Definition:Homogeneous Function

Definition

Let $V$ and $W$ be two vector spaces over a field $F$.

Let $f: V \to W$ be a function from $V$ to $W$.

Then $f$ is homogeneous of degree $n$ if and only if:

$f \left({\alpha \mathbf v}\right) = \alpha^n f \left({\mathbf v}\right)$

for all nonzero $\mathbf v \in V$ and $\alpha \in F$.

Degree

The element $n \in \N$ is the degree of $f$.

Zero Degree

A special case is when $n = 0$:

$f$ is a homogeneous function of degree zero if and only if:

$f \left({\alpha \mathbf v}\right) = \alpha^0 f \left({\mathbf v}\right) = f \left({\mathbf v}\right)$

Real Number Space

Another special case is when $f: \R^2 \to \R$ is a real function of two variables.

Let $f: \R^2 \to \R$ be a real-valued function of two variables.

$f \left({x, y}\right)$ is a homogeneous function of degree zero if and only if:

$\exists n \in \Z: \forall t \in \R: f \left({t x, t y}\right) = t^n f \left({x, y}\right)$