Positive-definite form

An expression

$$\sum _ {i,k= 1 } ^ { n } a _ {ik} x _ {i} x _ {k} ,$$

where $a _ {ik} = a _ {ki}$, which takes non-negative values for any real values $x _ {1} \dots x _ {n}$ and vanishes only for $x _ {1} = \dots = x _ {n} = 0$. 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

$$\sum _ { i= } 1 ^ { n } x _ {i} ^ {2} .$$

In order that a form

$$\sum _ {i,k= 1 } ^ { n } a _ {ik} x _ {i} x _ {k}$$

be positive definite, it is necessary and sufficient that $\Delta _ {1} > 0 \dots \Delta _ {n} > 0$, where

$$\Delta _ {k} = \left | \begin{array}{ccc} a _ {11} &\dots &a _ {1k} \\ \cdot &\dots &\cdot \\ a _ {k1} &\dots &a _ {kk} \\ \end{array} \ \right | .$$

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

$$f = \sum _ {i,k= 1 } ^ { n } a _ {ik} x _ {i} \overline{x}\; _ {k}$$

such that $a _ {ik} = \overline{a}\; _ {ki}$ and $f \geq 0$ for all values of $x _ {1} \dots x _ {n}$ and $f = 0$ only for $x _ {1} = \dots = x _ {n} = 0$ 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 $\| a _ {ik} \| _ {1} ^ {n}$ is a matrix such that $\sum _ {i,k= 1 } ^ {n} a _ {ik} x _ {i} \overline{x}\; _ {k}$ is a Hermitian positive-definite form; 2) a positive-definite kernel is a function $K( x, y) = K( y, x)$ such that

$$\int\limits _ {- \infty } ^ \infty \int\limits _ {- \infty } ^ \infty K( x, y) \phi ( x) \overline{ {\phi ( y) }}\; dx dy \geq 0$$

for every function $\phi ( x)$ with an integrable square; 3) a positive-definite function is a function $f( x)$ such that the kernel $K( x, y) = f( x- y)$ is positive definite. By Bochner's theorem, the class of continuous positive-definite functions $f( x)$ with $f( 0) = 1$ coincides with the class of characteristic functions of distributions of random variables (cf. Characteristic function).

A kernel that is semi-positive definite (non-negative definite) is one that satisfies $\int K( x, y) \phi ( x) \overline{ {\phi ( y) }}\; dx dy \geq 0$ for all $\phi \in L _ {2}$. Such a kernel is sometimes also simply called positive. However, the phrase "positive kernel" is also used for the weaker notion $K( x, y) \geq 0$( almost-everywhere). A positive kernel $\neq 0$ in the latter sense has at least one eigen value $> 0$ while a semi-positive definite kernel has all eigen values $\geq 0$.