Homogeneous function
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In mathematics, a homogeneous function is a function with multiplicative scaling behaviour: if the argument is multiplied by some factor, then the result is multiplied by some power of this factor. Examples are given by (the functions associated to) homogeneous polynomials.
Below are also given some generalizations of this definition, in particular positive homogeneity or positive scalability.
[edit] Formal definition
Formally, let
- <math> f: V \rarr W \qquad\qquad </math>
be a function between two vector spaces over a field <math> F \qquad\qquad</math>.
We say that <math> f \qquad\qquad</math> is homogeneous of degree <math> k \qquad\qquad</math> if the equation
- <math> f(\alpha \mathbf{v}) = \alpha^k f(\mathbf{v}) \qquad\qquad (*) </math>
holds for all <math> \alpha \isin F \qquad\qquad</math> and <math> \mathbf{v} \isin V \qquad\qquad</math>.
A function
- <math> f(\mathbf{x}) = f(x_1, x_2,..., x_n) \qquad\qquad </math>
that is homogeneous of degree <math> k \qquad\qquad</math> has first order partial derivatives which are of degree <math> k-1 \qquad\qquad</math>. Furthermore, it satisfies Euler's homogeneous function theorem, which states that
- <math> \mathbf{x} \cdot \nabla f(\mathbf{x}) = kf(\mathbf{x}) \qquad\qquad </math>
Written out in components, this is
- <math>
\sum_{i=1}^n x_i \frac{\partial f}{\partial x_i} (\mathbf{x}) = k f(\mathbf{x}). </math>
[edit] Generalizations
Sometimes, a function satisfying <math>(*)</math> for all positive <math>\alpha</math> is said to be positively homogeneous of degree <math>k</math>. (This makes sense only if there is a reasonable notion of "positivity" defined, such as in ordered fields.)
A similar definition, usually given for functions taking only positive values, positive homogeneity or positive scalability is defined by taking the absolute value of the factor. For example, seminorms are (positively) homogeneous of degree 1. (In that case, one must have an absolute value defined on the field.)
Even more generally, a function <math> f </math> is said to be homogeneous if the equation <math> f(\alpha \mathbf{v}) = g(\alpha) f(\mathbf{v}) </math> holds for some strictly increasing positive function <math> g </math>. (This, in turn, requires an order relation on the set of scalars.)
