Quadratic form

From Academic Kids

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:

<math>F(x) = ax^2<math>
<math>F(x,y) = ax^2 + by^2 + cxy<math>
<math>F(x,y,z) = ax^2 + by^2 + cz^2 + dxy + exz + fyz<math>

Note that general quadratic functions and quadratic equations are not examples of quadratic forms.

The cases where the theory is equivalent to symmetric bilinear forms

Taking with a slight change of notation

<math>F(x,y) = ax^2 + by^2 + 2cxy<math>

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

xt·M·x

in terms of a 2×2 matrix M with diagonal entries a and b, and off-diagonal entries c. Here the superscript xt denotes the transpose of a matrix.

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.

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.

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.

Quadratic form on a module or vector space

Let V be a module V over a commutative ring F; often V is a vector space over a field F.

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

  • Q(au) = a2 Q(u) for all aF and uV, 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 uV

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.

Two elements of V are called orthogonal if B(u, v)=0.

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.

If V is free of rank n we write the bilinear form B as a symmetric matrix B relative to some basis {ei} for V. The components of B are given by <math>B_{ij} = B(e_i,e_j)<math>. If 2 is invertible the quadratic form Q is then given by

<math>2 Q(u) = \mathbf{u}^T \mathbf{Bu} = \sum_{i,j=1}^{n}B_{ij}u^i u^j<math>

where ui are the components of u in this basis.

Some other properties of quadratic forms:

<math>Q(u+v) + Q(u-v) = 2Q(u) + 2Q(v)<math>
  • The vectors u and v are orthogonal with respect to B if and only if
<math>Q(u+v) = Q(u) + Q(v)<math>de:Quadratische Form

fr:Forme quadratique ja:二次形式

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