Quadratic form

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.

The cases where the theory is equivalent to symmetric bilinear forms

Taking with a slight change of notation

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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.

Related Topics:
Symmetric matrices - Symmetric bilinear form - Field - Characteristic

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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.

Related Topics:
Number theory - Lattice - Topology - Intersection theory - Lie group - Algebraic group

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~