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Concept
Quadratic form
Summary
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
is a quadratic form in the variables x and y. 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. If , and the quadratic form equals zero only when all variables are simultaneously zero, then it is a definite quadratic form; otherwise it is an isotropic quadratic form.
Quadratic forms occupy a central place in various branches of mathematics, including number theory, linear algebra, group theory (orthogonal groups), differential geometry (the Riemannian metric, the second fundamental form), differential topology (intersection forms of four-manifolds), Lie theory (the Killing form), and statistics (where the exponent of a zero-mean multivariate normal distribution has the quadratic form )
Quadratic forms are not to be confused with a quadratic 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 of homogeneous polynomials.
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:
where a, ..., f are the coefficients.
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 Zp. Binary quadratic forms have been extensively studied in number theory, in particular, in the theory of quadratic fields, continued fractions, and modular forms.
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