# Positive Definite Quadratic form

## Positive Definite Quadratic form

an expression of the form

(where a_{ik} = a_{ki}) that assumes nonnegative values for all real values of *x _{1}, x_{2}, …, x_{n}* and that vanishes only when

*x*= 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

_{1}= x_{2}= … = x_{n}by means of a linear transformation. In order for

to be a positive definite quadratic form, it is necessary and sufficient that Δ_{1} > 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

*x*, and

_{1}, x_{2}, …, x_{n}*f = 0*only when

*x*= 0; here overbar denotes the operation of complex conjugation.

_{1}= x_{2}= … = x_{n}The following concepts are also associated with positive definite quadratic forms: (1) positive definite matrix ǀǀa_{ik}ǀǀ^{n}, 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.