Positive-definite function: Difference between revisions

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== Definition 1 ==

Let <math>\mathbb{R}</math> be the set of [[real number]]s and <math>\mathbb{C}</math> be the set of [[complex number]]s.

A ''positive semi-definite function'' of a [[real number|real]] variable ''x'' is a [[complex number|complex]]-valued function <math> f: \mathbb{R} \to \mathbb{C} </math> such that for any real numbers ''x''₁, …, ''x''_''n'' the ''n''&thinsp;×&thinsp;''n'' [[matrix (mathematics)|matrix]]

A ''positive semi-definite function'' of a [[real number|real]] variable ''x'' is a [[complex number|complex]]-valued function <math> f: \mathbb{R} \to \mathbb{C} </math> such thatis called ''positive semi-definite'' if for anyall real numbers ''x''₁, …, ''x''_''n'' the ''n''&thinsp;×&thinsp;''n'' [[matrix (mathematics)|matrix]]

:<math> A = \left(a_{ij}\right)_{i,j=1}^n~, \quad a_{ij} = f(x_i - x_j) </math>

is a [[positive-definite matrix|positive ''semi-''definite]] (which requires ''A'' to be [[Hermitian matrix|Hermitian]];.{{citation thereforeneeded|date=June ''f''(−''x'') is the [[complex conjugate]] of ''f''(''x'')).2023}}

By definition, a positive semi-definite matrix, such as <math>A</math>, is [[Hermitian matrix|Hermitian]]; therefore ''f''(−''x'') is the [[complex conjugate]] of ''f''(''x'')).

In particular, it is necessary (but not sufficient) that

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==Definition 2==

Alternatively, a function <math>f : \reals^n \to \reals</math> is called ''positive-definite'' on a [[neighborhood (mathematics)|neighborhood]] ''D'' of the origin if <math>f(0) = 0</math> and <math>f(x) > 0</math> for every non-zero <math>x \in D</math>.<ref>{{cite book|last=Verhulst|first=Ferdinand|title=Nonlinear Differential Equations and Dynamical Systems|edition=2nd|publisher=Springer|year=1996|isbn=3-540-60934-2}}</ref><ref>{{cite book|last=Hahn|first=Wolfgang|title=Stability of Motion|url=https://archive.org/details/stabilityofmotio0000hahn|url-access=registration|publisher=Springer|year=1967}}</ref>

TheNote followingthat this definition conflicts with thedefinition 1, onegiven above.

In dynamical systems, a [[real number|real]]-valued, [[continuously differentiable function|continuously differentiable]] function ''f'' can be called ''positive-definite'' on a [[neighborhood (mathematics)|neighborhood]] ''D'' of the origin if <math>f(0) = 0</math> and <math>f(x) > 0</math> for every non-zero <math>x \in D</math>.<ref>{{cite book|last=Verhulst|first=Ferdinand|title=Nonlinear Differential Equations and Dynamical Systems|edition=2nd|publisher=Springer|year=1996|isbn=3-540-60934-2}}</ref><ref>{{cite book|last=Hahn|first=Wolfgang|title=Stability of Motion|url=https://archive.org/details/stabilityofmotio0000hahn|url-access=registration|publisher=Springer|year=1967}}</ref> In physics, the requirement that <math>f(0) = 0</math> mayis besometimes dropped (see, e.g., Corney and Olsen<ref>{{cite journal|first1=J. F.|last1=Corney|first2=M. K.|last2=Olsen|title=Non-Gaussian pure states and positive Wigner functions|journal=Physical Review A|date=19 February 2015|issn=1050-2947 |pages=023824|volume=91|issue=2|doi=10.1103/PhysRevA.91.023824|arxiv=1412.4868|bibcode=2015PhRvA..91b3824C|s2cid=119293595}}</ref>).

==See also==

* [[Positive definiteness]]

* [[Positive-definite kernel]]