In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
is a quadratic form in the variables and . The coefficients usually belong to a fixed field , such as the real or complex numbers, and one speaks of a quadratic form over . 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 of a zero-mean multivariate normal distribution has the quadratic form ).
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 variables. In the cases of one, two, and three variables they are called unary, binary, and ternary and have the following explicit form:
where , ..., are the coefficients.A tradition going back to Gauss dictates the use of manifestly even coefficients for the products of distinct variables, that is, in place of in binary forms and , , in place of , , in ternary forms. Both conventions occur in the literature.
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 or the -adic integers .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. 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 variables has important applications to algebraic topology.
Using homogeneous coordinates, a non-zero quadratic form in variables defines an -dimensional projective quadric in the -dimensional projective space. 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 and the origin:
A closely related notion with geometric overtones is a quadratic space, which is a pair , with a vector space over a field , and a quadratic form on . See 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 , where , are integers. This problem is related to the problem of finding Pythagorean triples, which appeared in the second millennium BCE.Babylonian Pythagoras
In 628, the Indian mathematician Brahmagupta wrote Brāhmasphuṭasiddhānta, which includes, among many other things, a study of equations of the form . He considered what is now called Pell's equation, , and found a method for its solution.Brahmagupta biography 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 matrix determines a quadratic form in variables by
where and where is a column vector.
Example Consider the case of quadratic forms in three variables . The matrix has the form
The above formula gives
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 , and . In particular, the quadratic form is defined by a unique symmetric matrix
This generalizes to any number of variables as follows.
General case Given a quadratic form over the real numbers, defined by the matrix , with indices and independently varying from through , the matrix
is symmetric, defines the same quadratic form as , and is the unique symmetric matrix that defines .
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
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"
where the associated symmetric matrix is diagonal. Moreover, the coefficients are determined uniquely up to a permutation.
If the change of variables is given by an invertible matrix that is not necessarily orthogonal, one can suppose that all coefficients are , , or . Sylvester's law of inertia states that the numbers of each , , and 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 , where these components count the number of s, number of s, and the number of s, respectively. Sylvester's law of inertia shows that this is a well-defined quantity attached to the quadratic form.
The case when all have the same sign is especially important: in this case the quadratic form is called positive definite (all ) or negative definite (all ). If none of the terms are , then the form is called ; this includes positive definite, negative definite, and isotropic quadratic form (a mix of and ); 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 (count of s, and of s) is often denoted as particularly in the physical theory of spacetime.
The discriminant of a quadratic form, concretely the class of the determinant of a representing matrix in (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, .
These results are reformulated in a different way below.
Let be a quadratic form defined on an -dimensional real vector space. Let be the matrix of the quadratic form in a given basis. This means that is a symmetric such that
where is the column vector of coordinates of in the chosen basis. Under a change of basis, the column is multiplied on the left by an invertible matrix , and the symmetric square is transformed into another symmetric square of the same size according to the formula
Any symmetric matrix can be transformed into a diagonal matrix
by a suitable choice of an orthogonal matrix , and the diagonal entries of are uniquely determined – this is Jacobi's theorem (see skew-symmetric matrix). If is allowed to be any invertible matrix then can be made to have only , , and on the diagonal, and the number of the entries of each type ( for , for , and for ) depends only on . This is one of the formulations of Sylvester's law of inertia and the numbers and 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 , Sylvester's law of inertia means that they are invariants of the quadratic form .
The quadratic form is positive definite if (similarly, negative definite if ) for every nonzero vector .If a non-strict inequality (with or ) holds then the quadratic form is called semidefinite. When assumes both positive and negative values, is an isotropic quadratic form. The theorems of Jacobi and Sylvester show that any positive definite quadratic form in variables can be brought to the sum of 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, conventionally notated as . This stands in contrast with the case of isotropic forms, when the corresponding group, the indefinite orthogonal group , is non-compact. Further, the isometry groups of and are the same (), but the associated Clifford algebras (and hence pin groups) are different.
Definitions A quadratic form over a field is a map from a finite-dimensional -vector space to such that for all , and the function is a bilinear form.
More concretely, an -ary quadratic form over a field is a homogeneous polynomial of degree 2 in variables with coefficients in :
This formula may be rewritten using matrices: let be the column vector with components , ..., and be the matrix over whose entries are the coefficients of . Then
A vector is a null vector if .
Two -ary quadratic forms and over are equivalent if there exists a nonsingular linear transformation such that
Let the characteristic of be different from . The coefficient matrix of may be replaced by the symmetric matrix with the same quadratic form, so it may be assumed from the outset that is symmetric. Moreover, a symmetric matrix is uniquely determined by the corresponding quadratic form. Under an equivalence , the symmetric matrix of and the symmetric matrix of are related as follows:
The associated bilinear form of a quadratic form is defined by
Thus, is a symmetric bilinear form over with matrix . Conversely, any symmetric bilinear form defines a quadratic form
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 variables are essentially the same.
Quadratic space
Given an -dimensional vector space over a field , a quadratic form on is a function that has the following property: for some basis, the function that maps the coordinates of to is a quadratic form. In particular, if with its standard basis, one has
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 , although the quadratic form depends on the choice of the basis.
A finite-dimensional vector space with a quadratic form is called a quadratic space.
The map is a homogeneous function of degree 2, which means that it has the property that, for all in and in :
When the characteristic of is not 2, the bilinear map over is defined:
This bilinear form is symmetric. That is, for all , in , and it determines : for all in .