# Positive-definite form

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An expression where , which takes non-negative values for any real values and vanishes only for . Therefore, a positive-definite form is a quadratic form of special type. Any positive-definite form can be converted by a linear transformation to the representation In order that a form be positive definite, it is necessary and sufficient that , where In any affine coordinate system, the distance of a point from the origin is expressed by a positive-definite form in the coordinates of the point.

A form such that and for all values of and only for is called a Hermitian positive-definite form.

The following concepts are related to the concept of a positive-definite form: 1) a positive-definite matrix is a matrix such that is a Hermitian positive-definite form; 2) a positive-definite kernel is a function such that for every function with an integrable square; 3) a positive-definite function is a function such that the kernel is positive definite. By Bochner's theorem, the class of continuous positive-definite functions with coincides with the class of characteristic functions of distributions of random variables (cf. Characteristic function).

#### Comments

A kernel that is semi-positive definite (non-negative definite) is one that satisfies for all . Such a kernel is sometimes also simply called positive. However, the phrase "positive kernel" is also used for the weaker notion (almost-everywhere). A positive kernel in the latter sense has at least one eigen value while a semi-positive definite kernel has all eigen values .