Homogeneous function: Difference between revisions

Revision as of 23:49, 7 September 2023

Revision as of 23:50, 7 September 2023

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The above definition extends to functions whose [[___domain of a function|___domain]] and [[codomain]] are [[vector space]]s over a [[Field (mathematics)|field]] {{mvar|F}}: a function <math>f : V \to W</math> between two {{mvar|F}}-vector spaces is ''homogeneous'' of degree <math>k</math> if

{{NumBlk|:|<math>f(s \mathbf{v}) = s^k f(\mathbf{v})</math>|{{EquationRef|1}}}}

for all nonzero <math>s \in F</math> and <math>v \in V.</math> This definition is often further generalized to functions whose ___domain is not {{mvar|V}}, but a [[cone (linear algebra)|cone]] in {{mvar|V}}, that is, a subset {{mvar|C}} of {{mvar|V}} such that <math>\mathbf{v}\in C</math> implies <math>(s \mathbf{v})\in C</math> for every nonzero scalar {{mvar|s}}.

In the case of [[functions of several real variables]] and [[real vector space]]s, a slightly more general form of homogeneity called '''positive homogeneity''' is often considered, by requiring only that the above identities hold for <math>s > 0,</math> and allowing any real number {{mvar|k}} as a degree of homogeneity. Every homogeneous real function is ''positively homogeneous''. The converse is not true, but is locally true in the sense that (for integer degrees) the two kinds of homogeneity cannot be distinguished by considering the behavior of a function near a given point.