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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.

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

1 Quadratic form on a vector space

Let V be a vector space V over a field F. For now we assume that F has characteristic different than 2. This is true, in particular, for the real and complex number fields which have characteristic 0. The case char(F) = 2 is somewhat exceptional, and will be treated separately.

A map Q : V → F is called a quadratic form on a V if there exists a symmetric bilinear form B : V × V → F such that

Q(u) = B(u,u) for all u ∈ V

B is called the associated bilinear form. Note that for any vectors u,v ∈ V

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

so we can recover the bilinear form B from Q:

This is an example of polarization of an algebraic form . There is then a 1-1 correspondence between quadratic forms on V and symmetric bilinear forms on V. Given one we can uniquely define the other.

If V has dimension 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 . The quadratic form Q is then given by

where uⁱ are the components of u in this basis. Note that Q(u) is a homogeneous polynomial of degree two in the coordinates of u and so agrees with our original definition.

Some other properties of quadratic forms:

for all a ∈ F and u ∈ V
Q obeys the parallelogram lawThe parallelogram law in elementary geometry In elementary geometry, the parallelogram law states that the sum of the squares of the lengths of the fours sides of a parallelogram equals the sum of the squares of the lengths of the two diagonals. In case t:

The vectors u and v are orthogonal with respect to B iffIn mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if . It is often, not always, written italicized: iff''. Although "P iff Q" is most standard, common alternative phrases include "P

2 Characteristic two

The theory of quadratic forms in characteristic two has quite a different flavor, essentially because division by 2 is not possible. It is no longer true that every quadratic form is of the form Q(u) = B(u,u) for a symmetric bilinear form B. Moreover, even if B exists it is not unique: since alternating forms are also symmetric in characteristic two, one can add any alternating form to B and get the same quadratic form.

A more general definition of a quadratic form which works for any characteristic is as follows. A quadractic form on a vector space V over a field F is as a map Q : V → F such that

for all a ∈ F and u ∈ V, and
is a bilinear form on V.

3 Generalizations

One can generalize the notion of a quadratic form to moduleAlgebra In abstract algebra, a module is a generalization of a vector space. In a vector space the set of scalars forms a field whereas in a module the scalars just form a ring. Much of the theory of modules consists of recovering desirable properties ofs over a commutative ringIn ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. This means that if a and b are any elements of the ring, and if the multiplication operation is written as then a '. Integral quadratic forms are important in number theory and topology.

Quadratic forms

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