View source for Positive-definite form

<!--
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

$$ 
\sum _ {i,k= 1 } ^ { n }  a _ {ik} x _ {i} x _ {k} ,
$$

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|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|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====
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====
<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>