Homogeneous function: Difference between revisions

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{{or section|date=December 2021}}

Let <math>f : X \to Y</math> be a map between two [[vector space]]s over a field <math>\mathbb{F}</math> (usually the [[real number]]s <math>\R</math> or [[complex number]]s <math>\Complex.</math>). If <math>S</math> is a set of scalars, such as <math>\Z, [0, \infty), \text{ or } \R</math> for example, then <math>f</math> is said to be {{em|{{visible anchor|homogeneous over}} <math>S</math>}} if

<math display=blockinline>f(s x) = s f(x) \qquad \text{</math> for every } <math>x \in X \text{</math> and scalar } <math>s \in S.</math>

For instance, every [[additive map]] between vector spaces is {{em|{{visible anchor|homogeneous over the rational numbers}}}} <math>S := \Q</math> although it [[Cauchy's functional equation|might not be {{em|{{visible anchor|homogeneous over the real numbers}}}}]] <math>S := \R.</math>

The following commonly encountered special cases and variations of this definition have their own terminology:

The following commonly encountered special cases have their own terminology:<ref group=note>For a property such as real homogeneity to even be well-defined, the fields <math>\mathbb{F}</math> and <math>\mathbb{G}</math> must both contain the real numbers. We will of course automatically make whatever assumptions on <math>\mathbb{F}</math> and <math>\mathbb{G}</math> are necessary in order for the scalar products below to be well-defined.</ref>

#({{em|{{visible anchor|Strict positive homogeneity|Strictly positive homogeneous|text=Strict}}}}) {{em|{{visible anchor|Positive homogeneity|Positive homogeneous|Positively homogeneous}}}}: <math>f(rx) = r f(x)</math> for all <math>x \in X</math> and all {{em|positive}} real <math>r > 0.</math>

#* This property is often also called {{em|{{visible anchor|Nonnegative homogeneity|Nonnegative homogeneous|Nonnegatively homogeneous|text=nonnegative homogeneity}}}} because for a function valued in a vector space or field, it is [[Logical equivalence|logically equivalent]] to: <math>f(rx) = r f(x)</math> for all <math>x \in X</math> and all {{em|non-negative}} real <math>r \geq 0.</math><ref group=proof>Assume that <math>f</math> is strictly positively homogeneous and valued in a vector space or a field. Then <math>f(0) = f(2 \cdot 0) = 2 f(0)</math> so subtracting <math>f(0)</math> from both sides shows that <math>f(0) = 0.</math> Writing <math>r := 0,</math> then for any <math>x \in X,</math> <math>f(r x) = f(0) = 0 = 0 f(x) = r f(x),</math> which shows that <math>f</math> is nonnegative homogeneous.</ref> However, for a function valued in the [[extended real numbers]] <math>[-\infty, \infty] = \R \cup \{\pm \infty\},</math> which appear in fields like [[convex analysis]], the multiplication <math>0 \cdot f(x)</math> will be undefined whenever <math>f(x) = \pm \infty</math> and so these statements are not necessarily interchangeable.<ref group=note>However, if such an <math>f</math> satisfies <math>f(rx) = r f(x)</math> for all <math>r > 0</math> and <math>x \in X,</math> then necessarily <math>f(0) \in \{\pm \infty, 0\}</math> and whenever <math>f(0), f(x) \in \R</math> are both real then <math>f(r x) = r f(x)</math> will hold for all <math>r \geq 0.</math></ref>