Positive-definite function: Difference between revisions

Revision as of 08:06, 20 June 2023

Revision as of 08:19, 20 June 2023

Line 3:

== 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 function <math> f: \mathbb{R} \to \mathbb{C} </math> is called ''positive semi-definite'' if for any{{clarify|reason="Any" here is ambiguous. Does this mean "for all sets of real numbers x1, …, xn" or "there exists a set of real numbers x1, …, xn"?}} 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 matrix]].{{citation (whichneeded}}

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

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