Quadratic form

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:

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

where a, …, f are the coefficients.[1]

The notation ${\displaystyle \langle a_{1},\ldots ,a_{n}\rangle }$ is often used for the quadratic form

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

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 Z or the p-adic integers Z_p.[2] 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. 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:

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

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.

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 x² + y², where x, y are integers. This problem is related to the problem of finding Pythagorean triples, which appeared in the second millennium B.C.[3]

In 628, the Indian mathematician Brahmagupta wrote Brāhmasphuṭasiddhānta, which includes, among many other things, a study of equations of the form x² − ny² = c. In particular he considered what is now called Pell's equation, x² − ny² = 1, and found a method for its solution.[4] 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.

Any n×n matrix A determines a quadratic form q_A in n variables by

${\displaystyle q_{A}(x_{1},\ldots ,x_{n})=\sum _{i=1}^{n}\sum _{j=1}^{n}a_{ij}{x_{i}}{x_{j}}=\mathbf {x} ^{\mathrm {T} }A\mathbf {x} ,}$

where ${\displaystyle A=(a_{ij})}$.

General case

Given a quadratic form ${\displaystyle q_{A},}$ defined by the matrix ${\displaystyle A=\left(a_{ij}\right),}$

the matrix

$${\displaystyle B=\left({\frac {a_{ij}+a_{ji}}{2}}\right)}$$

is symmetric, defines the same quadratic form as A, and is the unique symmetric matrix that defines ${\displaystyle q_{A}.}$

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.

A fundamental question is the classification of the real quadratic form under 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"

${\displaystyle \lambda _{1}{\tilde {x}}_{1}^{2}+\lambda _{2}{\tilde {x}}_{2}^{2}+\cdots +\lambda _{n}{\tilde {x}}_{n}^{2},}$

where the associated symmetric matrix is diagonal. Moreover, the coefficients λ₁, λ₂, ..., λ_n are determined uniquely up to a permutation.[5]

If the change of variables is given by an invertible matrix, that is not necessarily orthogonal, one can suppose that all coefficients λ_i 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 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 (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 (p, q) (denoting p 1s and q −1s) is often denoted as R^p,q particularly in the physical theory of spacetime.

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, ${\displaystyle (-1)^{n_{-}}.}$

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

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

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

${\displaystyle A\to B=S^{\mathrm {T} }AS.}$

Any symmetric matrix A can be transformed into a diagonal matrix

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

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.[6] 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) ≈ O(q, p)), but the associated Clifford algebras (and hence pin groups) are different.

A quadratic form over a field K is a map ${\displaystyle q:V\to K}$ from a finite-dimensional K-vector space to K such that ${\displaystyle q(av)=a^{2}q(v)}$ for all ${\displaystyle a\in K,v\in V}$ and the function ${\displaystyle q(u+v)-q(u)-q(v)}$ 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:

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

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

${\displaystyle q(x)=x^{\mathrm {T} }Ax.}$

A vector ${\displaystyle v=(x_{1},\ldots ,x_{n})}$ 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

${\displaystyle \psi (x)=\varphi (Cx).}$

Let the characteristic of K be different from 2.[7] The coefficient matrix A of q may be replaced by the symmetric matrix (A + A^T)/2 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 C, the symmetric matrix A of φ and the symmetric matrix B of ψ are related as follows:

${\displaystyle B=C^{\mathrm {T} }AC.}$

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

${\displaystyle b_{q}(x,y)={\tfrac {1}{2}}(q(x+y)-q(x)-q(y))=x^{\mathrm {T} }Ay=y^{\mathrm {T} }Ax.}$

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

${\displaystyle q(x)=b(x,x),}$

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 n variables are essentially the same.

Every quadratic form q in n variables over a field of characteristic not equal to 2 is equivalent to a diagonal form

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

Such a diagonal form is often denoted by ${\displaystyle \langle a_{1},\ldots ,a_{n}\rangle .}$ Classification of all quadratic forms up to equivalence can thus be reduced to the case of diagonal forms.

Using Cartesian coordinates in three dimensions, let ${\displaystyle \mathbf {x} =(x,y,z)^{\text{T}}}$, and let ${\displaystyle A}$ be a symmetric 3-by-3 matrix. Then the geometric nature of the solution set of the equation ${\displaystyle \mathbf {x} ^{\text{T}}A\mathbf {x} +\mathbf {b} ^{\text{T}}\mathbf {x} =1}$ depends on the eigenvalues of the matrix ${\displaystyle A}$.

If all eigenvalues of ${\displaystyle A}$ are non-zero, then the solution set is an ellipsoid or a hyperboloid. 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 ${\displaystyle \lambda _{i}=0}$, then the shape depends on the corresponding ${\displaystyle b_{i}}$. If the corresponding ${\displaystyle b_{i}\neq 0}$, then the solution set is a paraboloid (either elliptic or hyperbolic); if the corresponding ${\displaystyle b_{i}=0}$, then the dimension ${\displaystyle i}$ degenerates and does not come into play, and the geometric meaning will be determined by other eigenvalues and other components of ${\displaystyle \mathbf {b} }$. 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.

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.

• 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

• O'Meara, O.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
• Shafarevich, I. R.; A. O. Remizov (2012). Linear Algebra and Geometry. Springer. ISBN 978-3-642-30993-9.

• Cassels, J.W.S. (1978). Rational Quadratic Forms. London Mathematical Society Monographs. Vol. 13. Academic Press. ISBN 0-12-163260-1. Zbl 0395.10029.
• Kitaoka, Yoshiyuki (1993). Arithmetic of quadratic forms. Cambridge Tracts in Mathematics. Vol. 106. Cambridge University Press. ISBN 0-521-40475-4. Zbl 0785.11021.
• Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics. Vol. 67. American Mathematical Society. ISBN 0-8218-1095-2. MR 2104929. Zbl 1068.11023.
• Milnor, J.; Husemoller, D. (1973). Symmetric Bilinear Forms. Ergebnisse der Mathematik und ihrer Grenzgebiete. Vol. 73. Springer-Verlag. ISBN 3-540-06009-X. Zbl 0292.10016.
• O'Meara, O.T. (1973). Introduction to quadratic forms. Die Grundlehren der mathematischen Wissenschaften. Vol. 117. Springer-Verlag. ISBN 3-540-66564-1. Zbl 0259.10018.
• Pfister, Albrecht (1995). Quadratic Forms with Applications to Algebraic Geometry and Topology. London Mathematical Society lecture note series. Vol. 217. Cambridge University Press. ISBN 0-521-46755-1. Zbl 0847.11014.

Wikimedia Commons has media related to Quadratic forms.

• 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