Quadratic form

Template:Short description Template:For In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, <math display="block">4x^2 + 2xy - 3y^2</math>

is a quadratic form in the variables Template:Mvar and Template:Mvar. The coefficients usually belong to a fixed field Template:Mvar, such as the real or complex numbers, and one speaks of a quadratic form over Template:Mvar. Over the reals, a quadratic form is said to be definite if it takes the value zero only when all its variables are simultaneously zero; otherwise it is isotropic.

Quadratic forms occupy a central place in various branches of mathematics, including number theory, linear algebra, group theory (orthogonal groups), differential geometry (the Riemannian metric, the second fundamental form), differential topology (intersection forms of manifolds, especially four-manifolds), Lie theory (the Killing form), and statistics (where the exponent of a zero-mean multivariate normal distribution has the quadratic form <math>-\mathbf{x}^\mathsf{T}\boldsymbol\Sigma^{-1} \mathbf{x}</math>)

Quadratic forms are not to be confused with quadratic equations, which have only one variable and may include terms of degree less than two. A quadratic form is a specific instance of the more general concept of forms.

Introduction

Quadratic forms are homogeneous quadratic polynomials in Template:Math variables. In the cases of one, two, and three variables they are called unary, binary, and ternary and have the following explicit form: <math display="block">\begin{align}

     q(x) &= ax^2&&\textrm{(unary)} \\
   q(x,y) &= ax^2 + bxy + cy^2&&\textrm{(binary)} \\
 q(x,y,z) &= ax^2 + bxy + cy^2 + dyz + ez^2 + fxz&&\textrm{(ternary)}

\end{align}</math>

where Template:Math, ..., Template:Math are the coefficients.<ref>A tradition going back to Gauss dictates the use of manifestly even coefficients for the products of distinct variables, that is, Template:Math in place of Template:Math in binary forms and Template:Math, Template:Math, Template:Math in place of Template:Math, Template:Math, Template:Math in ternary forms. Both conventions occur in the literature.</ref>

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 the 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 Template:Math or the [[p-adic integer|Template:Math-adic integers]] Template:Math.<ref>away from 2, that is, 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.</ref> 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 Template:Math variables has important applications to algebraic topology.

Using homogeneous coordinates, a non-zero quadratic form in Template:Math variables defines an Template:Math-dimensional quadric in the Template:Math-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. 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 Template:Math and the origin: <math display="block">q(x,y,z) = d((x,y,z), (0,0,0))^2 = \left\|(x,y,z)\right\|^2 = x^2 + y^2 + z^2.</math>

A closely related notion with geometric overtones is a quadratic space, which is a pair Template:Math, with Template:Math a vector space over a field Template:Math, and Template:Math a quadratic form on V. See Template:Section link below for the definition of a quadratic form on a vector space.

History

The study of 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 Template:Math, where Template:Math, Template:Math are integers. This problem is related to the problem of finding Pythagorean triples, which appeared in the second millennium BCE.<ref>Babylonian Pythagoras</ref>

In 628, the Indian mathematician Brahmagupta wrote Brāhmasphuṭasiddhānta, which includes, among many other things, a study of equations of the form Template:Math. He considered what is now called Pell's equation, Template:Math, and found a method for its solution.<ref>Brahmagupta biography</ref> 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.

Associated symmetric matrix

Any Template:Math matrix Template:Math determines a quadratic form Template:Math in Template:Math variables by <math display="block">q_A(x_1,\ldots,x_n) = \sum_{i=1}^{n}\sum_{j=1}^{n}a_{ij}{x_i}{x_j} = \mathbf x^\mathsf{T} A \mathbf x, </math> where Template:Math.

Example

Consider the case of quadratic forms in three variables Template:Math. The matrix Template:Mvar has the form <math display="block">A=\begin{bmatrix}

 a&b&c\\d&e&f\\g&h&k

\end{bmatrix}.</math>

The above formula gives <math display="block">q_A(x,y,z)=ax^2 + ey^2 +kz^2 + (b+d)xy + (c+g)xz + (f+h)yz.</math>

So, two different matrices define the same quadratic form if and only if they have the same elements on the diagonal and the same values for the sums Template:Math, Template:Math and Template:Math. In particular, the quadratic form Template:Math is defined by a unique symmetric matrix <math display="block">A=\begin{bmatrix}

 a&\frac{b+d}2&\frac{c+g}2\\
 \frac{b+d}2&e&\frac{f+h}2\\
 \frac{c+g}2&\frac{f+h}2&k

\end{bmatrix}.</math>

This generalizes to any number of variables as follows.

General case

Given a quadratic form Template:Math over the real numbers, defined by the matrix Template:Math, the matrix <math display = block>B = \left(\frac{a_{ij}+a_{ji}} 2\right) = \frac{1} 2(A + A^\text{T})</math> is symmetric, defines the same quadratic form as Template:Mvar, and is the unique symmetric matrix that defines Template:Math.

So, over the real numbers (and, more generally, over a field of characteristic different from two), there is a one-to-one correspondence between quadratic forms and symmetric matrices that determine them.

Real quadratic forms

Template:See also A fundamental problem is the classification of real quadratic forms under a linear change of variables.

Jacobi proved that, for every real quadratic form, there is an orthogonal diagonalization; that is, an orthogonal change of variables that puts the quadratic form in a "diagonal form" <math display="block"> \lambda_1 \tilde x_1^2 + \lambda_2 \tilde x_2^2 + \cdots + \lambda_n \tilde x_n^2, </math> where the associated symmetric matrix is diagonal. Moreover, the coefficients Template:Math are determined uniquely up to a permutation.<ref>Maxime Bôcher (with E.P.R. DuVal)(1907) Introduction to Higher Algebra, § 45 Reduction of a quadratic form to a sum of squares via HathiTrust</ref>

If the change of variables is given by an invertible matrix that is not necessarily orthogonal, one can suppose that all coefficients Template:Math are 0, 1, or −1. Sylvester's law of inertia states that the numbers of each 0, 1, and −1 are invariants of the quadratic form, in the sense that any other diagonalization will contain the same number of each. The signature of the quadratic form is the triple Template:Math, where these components count the number of 0s, number of 1s, and the number of −1s, respectively. Sylvester's law of inertia shows that this is a well-defined quantity attached to the quadratic form.

The case when all Template:Math 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 Template:Visible anchor; this includes positive definite, negative definite, and isotropic quadratic form (a 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 Template:Math (denoting Template:Math 1s and Template:Math −1s) is often denoted as Template:Math particularly in the physical theory of spacetime.

The discriminant of a quadratic form, concretely the class of the determinant of a representing matrix in Template:Math (up to non-zero squares) can also be defined, and for a real quadratic form is a cruder invariant than signature, taking values of only "positive, zero, or negative". Zero corresponds to degenerate, while for a non-degenerate form it is the parity of the number of negative coefficients, Template:Math.

These results are reformulated in a different way below.

Let Template:Math be a quadratic form defined on an Template:Math-dimensional real vector space. Let Template:Math be the matrix of the quadratic form Template:Math in a given basis. This means that Template:Math is a symmetric Template:Math matrix such that <math display="block">q(v) = x^\mathsf{T} Ax,</math> where x is the column vector of coordinates of Template:Math in the chosen basis. Under a change of basis, the column Template:Math is multiplied on the left by an Template:Math invertible matrix Template:Math, and the symmetric square matrix Template:Math is transformed into another symmetric square matrix Template:Math of the same size according to the formula <math display="block"> A\to B=S^\mathsf{T}AS.</math>

Any symmetric matrix Template:Math can be transformed into a diagonal matrix <math display="block"> B=\begin{pmatrix} \lambda_1 & 0 & \cdots & 0\\ 0 & \lambda_2 & \cdots & 0\\ \vdots & \vdots & \ddots & 0\\ 0 & 0 & \cdots & \lambda_n \end{pmatrix}</math> by a suitable choice of an orthogonal matrix Template:Math, and the diagonal entries of Template:Math are uniquely determined – this is Jacobi's theorem. If Template:Math is allowed to be any invertible matrix then Template:Math can be made to have only 0, 1, and −1 on the diagonal, and the number of the entries of each type (Template:Math for 0, Template:Math for 1, and Template:Math for −1) depends only on Template:Math. This is one of the formulations of Sylvester's law of inertia and the numbers Template:Math and Template:Math 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 Template:Math, Sylvester's law of inertia means that they are invariants of the quadratic form Template:Math.

The quadratic form Template:Math is positive definite if Template:Math (similarly, negative definite if Template:Math) for every nonzero vector Template:Math.<ref>If a non-strict inequality (with ≥ or ≤) holds then the quadratic form Template:Math is called semidefinite.</ref> When Template:Math assumes both positive and negative values, Template:Math is an isotropic quadratic form. The theorems of Jacobi and Sylvester show that any positive definite quadratic form in Template:Math variables can be brought to the sum of Template:Math 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 Template:Math. This stands in contrast with the case of isotropic forms, when the corresponding group, the indefinite orthogonal group Template:Math, is non-compact. Further, the isometry groups of Template:Math and Template:Math are the same (Template:Math, but the associated Clifford algebras (and hence pin groups) are different.

Definitions

A quadratic form over a field Template:Math is a map Template:Math from a finite-dimensional Template:Math-vector space to Template:Math such that Template:Math for all Template:Math, Template:Math and the function Template:Math is bilinear.

More concretely, an Template:Math-ary quadratic form over a field Template:Math is a homogeneous polynomial of degree 2 in Template:Math variables with coefficients in Template:Math: <math display="block">q(x_1,\ldots,x_n) = \sum_{i=1}^{n}\sum_{j=1}^{n}a_{ij}{x_i}{x_j}, \quad a_{ij}\in K. </math>

This formula may be rewritten using matrices: let Template:Math be the column vector with components Template:Math, ..., Template:Math and Template:Math be the Template:Math matrix over Template:Math whose entries are the coefficients of Template:Math. Then <math display="block"> q(x) = x^\mathsf{T} A x. </math>

A vector Template:Math is a null vector if Template:Math.

Two Template:Math-ary quadratic forms Template:Math and Template:Math over Template:Math are equivalent if there exists a nonsingular linear transformation Template:Math such that <math display="block"> \psi(x) = \varphi(Cx). </math>

Let the characteristic of Template:Math be different from 2.Template:Refn The coefficient matrix Template:Math of Template:Math may be replaced by the symmetric matrix Template:Math with the same quadratic form, so it may be assumed from the outset that Template:Math is symmetric. Moreover, a symmetric matrix Template:Math is uniquely determined by the corresponding quadratic form. Under an equivalence Template:Math, the symmetric matrix Template:Math of Template:Math and the symmetric matrix Template:Math of Template:Math are related as follows: <math display="block"> B = C^\mathsf{T} A C. </math>

The associated bilinear form of a quadratic form Template:Math is defined by <math display="block"> b_q(x,y)=\tfrac{1}{2}(q(x+y)-q(x)-q(y)) = x^\mathsf{T}Ay = y^\mathsf{T}Ax. </math>

Thus, Template:Math is a symmetric bilinear form over Template:Math with matrix Template:Math. Conversely, any symmetric bilinear form Template:Math defines a quadratic form <math display="block"> q(x)=b(x,x),</math> and these two processes are the inverses of each other. As a consequence, over a field of characteristic not equal to 2, the theories of symmetric bilinear forms and of quadratic forms in Template:Math variables are essentially the same.

Quadratic space

Template:See also Given an Template:Math-dimensional vector space Template:Math over a field Template:Math, a quadratic form on Template:Math is a function Template:Math that has the following property: for some basis, the function Template:Math that maps the coordinates of Template:Math to Template:Math is a quadratic form. In particular, if Template:Math with its standard basis, one has <math display="block"> q(v_1,\ldots, v_n)= Q([v_1,\ldots,v_n])\quad \text{for} \quad [v_1,\ldots,v_n] \in K^n. </math>

The change of basis formulas show that the property of being a quadratic form does not depend on the choice of a specific basis in Template:Math, although the quadratic form Template:Math depends on the choice of the basis.

A finite-dimensional vector space with a quadratic form is called a quadratic space.

The map Template:Math is a homogeneous function of degree 2, which means that it has the property that, for all Template:Math in Template:Math and Template:Math in Template:Math: <math display="block"> Q(av) = a^2 Q(v). </math>

When the characteristic of Template:Math is not 2, the bilinear map Template:Math over Template:Math is defined: <math display="block"> B(v,w)= \tfrac{1}{2}(Q(v+w)-Q(v)-Q(w)).</math> This bilinear form Template:Math is symmetric. That is, Template:Math for all Template:Math, Template:Math in Template:Math, and it determines Template:Math: Template:Math for all Template:Math in Template:Math.

When the characteristic of Template:Math is 2, so that 2 is not a unit, it is still possible to use a quadratic form to define a symmetric bilinear form Template:Math. However, Template:Math can no longer be recovered from this Template:Math in the same way, since Template:Math for all Template:Math (and is thus alternating).<ref>This alternating form associated with a quadratic form in characteristic 2 is of interest related to the Arf invariant – Template:Citation.</ref> Alternatively, there always exists a bilinear form Template:Math (not in general either unique or symmetric) such that Template:Math.

The pair Template:Math consisting of a finite-dimensional vector space Template:Math over Template:Math and a quadratic map Template:Math from Template:Math to Template:Math is called a quadratic space, and Template:Math as defined here is the associated symmetric bilinear form of Template:Math. The notion of a quadratic space is a coordinate-free version of the notion of quadratic form. Sometimes, Template:Math is also called a quadratic form.

Template:AnchorTwo Template:Math-dimensional quadratic spaces Template:Math and Template:Math are isometric if there exists an invertible linear transformation Template:Math (isometry) such that <math display="block"> Q(v) = Q'(Tv) \text{ for all } v\in V.</math>

The isometry classes of Template:Math-dimensional quadratic spaces over Template:Math correspond to the equivalence classes of Template:Math-ary quadratic forms over Template:Math.

Generalization

Let Template:Math be a commutative ring, Template:Math be an Template:Math-module, and Template:Math be an Template:Math-bilinear form.Template:Refn A mapping Template:Math is the associated quadratic form of Template:Math, and Template:Math is the polar form of Template:Math.

A quadratic form Template:Math may be characterized in the following equivalent ways:

• There exists an Template:Math-bilinear form Template:Math such that Template:Math is the associated quadratic form.
• Template:Math for all Template:Math and Template:Math, and the polar form of Template:Math is Template:Math-bilinear.

Related concepts

Template:See also Two elements Template:Math and Template:Math of Template:Math are called orthogonal if Template:Math. The kernel of a bilinear form Template:Math consists of the elements that are orthogonal to every element of Template:Math. Template:Math is non-singular if the kernel of its associated bilinear form is Template:Math. If there exists a non-zero Template:Math in Template:Math such that Template:Math, the quadratic form Template:Math is isotropic, otherwise it is definite. This terminology also applies to vectors and subspaces of a quadratic space. If the restriction of Template:Math to a subspace Template:Math of Template:Math is identically zero, then Template:Math is totally singular.

The orthogonal group of a non-singular quadratic form Template:Math is the group of the linear automorphisms of Template:Math that preserve Template:Math: that is, the group of isometries of Template:Math into itself.

If a quadratic space Template:Math has a product so that Template:Math is an algebra over a field, and satisfies <math display="block">\forall x, y \isin A \quad Q(x y) = Q(x) Q(y) ,</math> then it is a composition algebra.

Equivalence of forms

Every quadratic form Template:Math in Template:Math variables over a field of characteristic not equal to 2 is equivalent to a diagonal form <math display="block">q(x)=a_1 x_1^2 + a_2 x_2^2+ \cdots +a_n x_n^2.</math>

Such a diagonal form is often denoted by Template:Math. Classification of all quadratic forms up to equivalence can thus be reduced to the case of diagonal forms.

Geometric meaning

Using Cartesian coordinates in three dimensions, let Template:Math, and let Template:Math be a symmetric 3-by-3 matrix. Then the geometric nature of the solution set of the equation Template:Math depends on the eigenvalues of the matrix Template:Math.

If all eigenvalues of Template:Math are non-zero, then the solution set is an ellipsoid or a hyperboloid.Template:Citation needed If all the eigenvalues are positive, then it is an ellipsoid; if all the eigenvalues are negative, then it is an imaginary ellipsoid (we get the equation of an ellipsoid but with imaginary radii); if some eigenvalues are positive and some are negative, then it is a hyperboloid.

If there exist one or more eigenvalues Template:Math, then the shape depends on the corresponding Template:Math. If the corresponding Template:Math, then the solution set is a paraboloid (either elliptic or hyperbolic); if the corresponding Template:Math, then the dimension Template:Math degenerates and does not come into play, and the geometric meaning will be determined by other eigenvalues and other components of Template:Math. When the solution set is a paraboloid, whether it is elliptic or hyperbolic is determined by whether all other non-zero eigenvalues are of the same sign: if they are, then it is elliptic; otherwise, it is hyperbolic.

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 Template:Math; equivalently, given a lattice Template:Math in a vector space Template:Math (over a field with characteristic 0, such as Template:Math or Template:Math), a quadratic form Template:Math is integral with respect to Template:Math if and only if it is integer-valued on Template:Math, meaning Template:Math if Template:Math.

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 Template:Math, represented by the symmetric matrix <math display="block">\begin{pmatrix}a & b\\ b&c\end{pmatrix}</math> This is the convention Gauss uses in Disquisitiones Arithmeticae.

In "twos out", binary quadratic forms are of the form Template:Math, represented by the symmetric matrix <math display="block">\begin{pmatrix}a & b/2\\ b/2&c\end{pmatrix}.</math>

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

An integral quadratic form whose image consists of all the positive integers is sometimes called universal. Lagrange's four-square theorem shows that Template:Math is universal. Ramanujan generalized this Template:Math and found 54 multisets Template:Math that can each generate all positive integers, namely, Template:Plainlist There are also forms whose image consists of all but one of the positive integers. For example, Template:Math 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
• Cubic form
• Discriminant of a quadratic form
• Hasse–Minkowski theorem
• Quadric
• Ramanujan's ternary quadratic form
• Square class
• Witt group
• Witt's theorem

References

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Further reading

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External links

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