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Disambiguate the definition in dynamic systems versus the most common contexts in machine learning, statistics, and pure math. |
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==In dynamical systems==
A [[real number|real]]-valued, continuously differentiable [[function (mathematics)|function]] ''f'' is '''positive definite''' on a neighborhood of the origin, ''D'', 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|publisher=Springer|year=1967}}</ref> This definition is in clear conflict with the one below.
A function is '''negative definite''' if the inequality is reversed. A function is '''semidefinite''' if the strong inequality is replaced with a weak (≤,≥0)▼
== Most common usage ==
A '''positive-definite function''' of a real variable ''x'' is a [[complex number|complex]]-valued function ''f'':'''R''' → '''C''' such that for any real numbers ''x''<sub>1</sub>, ..., ''x''<sub>n</sub> the ''n''×''n'' [[matrix (mathematics)|matrix]]
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(these inequalities follow from the condition for ''n''=1,2.)
▲A function is '''negative definite''' if the inequality is reversed. A function is '''semidefinite''' if the strong inequality is replaced with a weak (≤,≥0).
===Bochner's theorem===
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