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

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

Note that general Quadratic Function s and Quadratic Equation s are not examples of quadratic forms.


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'' 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 Form s 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 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 over a Commutative Ring ''F'';
often ''V'' is a Vector Space over a Field ''F''.

A map ''Q'' : ''V'' → ''F'' is called a quadratic form on a ''V'' if
  • ''Q''(''au'') = ''a''2 ''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''
:2''Q''(''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

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 Lattice s'''.
They are important in Number Theory and Topology .

Two elements ''u'' and ''v'' 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 {''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
:2 Q(u) = \mathbf{u}^T \mathbf{Bu} = \sum_{i,j=1}^{n}B_{ij}u^i u^j
where ''u''''i'' are the components of ''u'' in this basis.

Some other properties of quadratic forms:

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

    • ::Q(u+v) = Q(u) + Q(v)



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 v
ot=0. If we substitute the strict inequality by a \leq or \geq , it is said to be semidefinite.


ISOTROPIC SPACES


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


SEE ALSO



REFERENCES


  • O'Meara, T. Introduction to Quadratic Forms Springer-Verlag Berlin Heidelberg, 2000. ISBN 3-540-66564-1.