Namespaces
Variants
Actions

Difference between revisions of "Positive-definite form"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (tex encoded by computer)
m (Undo revision 48248 by Ulf Rehmann (talk))
Tag: Undo
Line 1: Line 1:
−
<!--
 
−
p0738801.png
 
−
$#A+1 = 29 n = 0
 
−
$#C+1 = 29 : ~/encyclopedia/old_files/data/P073/P.0703880 Positive\AAhdefinite form
 
−
Automatically converted into TeX, above some diagnostics.
 
−
Please remove this comment and the {{TEX|auto}} line below,
 
−
if TeX found to be correct.
 
−
-->
 
−
 
−
{{TEX|auto}}
 
−
{{TEX|done}}
 
−
 
 
An expression
 
An expression
  
−
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073880/p0738801.png" /></td> </tr></table>
−
\sum _ {i,k= 1 } ^ { n }  a _ {ik} x _ {i} x _ {k} ,
 
−
$$
 
  
−
where $  a _ {ik} = a _ {ki} $,  
+
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
−
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|quadratic form]] of special type. Any positive-definite form can be converted by a linear transformation to the representation
 
  
−
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073880/p0738805.png" /></td> </tr></table>
−
\sum _ { i= } 1 ^ { n }  x _ {i}  ^ {2} .
 
−
$$
 
  
 
In order that a form
 
In order that a form
  
−
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073880/p0738806.png" /></td> </tr></table>
−
\sum _ {i,k= 1 } ^ { n }  a _ {ik} x _ {i} x _ {k}  $$
 
  
−
be positive definite, it is necessary and sufficient that $  \Delta _ {1} > 0 \dots \Delta _ {n} > 0 $,  
+
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
−
where
 
  
−
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073880/p0738808.png" /></td> </tr></table>
−
\Delta _ {k}  = \left |
 
  
 
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.
 
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.
Line 41: Line 19:
 
A form
 
A form
  
−
$$
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073880/p0738809.png" /></td> </tr></table>
−
f  = \sum _ {i,k= 1 } ^ { n }  a _ {ik} x _ {i} \overline{x}\; _ {k}  $$
+
 
 +
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
  
−
such that  $  a _ {ik} = \overline{a}\; _ {ki} $
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073880/p07388018.png" /></td> </tr></table>
−
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} $
+
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]]).
−
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|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|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|Characteristic function]]).
 
  
 
====Comments====
 
====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 $
+
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" />.
−
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====
 
====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>
 
<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>

Revision as of 14:52, 7 June 2020

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. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Positive-definite_form&oldid=49368
This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
Retrieved from "https://encyclopediaofmath.org/index.php?title=Positive-definite_form&oldid=49368"