Quadratic form

In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables.

Quadratic forms are central objects in mathematics, occurring for instance in number theory, Riemannian geometry (as curvature), and Lie theory (via the Killing form).

They are also ubiquitous in physics and chemistry, as the energy of a system, particularly in relation to the L² norm, which leads to the use of Hilbert spaces.

1 Introduction
2 History
3 Symmetric forms
- 3.1 Details
4 Definition
- 4.1 Away from 2
- 4.2 Further definitions
5 Properties
6 Equivalence of Quadratic Forms
7 Integral quadratic form
- 7.1 Historical use
- 7.2 Universal quadratic forms
8 Real quadratic forms
9 See also
10 References
11 External Link

Introduction

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

$F(x) = ax^2$

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

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

The coefficients are elements of a ring. Away from 2, i. e. if 2 is invertible in the ring, quadratic forms are equivalent to symmetric bilinear forms (by the polarization identities), but at 2 they are different concepts; this distinction is particularly important for quadratic forms over the integers.

The term quadratic form is also often used to refer to a quadratic space, which is a pair (V,q) where V is a vector space over a field k, and q:V → k is a quadratic form on V. 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.

A quadratic form in 2 variables is called a binary quadratic form, and these are extensively studied in number theory (particularly in the theory of modular forms), together with their associated quadratic fields.

Note that general quadratic functions and quadratic equations are not examples of quadratic forms, as they are not always homogeneous: quadratic functions are functions on affine space, while quadratic forms are "functions" on projective space (properly, sections of $\mathcal{O}(2)$ , the square of the twisting sheaf).

Any non-zero quadratic form in n variables defines an (n-2)-dimensional quadric in projective space. In this way one may visualize 3-dimensional quadratic forms as conic sections.

History

The study of individual quadratic forms, in particular the question of whether a given integer can be the value of a quadratic form over the integers, dates back many centuries. One such case is Fermat's theorem on sums of two squares, which determines when an integer may be expressed in the form $x^2+y^2$ , where $x,\ y$ are integers. This problem is related to the problem of finding Pythagorean triples, which appeared in the second millenium B.C.^[1]

In 628, the Indian mathemetician Brahmagupta wrote Brahmasphutasiddhanta which includes, among many other things, a study equations of the form $x^2-ny^2 = c$ . In particular he considered what is now called Pell's equation, $x^2-ny^2 = 1$ , and found a method for it's solution.^[2] In Europe this problem was studied by Brouncker, Euler and Lagrange.

In 1801 Gauss published Disquisitiones Arithmeticae, a major portion of which was devoted to a complete theory of binary quadratic forms over the integers. Since then, the concept has been generalized, and the connections with quadratic number fields, the modular group, and other area of mathematics have been further elucidated.

Symmetric forms

When working over a ring where 2 is invertible (for instance, over a field of characteristic not equal to 2), a quadratic form is equivalent to a symmetric bilinear form, in this context often called simply a symmetric form. They are thus frequently confused, as in integral quadratic forms (below), or in higher Witt groups. However, they are distinct concepts, and the distinction is frequently important.

Intuitively, a symmetric form generalizes $xy$ , while a quadratic form generalizes $x^2$ , and one can pass between these via the polarization identities.

Given a quadratic form $Q$ , one obtains a symmetric form $B$ , called the associated symmetric form or associated bilinear form, via:

$B(u,v) = Q(u+v) - Q(u) - Q(v)$

This corresponds to:

$2xy = (x+y)^2 - x^2 - y^2$

Conversely, given a bilinear form $B$ (which need not be symmetric), one obtains a quadratic form via:

$Q(u) = B(u,u)$

This corresponds to:

$x^2 = x\cdot x$

If one composes these two operations, one gets multiplication by 2 (if one starts with either a quadratic form or a symmetric bilinear form); thus if 2 is invertible, these operations are invertible (the polarization identities); by analogy with

$xy = \frac{1}{2}\left((x+y)^2 - x^2 - y^2\right)$

one takes

$B(u,v) = \frac{1}{2}\left(Q(u+v) - Q(u) - Q(v)\right)$

which gives a 1-1 correspondence between quadratic forms on V and symmetric forms on V.

But if 2 is not invertible, symmetric forms and quadratic forms are different: some quadratic forms cannot be written in the form $B(u,u)$ , for example, over the integers, $Q(u)=x^2+xy+y^2$ , or more simply $Q(u)=xy$ .

Details

Let us describe this equivalence in the 2 dimensional case. Any 2 dimensional quadratic form may be written as

$F(x,y) = ax^2 + bxy + cy^2$ .

Let us write v = (x,y) for any vector in the vector space. The quadratic form F can be expressed in terms of matrices if we let M be the 2×2 matrix:

$M= \begin{bmatrix} a & b/2 \\ b/2 & c \end{bmatrix}.$

Then matrix multiplication gives us the following equality:

F(v)=v^T·M·v

Where the superscript v^T denotes the transpose of a matrix. Notice we have used that the characteristic is not 2, since we divided by 2 to define M. So we see the correspondence between 2 dimensional quadratic forms F and 2×2 symmetric matrices M, which correspond to symmetric forms.

This observation generalises quickly to forms in n variables and n×n symmetric matrices. For example, in the case of real-valued quadratic forms, the characteristic of the real numbers is 0, so real quadratic forms and real symmetric bilinear forms are the same objects, from different points of view.

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 uⁱ are the components of u in this basis.

Definition

For more details on this topic, see ε-quadratic form.

Let V be a module over a commutative ring R; often R is a field, such as the real numbers, in which case V is a vector space.

A quadratic form is an element of the symmetric square of the dual space,

$\mbox{Sym}^2\left(V^*\right)�:= V^* \otimes V^* / \langle v\otimes w - w\otimes v\rangle.$

This is precisely the coordinate-free formulation of "homogeneous degree 2 polynomial", as the symmetric algebra of $V^*$ corresponds to polynomials on $V$ .

Bilinear forms are the full tensor product $V^* \otimes V^*$ , and symmetric forms are the subspace of symmetric tensors. Note that the space of quadratic forms is a quotient of the space of bilinear forms, while symmetric forms are a subspace.

In terms of matrices, (we take $V$ to be 2-dimensional):

matrices $\begin{pmatrix}a & b\\c & d\end{pmatrix}$ correspond to bilinear forms
the subspace of symmetric matrices $\begin{pmatrix}a & b\\b & c\end{pmatrix}$ correspond to symmetric forms
the bilinear form $\begin{pmatrix}a & b\\c & d\end{pmatrix}$ yields the quadratic form $ax^2 + bxy+cyx + dy^2 = ax^2 + (b+c)xy + dy^2$ , which is a quotient map with kernel $\begin{pmatrix}0 & b\\-b & 0\end{pmatrix}$ .

One can likewise define quadratic forms corresponding to skew-symmetric forms, Hermitian forms, and skew-Hermitian forms; the general concept is ε-quadratic form.

Away from 2

When 2 is invertible in the ring R, one can define a quadratic form in terms of its associated symmetric form in the following way.

A map $Q\colon V \to R$ is called a quadratic form on V if

Q(av) = a² Q(v) for all $a \in R$ and $v \in V$ , and
B(u,v) = Q(u+v) − Q(u) − Q(v) is a bilinear form on V.

Here B is called the associated symmetric form; it is a symmetric bilinear form.

Further definitions

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.

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. A vector or a subspace of a quadratic space may also be referred to as isotropic. If $Q(V) = 0$ then $Q$ is called totally singular.

Properties

Some other properties of quadratic forms:

Q obeys the parallelogram law:

$Q(u+v) + Q(u-v) = 2Q(u) + 2Q(v)$

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

$Q(u+v) = Q(u) + Q(v)$

Equivalence of Quadratic Forms

Let (V,q) and (W,q') be two quadratic spaces over a field F. They are called equivalent if there exists an isomorphism of vector spaces $s\colon V \to W$ such that

$q'(s(x))=q(x)$

holds for all $x\in V.$ The isomorphism s is called an isometry from (V,q) to (W,q´). This notion of equivalence is an equivalence relation on quadratic forms.

Every quadratic form q on an n-dimensional F-vector space V is equivalent to a diagonal form

$Q(x)=a_1x_1^2+a_2x_2^2+...+a_nx_n^2$

where $x=(x_1,...,x_n)\in V.$ Such a diagonal form is often denoted by $\langle a_1,...,a_n\rangle$ .

It often occurs that two diagonal forms with different coefficients are equivalent. In general, it is not easy to decide whether two given diagonal forms are equivalent or not.
Every diagonal form q over an n-dimensional complex vector space is equivalent to a diagonal form of the shape $\langle 1,...,1,0,...,0\rangle$ where the coefficient 1 occurs r times. For a given q the number r is uniquely determined.

Every diagonal form q over an n-dimensional real vector space is equivalent to a diagonal form of the shape $\langle 1,...,1,-1,...,-1,0,...,0\rangle$ where the coefficient 1 occurs r times and the coefficient -1 occurs s. As in the complex case, for a given q the numbers r and s are uniquely determined. Also for a finite field F the classification of the equivalence classes of quadratic forms on finite dimensional vector spaces is simple.

The rational case is more complicated, but also solved because of the theorem of Hasse-Minkowski, an important result of Number Theory.

Integral quadratic form

Quadratic forms over the ring of integers are called integral quadratic forms or integral lattices. They are important in number theory and topology.

An integral quadratic form is one with integer coefficients, such as $x^2 + xy + y^2$ ; equivalently, given a lattice $\Lambda$ in a vector space $V$ (over a field with characteristic 0, such as $\mathbf{Q}$ or $\mathbf{R}$ ), a quadratic form $Q$ is integral with respect to $\Lambda$ if and only if it is integer-valued on $\Lambda$ , meaning $Q(x,y) \in \mathbf{Z}$ if $x,y \in \Lambda$ .

This is the current use of the term; in the past it was sometimes used differently, as detailed below.

Historical use

Historically there was some confusion and controversy over whether the notion of integral quadratic form should mean:

twos in: the quadratic form associated to a symmetric matrix with integer coefficients
twos out: a polynomial with integer coefficients (so the associated symmetric matrix may have half-integer coefficients off the diagonal)

This debate was due to the confusion of quadratic forms (represented by polynomials) and symmetric bilinear forms (represented by matrices), and "twos out" is now the accepted convention; "twos in" is instead the theory of integral symmetric bilinear forms (integral symmetric matrices).

In "twos in", binary quadratic forms are of the form $ax^2+2bxy+cy^2$ , represented by the symmetric matrix $\begin{pmatrix}a & b\\ b&c\end{pmatrix}$ ; this is the convention Gauss uses in Disquisitiones Arithmeticae.

In "twos out", binary quadratic forms are of the form $ax^2+bxy+cy^2$ , represented by the symmetric matrix $\begin{pmatrix}a & b/2\\ b/2&c\end{pmatrix}$ .

Several points of view mean that twos out has been adopted as the standard convention. Those include:

better understanding of the 2-adic theory of quadratic forms, the 'local' source of the difficulty;
the lattice point of view, which was generally adopted by the experts in the arithmetic of quadratic forms during the 1950s;
the actual needs for integral quadratic form theory in topology for intersection theory;
the Lie group and algebraic group aspects.

Universal quadratic forms

A quadratic form representing all positive integers is sometimes called universal.

Lagrange's four-square theorem shows that $w^2+x^2+y^2+z^2$ is universal.

Recently, the 15 and 290 theorems have completely characterized universal integral quadratic forms: if all coefficients are integers, then it represents all positive integers if and only if it represents all integers up through 290; if it has an integral matrix, it represents all positive integers if and only if it represents all integers up through 15.

Real quadratic forms

Assume $Q$ is a quadratic form defined on a real vector space.

It is said to be positive definite (resp. negative definite) if $Q(v)>0$ (resp. $Q(v)<0$ ) for every vector $v\ne 0.$
If we weaken the strict inequality to ≥ or ≤, the form $Q$ is said to be semidefinite.
If $Q(v)<0$ for some $v$ and $Q(v)>0$ for some other $v$ , $Q$ is said to be indefinite.

Let $A$ be the real symmetric matrix associated with $Q$ as described above, so for any column vector $v$ it holds that

$Q(v)=v^T Av.$

Then, $Q$ is positive (semi)definite, negative (semi)definite, indefinite, if and only if the matrix $A$ has the same properties (see positive-definite matrix). Ultimately, these properties can be characterized in terms of the eigenvalues of $A.$

References

O'Meara, T. (2000), Introduction to Quadratic Forms, Berlin, New York: Springer-Verlag, ISBN 978-3-540-66564-9
Conway, John Horton; Fung, Francis Y. C. (1997), The Sensual (Quadratic) Form, Carus Mathematical Monographs, The Mathematical Association of America, ISBN 978-0-88385-030-5