Quadratic form

In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,

4 x 2 + 2 x y − 3 y 2 {\displaystyle 4x^{2}+2xy-3y^{2}}

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. 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. 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 manifolds, especially four-manifolds), Lie theory (the Killing form), and statistics (where the exponent of e {\displaystyle e} of a zero-mean multivariate normal distribution has the quadratic form (−1/2)xT‍Σ−1‍x). Quadratic forms are not to be confused with quadratic 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 of forms.

Introduction 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:

q ( x ) = a x 2 (unary) q ( x , y ) = a x 2 + b x y + c y 2 (binary) q ( x , y , z ) = a x 2 + b x y + c y 2 + d y z + e z 2 + f x z (ternary) {\displaystyle {\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 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. 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 projective quadric in the (n − 1)-dimensional projective space. In this way one may visualize 3-dimensional real quadratic forms as conic sections. An example is given by the three-dimensional Euclidean space and the square of the Euclidean norm expressing the distance between a point with coordinates (x, y, z) and the origin:

q ( x , y , z ) = d ( ( x , y , z ) , ( 0 , 0 , 0 ) ) 2 = ‖ ( x , y , z ) ‖ 2 = x 2 + y 2 + z 2 . {\displaystyle q(x,y,z)=d((x,y,z),(0,0,0))^{2}=\left\|(x,y,z)\right\|^{2}=x^{2}+y^{2}+z^{2}.}

… excerpt ends here. Continue reading the full article.