Positive Definite Quadratic form

an expression of the form

(where a_ik = a_ki) that assumes nonnegative values for all real values of x₁, x₂, …, x_n and that vanishes only when x₁ = x₂ = … = x_n = 0. Thus, the positive definite quadratic form is a special case of a quadratic form. Any positive definite quadratic form can be reduced to the form

by means of a linear transformation. In order for

to be a positive definite quadratic form, it is necessary and sufficient that Δ₁ > 0…..Δ_n > 0, where

In any affine coordinate system the distance of a point from the origin is expressed by a positive definite quadratic form in the coordinates of the point.

A Hermitian positive definite quadratic form is the form

such that a_ik = ā_ki, f ≥ 0 for all values of x₁, x₂, …, x_n, and f = 0 only when x₁ = x₂ = … = x_n = 0; here overbar denotes the operation of complex conjugation.

The following concepts are also associated with positive definite quadratic forms: (1) positive definite matrix ǀǀa_ikǀǀⁿ, which is a matrix such that

is a Hermitian positive definite form; (2) positive definite kernel, which is a function K(x,y) = K(y,x) such that

for any function ξ(ξ) with integrable square; and (3) positive definite function, which is a function f(x) such that the kernel K(x,y,) = f(x — y) is positive definite. The class of continuous positive definite functions f(x) with f(0) = 1 coincides with the class of characteristic functions of the laws governing the distribution of random variables.