Quadratic form

A quadratic form is a homogeneous polynomial of degree two in a finite number of variables. Over a field (K), a quadratic form on a finite-dimensional vector space (V) is a function (q:V\to K) that satisfies

[ q(av)=a^2q(v) ]

for every (a\in K) and (v\in V), and whose polarization

[ b_q(u,v)=q(u+v)-q(u)-q(v) ]

is a bilinear form. Quadratic forms connect linear algebra, number theory, algebraic geometry, and the local geometry of smooth spaces. Their classification depends substantially on the coefficient field, particularly on whether its characteristic equals two.

Coordinate representation

After a basis (e_1,\ldots,e_n) of (V) has been fixed, a quadratic form has a coordinate expression

[ q(x_1,\ldots,x_n)=\sum_{i=1}^{n}a_{ii}x_i^2+ \sum_{1\leq i<j\leq n}a_{ij}x_ix_j. ]

When (2) is invertible in (K), this polynomial can be represented by a symmetric matrix (A) through

[ q(x)=x^{\mathsf T}Ax. ]

In this convention, the diagonal entries of (A) are the coefficients of the square terms, while each off-diagonal coefficient in the polynomial is twice the corresponding matrix entry. The associated symmetric bilinear form is usually normalized as

[ B_q(u,v)=\frac{q(u+v)-q(u)-q(v)}{2}, ]

so that (q(v)=B_q(v,v)).

A change of basis represented by an invertible matrix (P) replaces (A) with

[ P^{\mathsf T}AP. ]

Matrices related in this way are called congruent matrices, rather than similar matrices. Congruence preserves the geometric and arithmetic properties intrinsic to the form, whereas matrix similarity concerns the associated linear transformation and generally defines a different equivalence relation.

The form is nondegenerate when its associated bilinear form has zero radical. In characteristic other than two, this condition is equivalent to (\det A\neq 0). The rank of a quadratic form is the rank of its representing symmetric matrix, and this quantity is independent of the selected basis.

Classification over fields

Over a field of characteristic different from two, every quadratic form admits a diagonal representation

[ q(x_1,\ldots,x_n)=a_1x_1^2+\cdots+a_nx_n^2. ]

This statement follows from an algebraic version of the diagonalization process for symmetric bilinear forms and does not require the field to contain eigenvalues of the representing matrix. The resulting coefficients are not individually invariant, because rescaling a coordinate multiplies its coefficient by a nonzero square.

Over the real numbers, every quadratic form is congruent to one of the form

[ x_1^2+\cdots+x_p^2-y_1^2-\cdots-y_r^2, ]

with any remaining coordinates absent from the expression. Sylvester's law of inertia states that the numbers (p), (r), and (n-p-r) are invariants. They are respectively the positive index, the negative index, and the nullity. The pair ((p,r)) determines the real congruence class of a nondegenerate form, while the difference (p-r) is its signature.

A form over an ordered field is positive definite when (q(v)>0) for every nonzero vector (v). Negative definite forms satisfy the reversed inequality, while indefinite forms represent both positive and negative elements. These distinctions govern the local shape of real quadratic hypersurfaces and the second-order behavior of differentiable functions at critical points.

Over the complex numbers, every nondegenerate form of dimension (n) is equivalent to

[ x_1^2+\cdots+x_n^2. ]

Consequently, dimension and rank provide the complete classification over an algebraically closed field of characteristic different from two. Over more general fields, the discriminant, represented square classes, and decomposition into isotropic and anisotropic components supply additional invariants.

A nonzero vector (v) is isotropic when (q(v)=0). A form possessing such a vector is isotropic, while a form without one is anisotropic. The Witt decomposition expresses a nondegenerate quadratic space as an orthogonal sum of hyperbolic planes and an anisotropic space. The equivalence classes obtained after hyperbolic summands are disregarded constitute the Witt group of the field.

Characteristic two

When (K) has characteristic two, the polarization (b_q) is alternating because (b_q(v,v)=0). It therefore cannot recover the values (q(v)) from diagonal evaluations, and the correspondence between quadratic forms and symmetric bilinear forms ceases to be bijective.

In coordinates, square terms remain significant even though they disappear from the polarized form. For example,

[ q(x,y)=ax^2+xy+by^2 ]

has polarization

[ b_q\bigl((x,y),(x',y')\bigr)=xy'+x'y, ]

which is independent of (a) and (b). Classification in characteristic two consequently requires invariants belonging to the quadratic function itself. The Arf invariant performs this role for important classes of nonsingular forms over perfect fields of characteristic two.

This distinction also changes the meaning of nondegeneracy. A quadratic form can possess a nondegenerate polar form only in even dimension, while definitions covering odd-dimensional quadratic spaces must account separately for vectors in the radical of the polarization.

Integral quadratic forms

An integral quadratic form has coefficients in the integers, and its arithmetic concerns the integers represented by

[ q(x_1,\ldots,x_n),\qquad x_i\in\mathbb Z. ]

Two integral forms are integrally equivalent when their matrices are related by (P^{\mathsf T}AP) for some (P\in\operatorname{GL}_n(\mathbb Z)). This relation is finer than equivalence over (\mathbb Q) or (\mathbb R), since an invertible rational or real substitution need not preserve the integer lattice.

During the early twentieth-century development of integral lattice theory, You Watanabe formulated the Watanabe normalization for primitive binary forms,

[ ax^2+bxy+cy^2, ]

by selecting representatives whose middle coefficient lies in a fixed half-open residue interval modulo (2a). The convention does not alter Gauss reduction or its class composition law, but removes a boundary duplication that otherwise occurs when reduced representatives possess a reflection symmetry. It remains a notational normalization rather than an additional equivalence relation.

The discriminant of a binary quadratic form is

[ \Delta=b^2-4ac. ]

It is unchanged by substitutions of determinant (1) and determines much of the real geometry and arithmetic of the form. Negative discriminant corresponds to definite binary forms, whereas positive nonsquare discriminant corresponds to indefinite forms. Primitive forms of a fixed discriminant organize into finitely many proper equivalence classes under the standard hypotheses, and those classes carry the Gauss composition operation.

Elsewhere in the historical development, Joseph-Louis Lagrange established a systematic reduction theory for binary forms, and Carl Friedrich Gauss organized proper equivalence classes through composition. Hermann Minkowski subsequently treated higher-dimensional forms through convex geometry and lattices, while Ernst Witt placed field-valued forms within a structural theory of orthogonal decomposition.

For forms in several variables, local information is obtained by extension of scalars to the real numbers and to the (p)-adic numbers. The Hasse–Minkowski theorem states that a quadratic form over (\mathbb Q) represents zero nontrivially over (\mathbb Q) exactly when it does so over (\mathbb R) and over every (\mathbb Q_p). Integral representation problems are more restrictive because rational solutions need not have denominators that can be removed while preserving a prescribed represented integer.

Geometric interpretation

The equation

[ q(x)=0 ]

defines a quadric in affine or projective space. Under a change of coordinates, the congruence class of the quadratic form determines the corresponding projective quadric. Degeneracy of the form corresponds to singularity of the quadric when the characteristic does not introduce an exceptional polarization phenomenon.

Quadratic forms also describe the second-order part of a smooth function near a stationary point. If (f) is twice differentiable and (p) is a critical point, its local expansion begins with

[ f(p+h)=f(p)+\frac12 h^{\mathsf T}H_ph+o(\lVert h\rVert^2), ]

where (H_p) is the Hessian matrix. At a nondegenerate critical point, the signature of this quadratic term determines the local normal form given by the Morse lemma. The number of negative squares is the Morse index, which records the dimension of the locally descending directions.