Quadratic form
In
is a quadratic form in the variables x and y. The coefficients usually belong to a fixed field K, such as the real or complex numbers, and one speaks of a quadratic form over K. If K = R, and the quadratic form equals zero only when all variables are simultaneously zero, then it is a definite quadratic form; otherwise it is an isotropic quadratic form.
Quadratic forms occupy a central place in various branches of mathematics, including
Quadratic forms are not to be confused with a quadratic equation, which has only one variable and includes terms of degree two or less. A quadratic form is one case of the more general concept of homogeneous polynomials.
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:
where a, ..., f are the coefficients.[1]
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
Using
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. See § Definitions 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 x2 + y2, where x, y are integers. This problem is related to the problem of finding Pythagorean triples, which appeared in the second millennium BCE.[3]
In 628, the Indian mathematician
In 1801
Associated symmetric matrix
Any n × n matrix A determines a quadratic form qA in n variables by
Example
Consider the case of quadratic forms in three variables x, y, z. The matrix A 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 b + d, c + g and f + h. In particular, the quadratic form qA is defined by a unique symmetric matrix
This generalizes to any number of variables as follows.
General case
Given a quadratic form qA, defined by the matrix A = (aij), the matrix
So, over the real numbers (and, more generally, over a
Real quadratic forms
A fundamental problem is the classification of real quadratic forms under a
If the change of variables is given by an invertible matrix that is not necessarily orthogonal, one can suppose that all coefficients λi are 0, 1, or −1. Sylvester's law of inertia states that the numbers of each 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 (n0, n+, n−), where n0 is the number of 0s and n± is the number of ±1s. Sylvester's law of inertia shows that this is a well-defined quantity attached to the quadratic form.
The case when all λi have the same sign is especially important: in this case the quadratic form is called
The discriminant of a quadratic form, concretely the class of the determinant of a representing matrix in K / (K×)2 (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, (−1)n−.
These results are reformulated in a different way below.
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
Any symmetric matrix A can be transformed into a diagonal matrix
The quadratic form q is positive definite if q(v) > 0 (similarly, negative definite if q(v) < 0) for every nonzero vector v.[6] When q(v) assumes both positive and negative values, q is an isotropic 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 isotropic 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) ≈ O(q, p)), but the associated Clifford algebras (and hence pin groups) are different.
Definitions
A quadratic form over a field K is a map q : V → K from a finite-dimensional K-vector space to K such that q(av) = a2q(v) for all a ∈ K, v ∈ V and the function q(u + v) − q(u) − q(v) is bilinear.
More concretely, an n-ary quadratic form over a field K is a homogeneous polynomial of degree 2 in n variables with coefficients in K:
This formula may be rewritten using matrices: let x be the
A vector v = (x1, ..., xn) is a null vector if q(v) = 0.
Two n-ary quadratic forms φ and ψ over K are equivalent if there exists a nonsingular linear transformation C ∈ GL(n, K) such that
Let the
The associated bilinear form of a quadratic form q is defined by
Thus, bq is a symmetric bilinear form over K with matrix A. Conversely, any symmetric bilinear form b defines a quadratic form
Quadratic space
Given an n-dimensional vector space V over a field K, a quadratic form on V is a function Q : V → K that has the following property: for some basis, the function q that maps the coordinates of v ∈ V to Q(v) is a quadratic form. In particular, if V = Kn 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 V, although the quadratic form q depends on the choice of the basis.
A finite-dimensional vector space with a quadratic form is called a quadratic space.
The map Q is a homogeneous function of degree 2, which means that it has the property that, for all a in K and v in V:
When the characteristic of K is not 2, the bilinear map B : V × V → K over K is defined:
When the characteristic of K is 2, so that 2 is not a unit, it is still possible to use a quadratic form to define a symmetric bilinear form B′(x, y) = Q(x + y) − Q(x) − Q(y). However, Q(x) can no longer be recovered from this B′ in the same way, since B′(x, x) = 0 for all x (and is thus alternating).[8] Alternatively, there always exists a bilinear form B″ (not in general either unique or symmetric) such that B″(x, x) = Q(x).
The pair (V, Q) consisting of a finite-dimensional vector space V over K and a quadratic map Q from V to K is called a quadratic space, and B as defined here is the associated symmetric 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
The isometry classes of n-dimensional quadratic spaces over K correspond to the equivalence classes of n-ary quadratic forms over K.
Generalization
Let R be a commutative ring, M be an R-module, and b : M × M → R be an R-bilinear form.[9] A mapping q : M → R : v ↦ b(v, v) is the associated quadratic form of b, and B : M × M → R : (u, v) ↦ q(u + v) − q(u) − q(v) is the polar form of q.
A quadratic form q : M → R may be characterized in the following equivalent ways:
- There exists an R-bilinear form b : M × M → R such that q(v) is the associated quadratic form.
- q(av) = a2q(v) for all a ∈ R and v ∈ M, and the polar form of q is R-bilinear.
Related concepts
Two elements v and w of V are called
The orthogonal group of a non-singular quadratic form Q is the group of the linear automorphisms of V that preserve Q: that is, the group of isometries of (V, Q) into itself.
If a quadratic space (A, Q) has a product so that A is an algebra over a field, and satisfies
Equivalence of forms
Every quadratic form q in n variables over a field of characteristic not equal to 2 is equivalent to a diagonal form
Such a diagonal form is often denoted by ⟨a1, ..., an⟩. Classification of all quadratic forms up to equivalence can thus be reduced to the case of diagonal forms.
Geometric meaning
Using
If all
If there exist one or more eigenvalues λi = 0, then the shape depends on the corresponding bi. If the corresponding bi ≠ 0, then the solution set is a paraboloid (either elliptic or hyperbolic); if the corresponding bi = 0, then the dimension i degenerates and does not come into play, and the geometric meaning will be determined by other eigenvalues and other components of b. 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 x2 + xy + y2; equivalently, given a lattice Λ in a vector space V (over a field with characteristic 0, such as Q or R), a quadratic form Q is integral with respect to Λ if and only if it is integer-valued on Λ, meaning Q(x, y) ∈ Z if x, y ∈ Λ.
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 ax2 + 2bxy + cy2, represented by the symmetric matrix
In "twos out", binary quadratic forms are of the form ax2 + bxy + cy2, represented by the symmetric matrix
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.
- {1, 1, 1, d}, 1 ≤ d ≤ 7
- {1, 1, 2, d}, 2 ≤ d ≤ 14
- {1, 1, 3, d}, 3 ≤ d ≤ 6
- {1, 2, 2, d}, 2 ≤ d ≤ 7
- {1, 2, 3, d}, 3 ≤ d ≤ 10
- {1, 2, 4, d}, 4 ≤ d ≤ 14
- {1, 2, 5, d}, 6 ≤ d ≤ 10
There are also forms whose image consists of all but one of the positive integers. For example, {1, 2, 5, 5} 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
Notes
- Gaussdictates the use of manifestly even coefficients for the products of distinct variables, that is, 2b in place of b in binary forms and 2b, 2d, 2f in place of b, d, f in ternary forms. Both conventions occur in the literature.
- polarization identities), but at 2 they are different concepts; this distinction is particularly important for quadratic forms over the integers.
- ^ Babylonian Pythagoras
- ^ Brahmagupta biography
- ^ 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
- ^ If a non-strict inequality (with ≥ or ≤) holds then the quadratic form q is called semidefinite.
- ^ The theory of quadratic forms over a field of characteristic 2 has important differences and many definitions and theorems must be modified.
- ^ This alternating form associated with a quadratic form in characteristic 2 is of interest related to the Arf invariant – Irving Kaplansky (1974), Linear Algebra and Geometry, p. 27.
- ^ The bilinear form to which a quadratic form is associated is not restricted to being symmetric, which is of significance when 2 is not a unit in R.
References
- ISBN 978-3-540-66564-9
- ISBN 978-0-88385-030-5
- ISBN 978-3-642-30993-9.
Further reading
- Zbl 0395.10029.
- Kitaoka, Yoshiyuki (1993). Arithmetic of quadratic forms. Cambridge Tracts in Mathematics. Vol. 106. Cambridge University Press. Zbl 0785.11021.
- Zbl 1068.11023.
- Zbl 0292.10016.
- Zbl 0259.10018.
- Zbl 0847.11014.
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