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. Over the reals, a quadratic form is said to be definite if it takes the value zero only when all its variables are simultaneously zero; otherwise it is isotropic.
In the cases of one, two, and three variables they are called unary, binary, and ternary and have the following explicit form:
In linear algebra, analytic geometry, and in the majority of applications of quadratic forms, the coefficients are real or complex numbers.
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.
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.
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 problem is the classification of real quadratic forms under a linear change of variables.
[5] 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 shows that this is a well-defined quantity attached to the quadratic form.
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−.
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.
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 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.
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 column vector with components x1, ..., xn and A = (aij) be the n × n matrix over K whose entries are the coefficients of q.
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
Moreover, a symmetric matrix A is uniquely determined by the corresponding 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
A quadratic form q : M → R may be characterized in the following equivalent ways: Two elements v and w of V are called orthogonal if B(v, w) = 0.
Then the geometric nature of the solution set of the equation xTAx + bTx = 1 depends on the eigenvalues of the matrix A.
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.
Several points of view mean that twos out has been adopted as the standard convention.
Those include: An integral quadratic form whose image consists of all the positive integers is sometimes called universal.