Inmathematics, aquadratic form is apolynomial with terms all ofdegree two ("form" is another name for ahomogeneous polynomial). For example,
is a quadratic form in the variablesx andy. The coefficients usually belong to a fixedfieldK, such as thereal orcomplex numbers, and one speaks of a quadratic formoverK. Over the reals, a quadratic form is said to bedefinite if it takes the value zero only when all its variables are simultaneously zero; otherwise it isisotropic.
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 forms ofmanifolds, especiallyfour-manifolds),Lie theory (theKilling form), andstatistics (where the exponent of a zero-meanmultivariate normal distribution has the quadratic form)
Quadratic forms are not to be confused withquadratic 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 offorms.
Quadratic forms are homogeneous quadratic polynomials inn variables. In the cases of one, two, and three variables they are calledunary,binary, andternary and have the following explicit form:
wherea, ...,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 bereal orcomplex 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 integersZ or thep-adic integersZp.[2]Binary quadratic forms have been extensively studied innumber theory, in particular, in the theory ofquadratic fields,continued fractions, andmodular forms. The theory of integral quadratic forms inn variables has important applications toalgebraic topology.
Usinghomogeneous coordinates, a non-zero quadratic form inn variables defines an(n − 2)-dimensionalquadric in 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 space and thesquare of theEuclidean norm expressing thedistance between a point with coordinates(x,y,z) and the origin:
A closely related notion with geometric overtones is aquadratic space, which is a pair(V,q), withV avector space over a fieldK, andq :V →K a quadratic form onV. See§ Definitions below for the definition of a quadratic form on a vector space.
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 formx2 +y2, wherex,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 mathematicianBrahmagupta wroteBrāhmasphuṭasiddhānta, which includes, among many other things, a study of equations of the formx2 −ny2 =c. He considered what is now calledPell's equation,x2 −ny2 = 1, and found a method for its solution.[4] In Europe this problem was studied byBrouncker,Euler andLagrange.
In 1801Gauss publishedDisquisitiones Arithmeticae, a major portion of which was devoted to a complete theory ofbinary quadratic forms over 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.
Anyn ×n matrixA determines a quadratic formqA inn variables bywhereA = (aij).
Consider the case of quadratic forms in three variablesx,y,z. The matrixA 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 sumsb +d,c +g andf +h. In particular, the quadratic formqA is defined by a uniquesymmetric matrix
This generalizes to any number of variables as follows.
Given a quadratic formqA over the real numbers, defined by the matrix with indiciesi andj independently varying from1 throughn ≥ 2, the matrix
issymmetric, defines the same quadratic form asA, and is the unique symmetric matrix that definesqA.
So, over the real numbers (and, more generally, over afield ofcharacteristic different from two), there is aone-to-one correspondence between quadratic forms andsymmetric matrices that determine them.
A fundamental problem is the classification of real quadratic forms under alinear change of variables.
Jacobi proved that, for every real quadratic form, there is anorthogonal diagonalization; that is, anorthogonal change of variables that puts the quadratic form in a "diagonal form"where the associated symmetric matrix isdiagonal. Moreover, the coefficientsλ1,λ2, ...,λn are determined uniquelyup to apermutation.[5]
If the change of variables is given by aninvertible matrix that is not necessarily orthogonal, one can suppose that all coefficientsλi are0, +1, or−1.Sylvester's law of inertia states that the numbers of each0,+1, and−1 areinvariants of the quadratic form, in the sense that any other diagonalization will contain the same number of each. Thesignature of the quadratic form is the triple(n0,n+,n− ), where these components count the number of0s, 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 allλi 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 are0, 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 anondegeneratebilinear form. A real vector space with an indefinite nondegenerate quadratic form of index(p,q) (countp of+1s, andq of−1s) is often denoted asℝp,q particularly in the physical theory ofspacetime.
Thediscriminant of a quadratic form, concretely the class of the determinant of a representing matrix inK / (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.
Letq be a quadratic form defined on ann-dimensionalreal vector space. LetA be the matrix of the quadratic formq in a given basis. This means thatA is a symmetricn ×n matrix such thatwherex is the column vector of coordinates ofv in the chosen basis. Under a change of basis, the columnx is multiplied on the left by ann ×ninvertible matrixS, and the symmetric squarematrixA is transformed into another symmetric squarematrixB of the same size according to the formula
Any symmetric matrixA can be transformed into a diagonal matrixby a suitable choice of anorthogonal matrixS, and the diagonal entries ofB are uniquely determined – this is Jacobi's theorem (seeskew-symmetric matrix). IfS is allowed to beany invertible matrix thenB 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, andn− for −1) depends onlyonA. This is one of the formulations of Sylvester's law of inertia and the numbersn+ andn− are called thepositive andnegativeindices of inertia. Although their definition involved a choice of basis and consideration of the corresponding real symmetricmatrixA, Sylvester's law of inertia means that they are invariants of the quadraticformq.
The quadratic formq is positive definite ifq(v) > 0 (similarly, negative definiteifq(v) < 0) for every nonzero vectorv.[6] Whenq(v) assumes both positive and negative values,q is anisotropic quadratic form. The theorems of Jacobi andSylvester show that any positive definite quadratic form inn variables can be brought to the sum ofn squares by a suitable invertible linear transformation: geometrically, there is onlyone positive definite real quadratic form of every dimension. Itsisometry group is acompactorthogonal group, conventionally notated asO(n). This stands in contrast with the case of isotropic forms, when the corresponding group, theindefinite orthogonal groupO(p,q), is non-compact. Further, theisometry groups ofQ and−Q are the same(O(p,q) ≈ O(q,p) ), but the associatedClifford algebras (and hencepin groups) are different.
Aquadratic form over a fieldK is a mapq :V →K from a finite-dimensionalK-vector space toK such thatq(av) =a2q(v) for alla ∈K,v ∈V and the functionq(u +v) −q(u) −q(v) is abilinear form.
More concretely, ann-aryquadratic form over a fieldK is ahomogeneous polynomial of degree 2 inn variables with coefficients inK:
This formula may be rewritten using matrices: letx be thecolumn vector with componentsx1, ...,xn andA = (aij) be then ×n matrix overK whose entries are the coefficients ofq. Then
A vectorv = (x1, ...,xn) is anull vector ifq(v) = 0.
Twon-ary quadratic formsφ andψ overK areequivalent if there exists a nonsingular linear transformationC ∈GL(n,K) such that
Let thecharacteristic ofK be different from 2.[7] The coefficient matrixA ofq may be replaced by thesymmetric matrix(A +AT)/2 with the same quadratic form, so it may be assumed from the outset thatA is symmetric. Moreover, a symmetric matrixA is uniquely determined by the corresponding quadratic form. Under an equivalenceC, the symmetric matrixA ofφ and the symmetric matrixB ofψ are related as follows:
Theassociated bilinear form of a quadratic formq is defined by
Thus,bq is asymmetric bilinear form overK with matrixA. Conversely, any symmetric bilinear formb defines a quadratic formand 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 inn variables are essentially the same.
Given ann-dimensionalvector spaceV over a fieldK, aquadratic form onV is afunctionQ :V →K that has the following property: for some basis, the functionq that maps the coordinates ofv ∈V toQ(v) is a quadratic form. In particular, ifV =Kn with itsstandard basis, one has
Thechange of basis formulas show that the property of being a quadratic form does not depend on the choice of a specific basis inV, although the quadratic formq depends on the choice of the basis.
A finite-dimensional vector space with a quadratic form is called aquadratic space.
The mapQ is ahomogeneous function of degree 2, which means that it has the property that, for alla inK andv inV:
When the characteristic ofK is not 2, the bilinear mapB :V ×V →K overK is defined:This bilinear formB is symmetric. That is,B(x,y) =B(y,x) for allx,y inV, and it determinesQ:Q(x) =B(x,x) for allx inV.
When the characteristic ofK is 2, so that 2 is not aunit, it is still possible to use a quadratic form to define a symmetric bilinear formB′(x,y) =Q(x +y) −Q(x) −Q(y). However,Q(x) can no longer be recovered from thisB′ in the same way, sinceB′(x,x) = 0 for allx (and is thus alternating).[8] Alternatively, there always exists a bilinear formB″ (not in general either unique or symmetric) such thatB″(x,x) =Q(x).
The pair(V,Q) consisting of a finite-dimensional vector spaceV overK and a quadratic mapQ fromV toK is called aquadratic space, andB as defined here is the associated symmetric bilinear form ofQ. 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.
Twon-dimensional quadratic spaces(V,Q) and(V′,Q′) areisometric if there exists an invertible linear transformationT :V →V′ (isometry) such that
The isometry classes ofn-dimensional quadratic spaces overK correspond to the equivalence classes ofn-ary quadratic forms overK.
LetR be acommutative ring,M be anR-module, andb :M ×M →R be anR-bilinear form.[9] A mappingq :M →R :v ↦b(v,v) is theassociated quadratic form ofb, andB :M ×M →R : (u,v) ↦q(u +v) −q(u) −q(v) is thepolar form ofq.
A quadratic formq :M →R may be characterized in the following equivalent ways:
Two elementsv andw ofV are calledorthogonal ifB(v,w) = 0. Thekernel of a bilinear formB consists of the elements that are orthogonal to every element ofV.Q isnon-singular if the kernel of its associated bilinear form is{0}. If there exists a non-zerov inV such thatQ(v) = 0, the quadratic formQ isisotropic, otherwise it isdefinite. This terminology also applies to vectors and subspaces of a quadratic space. If the restriction ofQ to a subspaceU ofV is identically zero, thenU istotally singular.
The orthogonal group of a non-singular quadratic formQ is the group of the linear automorphisms ofV that preserveQ: that is, the group of isometries of(V,Q) into itself.
If a quadratic space(A,Q) has a product so thatA is analgebra over a field, and satisfies then it is acomposition algebra.
Every quadratic formq inn variables over a field of characteristic not equal to 2 isequivalent to adiagonal 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.
UsingCartesian coordinates in three dimensions, let and letA be asymmetric 3-by-3 matrix. Then the geometric nature of thesolution set of the equation depends on the eigenvalues of the matrixA.
If alleigenvalues ofA are non-zero, then the solution set is anellipsoid or 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 the eigenvalues are all equal and positive, then it is a sphere (special case of an ellipsoid with all axes equal, corresponding to the presence of equal eigenvalues).
If there exist one or more eigenvaluesλi = 0, then the shape depends on the correspondingbi. If the correspondingbi ≠ 0, then the solution set is aparaboloid (either elliptic or hyperbolic); if the correspondingbi = 0, then the dimensioni degenerates and does not come into play, and the geometric meaning will be determined by other eigenvalues and other components ofb. When the solution set is aparaboloid, 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 calledintegral quadratic forms, whereas the corresponding modules arequadratic lattices (sometimes, simplylattices). They play an important role innumber theory andtopology.
An integral quadratic form has integer coefficients, such asx2 +xy +y2; equivalently, given a latticeΛ in a vector spaceV (over a field with characteristic 0, such asQ orR), a quadratic formQ is integralwith respect toΛ if and only if it is integer-valued onΛ, meaningQ(x,y) ∈Z ifx,y ∈ Λ.
This is the current use of the term; in the past it was sometimes used differently, as detailed below.
Historically there was some confusion and controversy over whether the notion ofintegral quadratic form should mean:
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 formax2 + 2bxy +cy2, represented by the symmetric matrixThis is the conventionGauss uses inDisquisitiones Arithmeticae.
In "twos out", binary quadratic forms are of the formax2 +bxy +cy2, represented by the symmetric matrix
Several points of view mean thattwos out has been adopted as the standard convention. Those include:
An integral quadratic form whose image consists of all the positive integers is sometimes calleduniversal.Lagrange's four-square theorem shows thatw2 +x2 +y2 +z2 is universal.Ramanujan generalized thisaw2 +bx2 +cy2 +dz2 and found 54 multisets{a,b,c,d} that can each generate all positive integers, namely,
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 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.
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