Quadratic form

Inmathematics,aquadratic formis apolynomialwith terms all ofdegreetwo ( "form"is another name for ahomogeneous polynomial). For example, $4x^{2}+2xy-3y^{2}$

is a quadratic form in the variables $x$ and $y$ .The coefficients usually belong to a fixedfield $K$ ,such as therealorcomplexnumbers, 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 adefinite quadratic form;otherwise it is anisotropic quadratic form.

Quadratic forms occupy a central place in various branches of mathematics, includingnumber theory,linear algebra,group theory(orthogonal groups),differential geometry(theRiemannian metric,thesecond fundamental form),differential topology(intersection formsofmanifolds,especiallyfour-manifolds),Lie theory(theKilling form), andstatistics(where the exponent of a zero-meanmultivariate normal distributionhas the quadratic form $-\mathbf {x} ^{\mathsf {T}}{\boldsymbol {\Sigma }}^{-1}\mathbf {x}$ )

Quadratic forms are not to be confused with aquadratic 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 ofhomogeneous polynomials.

Introduction

Quadratic forms are homogeneous quadratic polynomials in $n$ variables. In the cases of one, two, and three variables they are calledunary,binary,andternaryand have the following explicit form: ${\begin{aligned}q(x)&=ax^{2}&&{\textrm {(unary)}}\\q(x,y)&=ax^{2}+bxy+cy^{2}&&{\textrm {(binary)}}\\q(x,y,z)&=ax^{2}+bxy+cy^{2}+dyz+ez^{2}+fxz&&{\textrm {(ternary)}}\end{aligned}}$

where $a$ ,..., $f$ are thecoefficients.^[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 berealorcomplex numbers,rational numbers,orintegers.Inlinear 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 certainfield.In the arithmetic theory of quadratic forms, the coefficients belong to a fixedcommutative ring,frequently the integers $Z$ or the $p$ -adic integers $Z p$ .^[2]Binary quadratic formshave been extensively studied innumber theory,in particular, in the theory ofquadratic fields,continued fractions,andmodular forms.The theory of integral quadratic forms in $n$ variables has important applications toalgebraic topology.

Usinghomogeneous coordinates,a non-zero quadratic form in $n$ variables defines an $(n - 2)$ -dimensionalquadricin the $(n - 1)$ -dimensionalprojective space.This is a basic construction inprojective geometry.In this way one may visualize 3-dimensional real quadratic forms asconic sections. An example is given by the three-dimensionalEuclidean spaceand thesquareof theEuclidean normexpressing thedistancebetween a point with coordinates $(x, y, z)$ and the origin: $q(x,y,z)=d((x,y,z),(0,0,0))^{2}=\left\|(x,y,z)\right\|^{2}=x^{2}+y^{2}+z^{2}.$

A closely related notion with geometric overtones is aquadratic space,which is a pair $(V, q)$ ,with $V$ avector spaceover a field $K$ ,and $q : V \to K$ a quadratic form onV.See§ Definitionsbelow 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 isFermat'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 findingPythagorean triples,which appeared in the second millennium BCE.^[3]

In 628, the Indian mathematicianBrahmaguptawroteBrāhmasphuṭasiddhānta,which includes, among many other things, a study of equations of the form $x 2 - ny 2 = c$ .He considered what is now calledPell's equation, $x 2 - ny 2 = 1$ ,and found a method for its solution.^[4]In Europe this problem was studied byBrouncker,EulerandLagrange.

In 1801GausspublishedDisquisitiones Arithmeticae,a major portion of which was devoted to a complete theory ofbinary quadratic formsover theintegers.Since then, the concept has been generalized, and the connections withquadratic number fields,themodular group,and other areas of mathematics have been further elucidated.

Associated symmetric matrix

Any $n \times n$ matrix $A$ determines a quadratic form $q A$ in $n$ variables by $q_{A}(x_{1},\ldots,x_{n})=\sum _{i=1}^{n}\sum _{j=1}^{n}a_{ij}{x_{i}}{x_{j}}=\mathbf {x} ^{\mathsf {T}}A\mathbf {x},$ where $A = (a ij)$ .

Example

Consider the case of quadratic forms in three variables $x, y, z$ .The matrix $A$ has the form $A={\begin{bmatrix}a&b&c\\d&e&f\\g&h&k\end{bmatrix}}.$

The above formula gives $q_{A}(x,y,z)=ax^{2}+ey^{2}+kz^{2}+(b+d)xy+(c+g)xz+(f+h)yz.$

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 $q A$ is defined by a uniquesymmetric matrix $A={\begin{bmatrix}a&{\frac {b+d}{2}}&{\frac {c+g}{2}}\\{\frac {b+d}{2}}&e&{\frac {f+h}{2}}\\{\frac {c+g}{2}}&{\frac {f+h}{2}}&k\end{bmatrix}}.$

This generalizes to any number of variables as follows.

General case

Given a quadratic form $q A$ ,defined by the matrix $A = (a ij)$ , the matrix $B=\left({\frac {a_{ij}+a_{ji}}{2}}\right)={\frac {1}{2}}(A+A^{\text{T}})$ issymmetric,defines the same quadratic form as $A$ ,and is the unique symmetric matrix that defines $q A$ .

So, over the real numbers (and, more generally, over afieldofcharacteristicdifferent from two), there is aone-to-one correspondencebetween quadratic forms andsymmetric matricesthat determine them.

Real quadratic forms

A fundamental problem is the classification of real quadratic forms under alinear change of variables.

Jacobiproved that, for every real quadratic form, there is anorthogonal diagonalization;that is, anorthogonal change of variablesthat puts the quadratic form in a "diagonal form" $\lambda _{1}{\tilde {x}}_{1}^{2}+\lambda _{2}{\tilde {x}}_{2}^{2}+\cdots +\lambda _{n}{\tilde {x}}_{n}^{2},$ where the associated symmetric matrix isdiagonal.Moreover, the coefficients $λ 1, λ 2,..., λ n$ are determined uniquelyup toapermutation.^[5]

If the change of variables is given by aninvertible matrixthat is not necessarily orthogonal, one can suppose that all coefficients $λ i$ are 0, 1, or −1.Sylvester's law of inertiastates that the numbers of each 0, 1, and −1 areinvariantsof the quadratic form, in the sense that any other diagonalization will contain the same number of each. Thesignatureof the quadratic form is the triple $(n 0, n +, n -)$ ,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 $λ i$ have the same sign is especially important: in this case the quadratic form is calledpositive definite(all 1) ornegative definite(all −1). If none of the terms are 0, then the form is callednondegenerate;this includes positive definite, negative definite, andisotropic quadratic form(a mix of 1 and −1); equivalently, a nondegenerate quadratic form is one whose associated symmetric form is anondegeneratebilinearform.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 ofspacetime.

Thediscriminant of a quadratic form,concretely the class of the determinant of a representing matrix in $K / (K \times) 2$ (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 $n$ -dimensionalrealvector space. Let $A$ be the matrix of the quadratic form $q$ in a given basis. This means that $A$ is a symmetric $n \times n$ matrix such that $q(v)=x^{\mathsf {T}}Ax,$ wherexis 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 $n \times n$ 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 $A\to B=S^{\mathsf {T}}AS.$

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 ( $n 0$ 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 thepositiveandnegativeindices 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 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 anisotropic quadratic form.The theorems of Jacobi andSylvestershow 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 onlyonepositive definite real quadratic form of every dimension. Itsisometry groupis acompactorthogonal group $O(n)$ .This stands in contrast with the case of isotropic forms, when the corresponding group, theindefinite 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 associatedClifford algebras(and hencepin groups) are different.

Definitions

Aquadratic formover a field $K$ is a map $q : V \to K$ from a finite-dimensional $K$ -vector space to $K$ such that $q (av) = a 2 q (v)$ for all $a \in K$ , $v \in V$ and the function $q (u + v) - q (u) - q (v)$ is bilinear.

More concretely, an $n$ -aryquadratic formover a field $K$ is ahomogeneous polynomialof degree 2 in $n$ variables with coefficients in $K$ : $q(x_{1},\ldots,x_{n})=\sum _{i=1}^{n}\sum _{j=1}^{n}a_{ij}{x_{i}}{x_{j}},\quad a_{ij}\in K.$

This formula may be rewritten using matrices: let $x$ be thecolumn vectorwith components $x 1$ ,..., $x n$ and $A = (a ij)$ be the $n \times n$ matrix over $K$ whose entries are the coefficients of $q$ .Then $q(x)=x^{\mathsf {T}}Ax.$

A vector $v = (x 1,..., x n)$ is anull vectorif $q (v) = 0$ .

Two $n$ -ary quadratic forms $φ$ and $ψ$ over $K$ areequivalentif there exists a nonsingular linear transformation $C \in GL (n, K)$ such that $\psi (x)=\varphi (Cx).$

Let thecharacteristicof $K$ be different from 2.^[7]The coefficient matrix $A$ of $q$ may be replaced by thesymmetric 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: $B=C^{\mathsf {T}}AC.$

Theassociated bilinear formof a quadratic form $q$ is defined by $b_{q}(x,y)={\tfrac {1}{2}}(q(x+y)-q(x)-q(y))=x^{\mathsf {T}}Ay=y^{\mathsf {T}}Ax.$

Thus, $b q$ is asymmetric bilinear formover $K$ 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 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.

Quadratic space

Given an $n$ -dimensionalvector space $V$ over a field $K$ ,aquadratic formon $V$ is afunction $Q : V \to K$ that has the following property: for some basis, the function $q$ that maps the coordinates of $v \in V$ to $Q (v)$ is a quadratic form. In particular, if $V = K n$ with itsstandard basis,one has $q(v_{1},\ldots,v_{n})=Q([v_{1},\ldots,v_{n}])\quad {\text{for}}\quad [v_{1},\ldots,v_{n}]\in K^{n}.$

Thechange of basisformulas 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 aquadratic space.

The map $Q$ is ahomogeneous functionof degree 2, which means that it has the property that, for all $a$ in $K$ and $v$ in $V$ : $Q(av)=a^{2}Q(v).$

When the characteristic of $K$ is not 2, the bilinear map $B : V \times V \to K$ over $K$ is defined: $B(v,w)={\tfrac {1}{2}}(Q(v+w)-Q(v)-Q(w)).$ This bilinear form $B$ is symmetric. That is, $B (x, y) = B (y, x)$ for all $x$ , $y$ in $V$ ,and it determines $Q$ : $Q (x) = B (x, x)$ for all $x$ in $V$ .

When the characteristic of $K$ is 2, so that 2 is not aunit,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 aquadratic 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 $(V', Q')$ areisometricif there exists an invertible linear transformation $T : V \to V'$ (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 $K$ .

Generalization

Let $R$ be acommutative ring, $M$ be an $R$ -module,and $b : M \times M \to R$ be an $R$ -bilinear form.^[9]A mapping $q : M \to R : v \mapsto b (v, v)$ is theassociated quadratic formof $b$ ,and $B : M \times M \to R :(u, v) \mapsto q (u + v) - q (u) - q (v)$ is thepolar formof $q$ .

A quadratic form $q : M \to R$ may be characterized in the following equivalent ways:

There exists an $R$ -bilinear form $b : M \times M \to R$ such that $q (v)$ is the associated quadratic form.
$q (av) = a 2 q (v)$ for all $a \in R$ and $v \in M$ ,and the polar form of $q$ is $R$ -bilinear.

Related concepts

Two elements $v$ and $w$ of $V$ are calledorthogonalif $B (v, w) = 0$ .Thekernelof a bilinear form $B$ consists of the elements that are orthogonal to every element of $V$ . $Q$ isnon-singularif 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 $Q$ isisotropic,otherwise it isdefinite.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, then $U$ istotally singular.

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 analgebra over a field,and satisfies $\forall x,y\in A\quad Q(xy)=Q(x)Q(y),$ then it is acomposition algebra.

Equivalence of forms

Every quadratic form $q$ in $n$ variables over a field of characteristic not equal to 2 isequivalentto adiagonal form $q(x)=a_{1}x_{1}^{2}+a_{2}x_{2}^{2}+\cdots +a_{n}x_{n}^{2}.$

Such a diagonal form is often denoted by $⟨ a 1,..., a n ⟩$ . Classification of all quadratic forms up to equivalence can thus be reduced to the case of diagonal forms.

Geometric meaning

UsingCartesian coordinatesin three dimensions, let $x = (x, y, z) T$ ,and let $A$ be asymmetric3-by-3 matrix. Then the geometric nature of thesolution setof the equation $x T A x + b T x = 1$ depends on the eigenvalues of the matrix $A$ .

If alleigenvaluesof $A$ are non-zero, then the solution set is anellipsoidor ahyperboloid.^{[citation needed]}If all the eigenvalues are positive, then it is an ellipsoid; if all the eigenvalues are negative, then it is animaginary 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 $λ i = 0$ ,then the shape depends on the corresponding $b i$ .If the corresponding $b i \neq 0$ ,then the solution set is aparaboloid(either elliptic or hyperbolic); if the corresponding $b i = 0$ ,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 calledintegral quadratic forms,whereas the corresponding modules arequadratic lattices(sometimes, simplylattices). They play an important role innumber theoryandtopology.

An integral quadratic form has integer coefficients, such as $x 2 + xy + y 2$ ;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 integralwith respect to $Λ$ if and only if it is integer-valued on $Λ$ ,meaning $Q (x, y) \in Z$ if $x, y \in Λ$ .

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 ofintegral quadratic formshould 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 $ax 2 + 2 bxy + cy 2$ ,represented by the symmetric matrix ${\begin{pmatrix}a&b\\b&c\end{pmatrix}}$ This is the conventionGaussuses inDisquisitiones 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 thattwos outhas been adopted as the standard convention. Those include:

better understanding of the 2-adic theory of quadratic forms, the 'local' source of the difficulty;
thelatticepoint 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 intopologyforintersection theory;
theLie groupandalgebraic groupaspects.

Universal quadratic forms

An integral quadratic form whose image consists of all the positive integers is sometimes calleduniversal.Lagrange's four-square theoremshows that $w 2 + x 2 + y 2 + z 2$ is universal.Ramanujangeneralized this $aw 2 + bx 2 + cy 2 + dz 2$ and found 54 multisets ${a, b, c, d}$ that can each generate all positive integers, namely,

${1, 1, 1, d}, 1 \leq d \leq 7$
${1, 1, 2, d}, 2 \leq d \leq 14$
${1, 1, 3, d}, 3 \leq d \leq 6$
${1, 2, 2, d}, 2 \leq d \leq 7$
${1, 2, 3, d}, 3 \leq d \leq 10$
${1, 2, 4, d}, 4 \leq d \leq 14$
${1, 2, 5, d}, 6 \leq d \leq 10$

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, the15 and 290 theoremshave 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.

Notes

^A tradition going back toGaussdictates the use of manifestly even coefficients for the products of distinct variables, that is, $2 b$ in place of $b$ in binary forms and $2 b$ , $2 d$ , $2 f$ in place of $b$ , $d$ , $f$ in ternary forms. Both conventions occur in the literature.
^away from 2,that is, if 2 is invertible in the ring, quadratic forms are equivalent tosymmetric bilinear forms(by thepolarization 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 squaresviaHathiTrust
^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 theArf 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

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.;Remizov, A. O. (2012).Linear Algebra and Geometry.Springer.ISBN 978-3-642-30993-9.

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

[1] A tradition going back toGaussdictates the use of manifestly even coefficients for the products of distinct variables, that is, $2 b$ in place of $b$ in binary forms and $2 b$ , $2 d$ , $2 f$ in place of $b$ , $d$ , $f$ in ternary forms. Both conventions occur in the literature.

[2] way from 2,that is, if 2 is invertible in the ring, quadratic forms are equivalent tosymmetric bilinear forms(by thepolarization identities), but at 2 they are different concepts; this distinction is particularly important for quadratic forms over the integers.

[3] Babylonian Pythagoras

[4] Brahmagupta biography

[5] Maxime Bôcher(with E.P.R. DuVal)(1907)Introduction to Higher Algebra,§ 45 Reduction of a quadratic form to a sum of squaresviaHathiTrust

[6] If a non-strict inequality (with ≥ or ≤) holds then the quadratic form $q$ is called semidefinite.

[7] The theory of quadratic forms over a field of characteristic 2 has important differences and many definitions and theorems must be modified.

[8] This alternating form associated with a quadratic form in characteristic 2 is of interest related to theArf invariant–Irving Kaplansky (1974),Linear Algebra and Geometry,p. 27.

[9] 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 $R$ .

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