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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 .

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)
How to Cite This Entry:
Positive-definite form. BSE-3 (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Positive-definite_form&oldid=12091
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098