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.
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).
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 .
|[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)|
Positive-definite form. BSE-3 (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Positive-definite_form&oldid=12091