In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example, the distance between two points in three-dimensional Euclidean space is found by taking the square root of a quadratic form involving six variables, the three coordinates of each of the two points. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Interactive Mathematics Miscellany and Puzzles — A collection of articles on various math topics, with interactive Java... In mathematics, homogeneous has a variety of meanings. ... This article is about the term degree as used in mathematics. ... For distance between people, see proxemics. ... In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...

Quadratic forms in one, two, and three variables are given by:

F(x) = ax²

F(x,y) = ax² + by² + cxy

F(x,y,z) = ax² + by² + cz² + dxy + exz + fyz

Note that general quadratic functions and quadratic equations are not examples of quadratic forms. f(x) = x2 - x - 2 In mathematics, a quadratic function is a polynomial function of the form , where is nonzero. ... Graph of a quadratic function: y = x2 - x - 2 = (x+1)(x-2) The x-coordinates of the points where the graph crosses the x-axis, x = -1 and x = 2, are the roots of the quadratic equation: x2 - x - 2 = 0 In mathematics, a quadratic equation is a polynomial...

The cases where the theory is equivalent to symmetric bilinear forms

Taking with a slight change of notation

F(x,y) = ax² + by² + 2cxy

it is easy to see that F can be written in terms of a vector x = (x,y) as

x^T·M·x

in terms of a 2×2 matrix M with diagonal entries a and b, and off-diagonal entries c. Here the superscript x^T denotes the transpose of a matrix. In mathematics, and in particular linear algebra, the transpose of a matrix is another matrix, produced by turning rows into columns and vice versa. ...

This observation generalises quickly to forms in n variables and n×n symmetric matrices. It can be used to show that the theory of quadratic forms coincides with that of symmetric bilinear forms, provided that the change of notation is harmless. As it involves only replacing each coefficient not in front of a squared variable by halving it, it is innocuous in most cases: unless the scalars are a field of characteristic 2, we can do this over any field. For example, the most common case of real-valued quadratic forms presents no difficulty, and to talk about real quadratic forms or real symmetric bilinear forms based on symmetric matrices is to discuss the same objects from different points of view. In linear algebra, a symmetric matrix is a matrix that is its own transpose. ... In mathematics, a bilinear form on a vector space V over a field F is a mapping V × V → F which is linear in both arguments. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0 (where n1R is defined as 1R + ... + 1R with n summands). ...

It has long been known, particularly from some aspects of number theory, that this is not the complete story. In fact there has been, historically speaking, some controversy over whether the notion of integral quadratic form should be presented with twos in (i.e., based on integral symmetric matrices) or twos out. Several points of view mean that twos out has been adopted as the standard convention. Those include: (i) better understanding of the 2-adic theory of quadratic forms, the 'local' source of the difficulty; (ii) the lattice point of view, which was generally adopted by the experts in the arithmetic of quadratic forms during the 1950s; (iii) the actual needs for integral quadratic form theory in topology for intersection theory; and (iv) the Lie group and algebraic group aspects. Traditionally, number theory is the branch of pure mathematics concerned with the properties of integers. ... See lattice for other meanings of this term, both within and without mathematics. ... Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ... In mathematics, intersection theory is a branch of algebraic geometry, where subvarieties are intersected on an algebraic variety, and of algebraic topology, where intersections are computed within the cohomology ring. ... In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ... In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety. ...

The rest of this article proceeds with the accepted way to handle the issue, which therefore has particular relevance to working over some ring R in which 2 is not a unit. In mathematics, a unit in a ring R is an element u such that there is v in R with uv = vu = 1R. That is, u is an invertible element of the multiplicative monoid of R. The units of R form a group U(R) under multiplication, the group of...

Quadratic form on a module or vector space

Let V be a module over a commutative ring F; often V is a vector space over a field F. In abstract algebra, the notion of a module over a ring is the common generalizations of two of the most important notions in algebra, vector space (where we take the ring to be a particular field), and abelian group (where we take the ring to be the ring of integers). ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...

A map Q : V → F is called a quadratic form on a V if

• Q(au) = a² Q(u) for all a ∈ F and u ∈ V, and
• B(u,v) = Q(u+v) − Q(u) − Q(v) is a bilinear form on V.

B is called the associated bilinear form. Note that for any vector u ∈ V In mathematics, a bilinear form on a vector space V over a field F is a mapping V × V → F which is linear in both arguments. ...

2Q(u) = B(u,u)

so if 2 is invertible in F we can recover the quadratic form from the symmetric bilinear form B by

Q(u) = B(u,u)/2.

When 2 is invertible this gives a 1-1 correspondence between quadratic forms on V and symmetric bilinear forms on V. If B is any symmetric bilinear form then B(u,u) is always a quadratic form. This is sometimes used as the definition of a quadratic form, but if 2 is not invertible this definition is wrong as not all quadratic forms can be obtained like this.

Quadratic forms over the ring of integers are called integral quadratic forms or integral lattices. They are important in number theory and topology. See lattice for other meanings of this term, both within and without mathematics. ... Traditionally, number theory is the branch of pure mathematics concerned with the properties of integers. ... Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ...

Two elements u and v of V are called orthogonal if B(u, v)=0. In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ...

The kernel of the bilinear form B consists of the elements that are orthogonal to all elements of V, and the kernel of the quadratic form Q consists of all elements u of the kernel of B with Q(u)=0. If 2 is invertible then Q and its associated bilinear form B have the same kernel.

The bilinear form B is called non-singular if its kernel is 0, and the quadratic form Q is called non-singular if its kernel is 0.

The orthogonal group of a non-singular quadratic form Q is the group of automorphisms of V that preserve the quadratic form Q. In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ...

If V is free of rank n we write the bilinear form B as a symmetric matrix B relative to some basis {e_i} for V. The components of B are given by B_ij = B(e_i,e_j). If 2 is invertible the quadratic form Q is then given by In mathematics, the dimension of a vector space V is the cardinality (i. ... In linear algebra, a symmetric matrix is a matrix that is its own transpose. ... In linear algebra, a basis is a minimum set of vectors that, when combined, can address every vector in a given space. ...

where uⁱ are the components of u in this basis.

Some other properties of quadratic forms:

• Q obeys the parallelogram law:

Q(u + v) + Q(u − v) = 2Q(u) + 2Q(v)

• The vectors u and v are orthogonal with respect to B if and only if

Q(u + v) = Q(u) + Q(v)

The parallelogram law in elementary geometry In elementary geometry, the parallelogram law states that the sum of the squares of the lengths of the four sides of a parallelogram equals the sum of the squares of the lengths of the two diagonals. ...

Definiteness of a quadratic form

If a quadratic form Q is defined on a real vector space, it is said to be positive (resp. negative) definite if Q(v) > 0 (resp. Q(v) < 0) for every vector . If we substitute the strict inequality by a or , it is said to be semidefinite.

Isotropic Spaces

A quadradic form Q is called isotropic when there is a non-zero v in V such that Q(v) = 0. Otherwise it is called anisotropic.