Homogeneous function: Difference between revisions

Revision as of 03:10, 16 January 2022

Revision as of 03:52, 17 January 2022

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There are two commonly used definitions. The general one works for vector spaces over arbitrary [[field (mathematics)|fields]], and is restricted to degrees of homogeneity that are [[integer]]s.

The second one supposes to work over the field of [[real number]]s, or, more generally, over an [[ordered field]]. This definition restricts to positive values the scaling factor that occurs in the definition, and is therefore called ''positive homogeneity'', the qualificative ''positive'' being often ommittedomitted when there is no risk of confusion. Positive homogeneity leads to consider more functions as homogeneous. For example, the [[absolute value]] and all [[norm (mathematics)|norms]] are positively homogeneous functions that are not homogeneous.

The restriction of the scaling factor to real positive values allows also considering homogeneous functions whose degree of homogeneity is any real number.