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).

Comments

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 $.

References

[a1]	E. Lukacs, "Characteristic functions" , Griffin (1970)
[a2]	P.P. Zabreiko (ed.) A.I. Koshelev (ed.) M.A. Krasnoselskii (ed.) S.G. Mikhlin (ed.) L.S. Rakovshchik (ed.) V.Ya. Stet'senko (ed.) T.O. Shaposhnikova (ed.) R.S. Anderssen (ed.) , Integral equations - a reference text , Noordhoff (1975) pp. Chapt. III, §3 (Translated from Russian)
[a3]	H. Hochstadt, "Integral equations" , Wiley (1973) pp. 255ff
[a4]	F.R. [F.R. Gantmakher] Gantmacher, "The theory of matrices" , I-II , Chelsea, reprint (1959) pp. Chapt. X (Translated from Russian)