User:Lethe/Quadratic form

In mathematics, a quadratic form on a vector space V over the field K is a mapping Q: V|->K that satisfies

Q(αx)=α²Q(x)
B(x,y)=Q(x+y)-Q(x)-Q(y) is a bilinear form on V

where x, y are in V and α is in K. B is called the bilinear form associated with the quadratic form Q, and B can be seen to be symmetric. Conversely, to any bilinear form B on V, one finds an associated quadratic form Q(x)=B(x,x).

If the characteristic of K is not 2, then we may put a_ik=a_ki=c_ik/2 for i<k and a_ii=c_ii, then the matrix A=(a_ik) is called the matrix of the quadratic form Q. Note that this is a symmetric matrix, reflecting the fact that B is a symmetric bilinear form.

Using matrix notation, we may write Q(x)=x^tA x and B(x,y)=y^tA x

Relation with bilinear forms

To express the quadratic form concept in linear algebra terms, we can note that for any bilinear form B on a vector space V of finite dimension, the expression B(v,v) for v in V will be a quadratic form in the co-ordinates of v with respect to a fixed basis. If F is the underlying field, then this is in fact the general quadratic form over F, unless the characteristic of F is 2. Provided we can divide by 2 in F there is no problem in writing down a matrix representing B, to give rise to any fixed quadratic form: we can choose B to be symmetric.

In fact under that condition there is a 1-1 correspondence between quadratic forms Q and symmetric bilinear forms B (an example of polarization of an algebraic form). For the purposes of quadratic form theory over rings in general, such as the integral quadratic forms important in number theory and topology, one must start with a more careful definition to avoid problems caused by division by 2.

If V is finite dimensional (of dimension n), the vector x may be expressed in terms of a basis for V, with components x₁,...,x_n, then Q can also be expressed as a homogeneous polynomial of degree two:

Q(x_{1},...,x_{n})=\sum _{1\leq i\leq k\leq n}c_{ik}x_{i}x_{k}

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 co-ordinates of each of the two points.

Examples

Two variables:

F(x,y)=ax^{2}+by^{2}+2cxy

Three variables:

F(x,y,z)=ax^{2}+by^{2}+cz^{2}+2dxy+2exz+2fyz