Positive-definite form
An expression
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 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 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 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).
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
[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. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Positive-definite_form&oldid=54878