines an (n−2)-dimensional in the (n−1)-dimensional. This is a basic construction in. In this way one may visualize 3-dimensional real quadratic forms as. An example is given by the three-dimensional and the square of the  expressing the  between a point with coordinates (x, y, z) and the origin:
 * $$q(x,y,z) = d((x,y,z), (0,0,0))^2 = \|(x,y,z)\|^2 = x^2 + y^2 + z^2.$$

A closely related notion with geometric overtones is a quadratic space, which is a pair (V, q), with V a over a field K, and q : V → K a quadratic form on V.

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, which determines when an integer may be expressed in the form x2 + y2, where x, y are integers. This problem is related to the problem of finding s, which appeared in the second millennium B.C.

In 628, the Indian mathematician wrote  which includes, among many other things, a study of equations of the form x2 − ny2 = c. In particular he considered what is now called, x2 − ny2 = 1, and found a method for its solution. In Europe this problem was studied by, and.

In 1801 published , a major portion of which was devoted to a complete theory of s over the s. Since then, the concept has been generalized, and the connections with s, the, and other areas of mathematics have been further elucidated.

Real quadratic forms
Any n×n real A determines a quadratic form qA 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.

An important question in the theory of quadratic forms is how to simplify a quadratic form q by a homogeneous linear change of variables. A fundamental theorem due to asserts that a real quadratic form q has an.
 * $$ \lambda_1 \tilde x_1^2 + \lambda_2 \tilde x_2^2 + \cdots + \lambda_n \tilde x_n^2, $$

so that the corresponding symmetric matrix is, and this is accomplished with a change of variables given by an – in this case the coefficients λ1, λ2, ..., λn are determined uniquely up to a permutation.

There always exists a change of variables given by an invertible matrix, not necessarily orthogonal, such that the coefficients λi are 0, 1, and −1. states that the numbers of each 1 and −1 are 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 (n0, n+, n−), where n0 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 λi have the same sign is especially important: in this case the quadratic form is called  (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 (a mix of 1 and −1); equivalently, a nondegenerate quadratic form is one whose associated symmetric form is a. A real vector space with an indefinite nondegenerate quadratic form of index (p, q) (denoting p 1s and q −1s) is often denoted as Rp,q particularly in the physical theory of.

The, 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, $$(-1)^{n_{-}}.$$

These results are reformulated in a different way below.

Let q be a quadratic form defined on an n-dimensional 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 S, and the symmetric square matrix A is transformed into another symmetric square matrix B of the same size according to the formula


 * $$ A\to B=S^\mathrm{T}AS.$$

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 (n0 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. 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 is a   O(n). This stands in contrast with the case of indefinite forms, when the corresponding group, the O(p, q), is non-compact. Further, the isometry groups of Q and −Q are the same (O(p, q) ≈ O(q, p)), but the associated s (and hence s) 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 $$q(av) = a^2q(v)$$ for all $$ a \in K, v \in V$$ and the function $$q(u+v) - q(u) - q(v)$$ is bilinear.

More concretely, an n-ary quadratic form over a field K is a 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 with components x1, ..., xn and A = (aij) be the n×n matrix over K whose entries are the coefficients of q. Then


 * $$ q(x)=x^\mathrm{T}Ax. $$

A vector $$v = (x_1,\ldots,x_n)$$ is a if q(v) = 0.

Two n-ary quadratic forms φ and ψ over K are equivalent if there exists a nonsingular linear transformation C ∈ (n, K) such that


 * $$ \psi(x)=\varphi(Cx). $$

Let the characteristic of K be different from 2. The coefficient matrix A of q may be replaced by the (A + AT)/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, bq is a 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 n-dimensional coordinate space Kn into K:


 * $$ Q(v)=q(v), \quad v=[v_1,\ldots,v_n]^\mathrm{T}\in K^n. $$

The map Q is a 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 bilinear map B : V × V → K over K below is defined:
 * $$ B(v,w)= \tfrac{1}{2}(Q(v+w)-Q(v)-Q(w)).$$

This bilinear form B is symmetric, i.e. B(x, y) = B(y, x) for all x, y in V, and it determines Q: Q(x) = B(x, x) for all x in V.

When the characteristic of K is 2, so that 2 is not a, 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). 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 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 (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 n-dimensional quadratic spaces over K correspond to the equivalence classes of n-ary quadratic forms over K.

Generalization
Let R be a, M be an R- and b : M × M → R be an R-bilinear form. A mapping q : M → R : v ↦ b(v, v) is the associated quadratic form of b, and B : M × M → R : (u, v) ↦ q(u + v) − q(u) − q(v) is the polar form of q.

A quadratic form q : M → R may be characterized in the following equivalent ways:
 * There exists an R-bilinear form b : M × M → R such that q(v) is the associated quadratic form.
 * q(av) = a2q(v) for all a ∈ R and v ∈ M, and the polar form of q is R-bilinear.

Related concepts
Two elements v and w of V are called  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 non-singular if the kernel of its associated bilinear form is {0}. If there exists a non-zero v in V such that Q(v) = 0, the quadratic form Q is , 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 of a non-singular 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.

If a quadratic space (A, Q) has a product so that A is an, and satisfies
 * $$\forall x, y \isin A \quad Q(x y) = Q(x) Q(y) ,$$ then it is a.

Equivalence of forms
Every quadratic form q in n variables over a field of characteristic not equal to 2 is 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 in three dimensions, let $$\mathbf{x} = (x,y,z)^\text{T}$$, and let $$A$$ be a  3-by-3 matrix. Then the geometric nature of the 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 s of $$A$$ are non-zero, then the solution set is an or a. 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 (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 $$\mathbf{b}$$. 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 s). They play an important role in and.

An integral quadratic form has integer coefficients, such as x2 + xy + y2; 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 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: 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).
 * 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)

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 uses in .

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 2-adic theory of quadratic forms, the 'local' source of the difficulty;
 * the 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 for ;
 * the and  aspects.

Universal quadratic forms
An integral quadratic form whose image consists of all the positive integers is sometimes called universal. shows that $$w^2+x^2+y^2+z^2$$ is universal. 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 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.