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.

References in periodicals archive ?
Solving such an inverse problem requires finding the potential energy of the system using data obtained from observations of the oscillations in it, they say, and because the oscillations are small, the potential energy is given by a positive definite quadratic form, the matrix of which is called the matrix of potential energy.
With the purpose to analyse the properties of the linear mapping (2.1), we may form a positive definite quadratic form function K(p, q) on E x E simply by taking
As about the possibilities in the product spaces, due to Schur's Theorem, we would like to recall that for any two positive definite quadratic form functions [K.sub.1](p, q) and [K.sub.2](p, q) on E x E, the usual product K(p, q) = [K.sub.1](p, q)[K.sub.2](p, q) is again a positive definite quadratic form function on E.
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Hence, the image space that means the existence of the solution of (3.4) or (3.8) is characterized as the reproducing kernel Hilbert space admitting the positive definite quadratic form function
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are the components of a positive definite quadratic form on [R.sup.m], [3],
[Mathematical Expression Omitted] is linearly spanned by theta series attached to quadratic forms in the genus of a specific totally positive definite quadratic form Q.
Computational geometry of positive definite quadratic forms; polyhedral reduction theories, algorithms, and applications.