Quadratic form

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

$4x^{2}+2xy-3y^{2}\,\!$

is a quadratic form in the variables x and y.

Quadratic forms are central objects in mathematics, occurring for instance in number theory, geometry (Riemannian metric), algebraic topology (intersection forms on homology), Group Theory (Orthogonal group), and Lie theory (the Killing form).

Introduction

Quadratic forms are homogeneous quadratic polynomials in n variables. In the cases of one, two, and three variables they are called unary, binary, and ternary and have the following explicit form:

$q(x)=ax^{2}\quad {\textrm {(unary)}}$

$q(x,y)=ax^{2}+bxy+cy^{2}\quad {\textrm {(binary)}}$

$q(x,y,z)=ax^{2}+by^{2}+cz^{2}+dxy+exz+fyz\quad {\textrm {(ternary)}}$,

where a,…,f are the coefficients. Note that general quadratic functions, such as ax²+bx+c, are not examples of quadratic forms, as they may not be homogeneous.

The theory of quadratic forms and methods used in their study depend in a large measure on the nature of the coefficients, which may be real or complex numbers, rational numbers, or integers. In linear algebra, analytic geometry, and in the majority of applications of quadratic forms, the coefficients are real or complex numbers. In algebraic theory of quadratic forms, the coefficients are elements of a certain field. In the arithmetic theory of quadratic forms, the coefficients belong to a fixed commutative ring, frequently the integers Z or the p-adic integers Z_p. Binary quadratic forms have been extensively studied in number theory, in particular, in the theory of quadratic fields, continued fractions, and modular forms. The theory of integral quadratic forms in n variables has important applications to algebraic topology.

Using homogeneous coordinates, a non-zero quadratic form in n variables defines an (n−2)-dimensional quadric in the (n−1)-dimensional projective space. This is a basic construction in projective geometry. In this way one may visualize 3-dimensional real quadratic forms as conic sections.

A closely related notion with geometric overtones is a quadratic space, which is a pair (V,q), with V a vector space over a field k, and q:V → k a quadratic form on V. An example is given by the three-dimensional Euclidean space and the square of the Euclidean norm expressing the distance between a point with coordinates (x,y,z) and the origin:

$q(x,y,z)=d((x,y,z),(0,0,0))^{2}=\|(x,y,z)\|^{2}=x^{2}+y^{2}+z^{2}.$

History

The study of particular 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 millennium B.C.

In 628, the Indian mathematician Brahmagupta wrote Brahmasphutasiddhanta which includes, among many other things, a study of 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 its solution. 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 areas of mathematics have been further elucidated.

Real quadratic forms

See also: Sylvester's law of inertia and Definite form

Any n×n real symmetric matrix A determines a quadratic form q_A in n variables by the formula

$q_{A}(x_{1},\ldots ,x_{n})=\sum _{i,j=1}^{n}a_{ij}{x_{i}}{x_{j}}.$

Conversely, given a quadratic form in n variables, its coefficients can be arranged into an n×n symmetric matrix. One of the most important questions in the theory of quadratic forms is how much can one simplify a quadratic form q by a homogeneous linear change of variables. A fundamental theorem due to Jacobi asserts that q can be brought to a diagonal form

$\lambda _{1}x_{1}^{2}+\lambda _{2}x_{2}^{2}+\cdots +\lambda _{n}x_{n}^{2},$

so that the corresponding symmetric matrix is diagonal, and this is even possible to accomplish with a change of variables given by an orthogonal matrix – in this case the coefficients λ₁, λ₂, …, λ_n are in fact determined uniquely up to a permutation. If the change of variables is given by an invertible matrix, not necessarily orthogonal, then the coefficients λ_i can be made to be 0,1, and −1. Sylvester's law of inertia states that the numbers of 1 and −1 are invariants of the quadratic form, in the sense that any other diagonalization will contain them in the same quantities. The case when all λ_i have the same sign is especially important: in this case the quadratic form is called positive definite (all 1) or negative definite (all −1); if none of the terms are 0 then the form is called nondegenerate; this includes positive definite, negative definite, and indefinite (mix of 1 and −1); equivalently, a nondegenerate quadratic form is one whose associated symmetric form is a nondegenerate bilinear form. A real vector space with an indefinite nondegenerate quadratic form of index $(p,q)$ (p 1s, q −1s) is often denoted as $\mathbf {R} ^{p,q},$ particularly in the physical theory of space-time.

Below we reformulate these results in a different way.

Let q be a quadratic form defined on an n-dimensional real vector space. Let A be the matrix of the quadratic form q in a given basis. This means that A is a symmetric n×n matrix such that

$q(v)=x^{T}Ax,$

where x is the column vector of coordinates of v in the chosen basis. Under a change of basis, the column x is multiplied on the left by an n×n invertible matrix S, and the symmetric square matrix A is transformed into another symmetric square matrix B of the same size according to the formula

$A\to B=SAS^{T}.$

Any symmetric matrix A can be transformed into a diagonal matrix

$B={\begin{pmatrix}\lambda _{1}&0&\cdots &0\\0&\lambda _{2}&\cdots &0\\\vdots &\vdots &\ddots &0\\0&0&\cdots &\lambda _{n}\end{pmatrix}}$

by a suitable choice of an orthogonal matrix S, and the diagonal entries of B are uniquely determined — this is Jacobi's theorem. If S is allowed to be any invertible matrix then B can be made to have only 0,1, and −1 on the diagonal, and the number of the entries of each type (n₀ for 0, n₊ for 1, and n₋ for −1) depends only on A. This is one of the formulations of Sylvester's law of inertia and the numbers n₊ and n₋ are called the positive and negative indices of inertia. Although their definition involved a choice of basis and consideration of the corresponding real symmetric matrix A, Sylvester's law of inertia means that they are invariants of the quadratic form q.

The quadratic form q is positive definite (resp., negative definite) if q(v)>0 (resp., q(v)<0) for every nonzero vector v. When q(v) assumes both positive and negative values, q is an indefinite quadratic form. The theorems of Jacobi and Sylvester show that any positive definite quadratic form in n variables can be brought to the sum of n squares by a suitable invertible linear transformation: geometrically, there is only one positive definite real quadratic form of every dimension. Its isometry group is a compact orthogonal group O(n). This stands in contrast with the case of indefinite forms, when the corresponding group, the indefinite orthogonal group O(p,q), is non-compact. Further, the isometry groups of Q and −Q are the same ($O(p,q)\approx O(q,p)$), but the associated Clifford algebras (and hence Pin groups) are different.

Definitions

An n-ary quadratic form over a field K is a homogeneous polynomial of degree 2 in n variables with coefficients in K:

$q(x_{1},\ldots ,x_{n})=\sum _{i,j=1}^{n}a_{ij}{x_{i}}{x_{j}},\quad a_{ij}\in K.$

This formula may be rewritten using matrices: let x be the column vector with components x₁, …, x_n and A = (a_ij) be the n×n matrix over K whose entries are the coefficients of q. Then

$q(x)=x^{t}Ax.$

Two n-ary quadratic forms φ and ψ over K are equivalent if there exists a nonsingular linear transformation T ∈GL_n(K) such that

$\psi (x)=\varphi (Tx).$

Let us assume that the characteristic of K is different from 2. 

(The theory of quadratic forms over a field of characteristic 2 has important differences and many definitions and theorems have to be modified.) The coefficient matrix A of q may be replaced by the symmetric matrix 1/2(A + A^t) with the same quadratic form, so it may be assumed from the outset that A is symmetric. Moreover, a symmetric matrix A is uniquely determined by the corresponding quadratic form. Under an equivalence T, the symmetric matrix A of φ and the symmetric matrix B of ψ are related as follows:

$B=T^{t}AT.$

The associated bilinear form of a quadratic form q is defined by

$b_{q}(x,y)={\frac {1}{2}}(q(x+y)-q(x)-q(y))=x^{t}Ay=y^{t}Ax.$

Thus, b_q is a symmetric bilinear form over K with matrix A. Conversely, any symmetric bilinear form b defines a quadratic form

$q(x)=b(x,x)$

and these two processes are the inverses of one another. As a consequence, over a field of characteristic not equal to 2, the theories of symmetric bilinear forms and of quadratic forms in n variables are essentially the same.

Quadratic spaces

A quadratic form q in n variables over Kinduces a map from the n-dimensional cooordinate space Kⁿ into K:

$Q(v)=q(v),\quad v=[v_{1},\ldots ,v_{n}]^{t}\in K^{n}.$

The map Q is a quadratic map, which means that it has the properties:

• $Q(av)=a^{2}Q(v)\quad {\text{ for all }}a\in K,v\in V.$

• The map B_Q: V×V → K defined below is bilinear over K:

$B_{q}(v,w)={\frac {1}{2}}(Q(v+w)-Q(v)-Q(w)).$

The pair (V,Q) consisting of a finite-dimensional vector space V over K and a quadratic map from V to K is called a quadratic space and B_Q is the associated bilinear form of Q. The notion of a quadratic space is a coordinate-free version of the notion of quadratic form. Sometimes, Q is also called a quadratic form.

Two n-dimensional quadratic spaces (V,Q) and (V^′, Q^′) are isometric if there exists an invertible linear transformation T: V →V^′ (isometry) such that

$Q(v)=Q'(Tv){\text{ for all }}v\in V.$

The isometry classes of n-dimensional quadratic spaces over K correspond to the equivalence classes of n-ary quadratic forms over K.

Further definitions

See also: Isotropic quadratic form

Two elements v and w of V are called orthogonal if B(v, w)=0. The kernel of a bilinear form B consists of the elements that are orthogonal to each elements of V. Q is non-singular if the kernel of its associated bilinear form is 0. If there exists a non-zero v in V such that Q(v) = 0, the quadratic form Q is isotropic, otherwise it is anisotropic. This terminology also applies to vectors and subspaces of a quadratic space. If the restriction of Q to a subspace U of V is identically zero, U is totally singular.

The orthogonal group of a non-singular quadratic form Q is the group of the linear automorphisms of V that preserve Q, i.e. the group of isometries of (V, Q) into itself.

Symmetric forms

See also: Homogeneous polynomial § Symmetric tensors

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. See homogeneous polynomial: symmetric tensors for generalization to higher dimensions.

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:

$B(u,v)=2^{-1}(Q(u+v)-Q(u)-Q(v)).$

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:

$F(\mathbf {v} )=\mathbf {v} ^{\top }\cdot M\cdot \mathbf {v}$

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

$2Q(u)=\mathbf {u} ^{T}\mathbf {Bu} =\sum _{i,j=1}^{n}B_{ij}u^{i}u^{j}$

where uⁱ are the components of u in this basis.

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

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

• Gradient of a quadratic form:

$Q({\vec {x}})={\vec {x}}^{T}\cdot A\cdot {\vec {x}}\longrightarrow {\vec {\nabla }}Q({\vec {x}})=(A+A^{T})\cdot {\vec {x}}$

where A is a square matrix (not necessary symmetric).

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.

When the characteristic of F is not 2, every quadratic form q on an n-dimensional F-vector space V is equivalent to a diagonal form

$Q(x)=a_{1}x_{1}^{2}+a_{2}x_{2}^{2}+\cdots +a_{n}x_{n}^{2}$

where $x=(x_{1},\dots ,x_{n})\in V.$ Such a diagonal form is often denoted by $\langle a_{1},\dots ,a_{n}\rangle$.

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 $\langle 1,\dots ,1,0,\dots ,0\rangle$ 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 $\langle 1,\dots ,1,-1,\dots ,-1,0,\dots ,0\rangle$ 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 forms

Quadratic forms over the ring of integers are called integral quadratic forms, whereas the corresponding modules are quadratic lattices (sometimes, simply lattices). They play an important role in number theory and topology.

An integral quadratic form has 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 of the positive integers is sometimes called universal. Lagrange's four-square theorem shows that $w^{2}+x^{2}+y^{2}+z^{2}$ is universal. Ramanujan generalized this to $aw^{2}+bx^{2}+cy^{2}+dz^{2}$ and found 54 {a,b,c,d} such that it can generate all positive integers, namely,

{1,1,1,d}; d = 1-7

{1,1,2,d}; d = 2-14

{1,1,3,d}; d = 3-6

{1,2,2,d}; d = 2-7

{1,2,3,d}; d = 3-10

{1,2,4,d}; d = 4-14

{1,2,5,d}; d = 6-10

There are also forms that can express nearly all positive integers except one, such as {1,2,5,5} which has 15 as the exception. 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.

See also

• ε-quadratic form
• Quadratic form (statistics)
• Discriminant#Discriminant of a quadratic form
• Witt group
• Witt's theorem
• Hasse–Minkowski theorem
• Orthogonal group

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

External links

• A.V.Malyshev (2001) [1994], "Quadratic form", Encyclopedia of Mathematics, EMS Press
• A.V.Malyshev (2001) [1994], "Binary quadratic form", Encyclopedia of Mathematics, EMS Press