You are here
Homehomogeneous function
Primary tabs
homogeneous function
Definition 1.
Suppose $V,\,W$ are a vector spaces over $\mathbbmss{R}$, and $f\colon V\to W$ is a mapping.

If there exists an $r\in\mathbbmss{R}$, such that
$f(\lambda v)=\lambda^{r}f(v)$ for all $\lambda\in\mathbbmss{R}$ and $v\in V$, then $f$ is a homogeneous function of degree $r$.

If there exists an $r\in\mathbbmss{R}$, such that
$f(\lambda v)=\lambda^{r}f(v)$ for all $\lambda\in\mathbbmss{R}$ and $v\in V$, then $f$ is absolutely homogeneous function of degree $r$.

If there exists an $r\in\mathbbmss{R}$, such that
$f(\lambda v)=\lambda^{r}f(v)$ for all $\lambda\geq 0$ and $v\in V$, then $f$ is a positively homogeneous function of degree $r$.
Notes
For any homogeneous function as above, $f(0)=0$.
Related:
HomogeneousPolynomial, SubLinear
Synonym:
positively homogeneous function of degree, homogeneous function of degree,positively homogeneous function
Type of Math Object:
Definition
Major Section:
Reference
Groups audience:
Mathematics Subject Classification
1500 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
 Corrections