Positive Definite Quadratic form

Positive Definite Quadratic form

 

an expression of the form

(where aik = aki) that assumes nonnegative values for all real values of x1, x2, …, xn and that vanishes only when x1 = x2 = … = xn = 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…..Δn > 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 aik = āki, f ≥ 0 for all values of x1, x2, …, xn, and f = 0 only when x1 = x2 = … = xn = 0; here overbar denotes the operation of complex conjugation.

The following concepts are also associated with positive definite quadratic forms: (1) positive definite matrix ǀǀaikǀǀn, 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.

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are the components of a positive definite quadratic form on [R.