|
|
|
|
homogeneous function
|
(Definition)
|
|
Definition 1 Suppose $V,\,W$ are a vector spaces over
 , and $f\colon V \to W$ is a mapping.
- If there exists an
 , such that $$ f(\lambda v) = \lambda^r f(v) $$ for all
 and $v\in V$ , then $f$ is a homogeneous function of degree $r$ .
- If there exists an
 , such that $$ f(\lambda v) = |\lambda|^r f(v) $$ for all
 and $v\in V$ , then $f$ is absolutely homogeneous function of degree $r$ .
- If there exists an
 , such that $$ f(\lambda v) = \lambda^r f(v) $$ for all $\lambda \ge 0$ and $v\in V$ , then $f$ is a positively homogeneous function of degree $r$ .
For any homogeneous function as above, $f(0)=0$ .
When the type of homegeneity is clear one simply talks about $r$ -homogeneous functions.
|
Anyone with an account can edit this entry. Please help improve it!
"homogeneous function" is owned by matte. [ full author list (2) ]
|
|
(view preamble | get metadata)
See Also: homogeneous polynomial, subadditive
| Other names: |
positively homogeneous function of degree, homogeneous function of degree, positively homogeneous function |
|
|
Cross-references: functions, clear, mapping, vector spaces
There are 12 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 15957 times total.
Classification:
| AMS MSC: | 15-00 (Linear and multilinear algebra; matrix theory :: General reference works ) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
