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homogeneous function (Definition)
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
    $\displaystyle 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
    $\displaystyle f(\lambda v) = \vert\lambda\vert^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
    $\displaystyle f(\lambda v) = \lambda^r f(v) $
    for all $ \lambda \ge 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$.

When the type of homegeneity is clear one simply talks about $ r$-homogeneous functions.



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"homogeneous function" is owned by matte. [ full author list (2) ]
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See Also: homogeneous polynomial, subadditive

Other names:  positively homogeneous function of degree, homogeneous function of degree, positively homogeneous function

Attachments:
derivative of homogeneous function (Theorem) by matte
Euler's theorem on homogeneous functions (Theorem) by CWoo
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Cross-references: functions, clear, mapping, vector spaces
There are 11 references to this entry.

This is version 5 of homogeneous function, born on 2004-10-17, modified 2008-06-08.
Object id is 6381, canonical name is HomogeneousFunction.
Accessed 10625 times total.

Classification:
AMS MSC: 15-00 (Linear and multilinear algebra; matrix theory :: General reference works )
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