View source for Positive-definite form

An expression

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073880/p0738802.png" />, which takes non-negative values for any real values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073880/p0738803.png" /> and vanishes only for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073880/p0738804.png" />. Therefore, a positive-definite form is a [[Quadratic form|quadratic form]] of special type. Any positive-definite form can be converted by a linear transformation to the representation

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In order that a form

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be positive definite, it is necessary and sufficient that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073880/p0738807.png" />, where

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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

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such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073880/p07388010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073880/p07388011.png" /> for all values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073880/p07388012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073880/p07388013.png" /> only for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073880/p07388014.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073880/p07388015.png" /> is a matrix such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073880/p07388016.png" /> is a Hermitian positive-definite form; 2) a [[Positive-definite kernel|positive-definite kernel]] is a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073880/p07388017.png" /> such that

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for every function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073880/p07388019.png" /> with an integrable square; 3) a [[Positive-definite function|positive-definite function]] is a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073880/p07388020.png" /> such that the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073880/p07388021.png" /> is positive definite. By Bochner's theorem, the class of continuous positive-definite functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073880/p07388022.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073880/p07388023.png" /> coincides with the class of characteristic functions of distributions of random variables (cf. [[Characteristic function|Characteristic function]]).



====Comments====
A kernel that is semi-positive definite (non-negative definite) is one that satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073880/p07388024.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073880/p07388025.png" />. Such a kernel is sometimes also simply called positive. However, the phrase  "positive kernel"  is also used for the weaker notion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073880/p07388026.png" /> (almost-everywhere). A positive kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073880/p07388027.png" /> in the latter sense has at least one eigen value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073880/p07388028.png" /> while a semi-positive definite kernel has all eigen values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073880/p07388029.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Lukacs,   "Characteristic functions" , Griffin  (1970)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Hochstadt,   "Integral equations" , Wiley  (1973)  pp. 255ff</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  F.R. [F.R. Gantmakher] Gantmacher,   "The theory of matrices" , '''I-II''' , Chelsea, reprint  (1959)  pp. Chapt. X  (Translated from Russian)</TD></TR></table>