Positive-definite function: Difference between revisions

Revision as of 07:06, 17 March 2016

Revision as of 07:07, 17 March 2016

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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>

A function is '''negative definite''' if the inequality is reversed. A function is '''semidefinite''' if the strong inequality is replaced with a weak (<math> \geq\,</math> or <math> \leq\,</math>) one.siva0

==In analysisSIVA

A '''positive-definite function''' of a real variable ''x'' is a [[complex number|complex]]-valued function ''f'':'''R''' &rarr; '''C''' such that for any real numbers ''x''₁, ..., ''x''_n the ''n''&times;''n'' [[matrix (mathematics)|matrix]]