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{{More footnotes|date=July 2018}}
{{for|homogeneous linear maps|Graded vector space#Homomorphisms}}
In [[mathematics]], a '''homogeneous function''' is a [[function of several variables]] such that
:<math>f(sx_1,\ldots, sx_n)=s^k f(x_1,\ldots, x_n)</math>
for every <math>x_1, \ldots, x_n,</math> and <math>s\ne 0.</math> This is also referred to a ''{{mvar|k}}th-degree'' or ''{{mvar|k}}th-order'' homogeneous function.
For example, a [[homogeneous polynomial]] of degree {{mvar|k}} defines a homogeneous function of degree {{mvar|k}}.
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The above definition extends to functions whose [[___domain of a function|___domain]] and [[codomain]] are [[vector space]]s over a [[Field (mathematics)|field]] {{mvar|F}}: a function <math>f : V \to W</math> between two {{mvar|F}}-vector spaces is ''homogeneous'' of degree <math>k</math> if
{{NumBlk|:|<math>f(s \mathbf{v}) = s^k f(\mathbf{v})</math>|{{EquationRef|1}}}}
for all nonzero <math>s \in F</math> and <math>v \in V.</math> This definition is often further generalized to functions whose ___domain is not {{mvar|V}}, but a [[cone (linear algebra)|cone]] in {{mvar|V}}, that is, a subset {{mvar|C}} of {{mvar|V}} such that <math>\mathbf{v}\in C</math> implies <math>s \mathbf{v}\in C</math> for every nonzero scalar {{mvar|s}}.
In the case of [[functions of several real variables]] and [[real vector space]]s, a slightly more general form of homogeneity called '''positive homogeneity''' is often considered, by requiring only that the above identities hold for <math>s > 0,</math> and allowing any real number {{mvar|k}} as a degree of homogeneity. Every homogeneous real function is ''positively homogeneous''. The converse is not true, but is locally true in the sense that (for integer degrees) the two kinds of homogeneity cannot be distinguished by considering the behavior of a function near a given point.
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There are two commonly used definitions. The general one works for vector spaces over arbitrary [[field (mathematics)|fields]], and is restricted to degrees of homogeneity that are [[integer]]s.
The second one supposes to work over the field of [[real number]]s, or, more generally, over an [[ordered field]]. This definition restricts to positive values the scaling factor that occurs in the definition, and is therefore called ''positive homogeneity'', the qualificative ''positive'' being often omitted when there is no risk of confusion. Positive homogeneity leads to
The restriction of the scaling factor to real positive values allows also considering homogeneous functions whose degree of homogeneity is any real number.
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More generally, every [[norm (mathematics)|norm]] and [[seminorm]] is a positively homogeneous function of degree {{math|1}} which is not a homogeneous function. As for the absolute value, if the norm or semi-norm is defined on a vector space over the complex numbers, this vector space has to be considered as vector space over the real number for applying the definition of a positively homogeneous function.
===Linear
Any [[linear map]] <math>f : V \to W</math> between [[vector space]]s over a [[field (mathematics)|field]] {{mvar|F}} is homogeneous of degree 1, by the definition of linearity:
<math display="block">f(\alpha \mathbf{v}) = \alpha f(\mathbf{v})</math>
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===Non-examples===
The homogeneous [[real functions]] of a single variable have the form <math>x\mapsto cx^k</math> for some constant {{mvar|c}}. So, the [[affine function]] <math>x\mapsto x+5,</math> the [[natural logarithm]] <math>x\mapsto \ln(x),</math> and the [[exponential function]] <math>x\mapsto e^x</math> are not homogeneous.
== Euler's theorem ==
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The converse is proved by integrating a simple [[differential equation]].
Let <math>\mathbf{x}</math> be in the interior of the ___domain of {{mvar|f}}. For {{mvar|s}} sufficiently close
<math display="inline"> g(s) = f(s \mathbf{x})</math> is well defined. The partial differential equation implies that
<math display=block>
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The following commonly encountered special cases and variations of this definition have their own terminology:
#({{em|{{visible anchor|Strict positive homogeneity|Strictly positive homogeneous|text=Strict}}}}) {{em|{{visible anchor|Positive homogeneity|Positive homogeneous|Positively homogeneous}}}}:{{sfn|Schechter|1996|pp=313-314}} <math>f(rx) = r f(x)</math> for all <math>x \in X</math> and all {{em|positive}} real <math>r > 0.</math>
#* When the function <math>f</math> is valued in a vector space or field, then this property is [[Logical equivalence|logically equivalent]]<ref group=proof name=posHomEquivToNonnegHom /> to {{em|{{visible anchor|Nonnegative homogeneity|Nonnegative homogeneous|Nonnegatively homogeneous|text=nonnegative homogeneity}}}}, which by definition means:{{sfn|Kubrusly|2011|p=200}}
#* This property is used in the definition of a [[sublinear function]].{{sfn|Schechter|1996|pp=313-314}}{{sfn|Kubrusly|2011|p=200}}
#* [[Minkowski functional]]s are exactly those non-negative extended real-valued functions with this property.
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#{{em|{{visible anchor|Homogeneity|Homogeneous}}}}:{{sfn|Kubrusly|2011|p=55}} <math>f(sx) = s f(x)</math> for all <math>x \in X</math> and all scalars <math>s \in \mathbb{F}.</math>
#* It is emphasized that this definition depends on the scalar field <math>\mathbb{F}</math> underlying the ___domain <math>X.</math>
#* This property is used in the definition of [[linear functional]]s and [[linear map]]s.{{sfn|Kubrusly|2011|p=200}}
#{{em|[[Semilinear map|{{visible anchor|Conjugate homogeneity|Conjugate homogeneous}}]]}}:{{sfn|Kubrusly|2011|p=310}} <math>f(sx) = \overline{s} f(x)</math> for all <math>x \in X</math> and all scalars <math>s \in \mathbb{F}.</math>
#* If <math>\mathbb{F} = \Complex</math> then <math>\overline{s}</math> typically denotes the [[complex conjugate]] of <math>s</math>. But more generally, as with [[semilinear map]]s for example, <math>\overline{s}</math> could be the image of <math>s</math> under some distinguished automorphism of <math>\mathbb{F}.</math>
#* Along with [[Additive map|additivity]], this property is assumed in the definition of an [[antilinear map]]. It is also assumed that one of the two coordinates of a [[sesquilinear form]] has this property (such as the [[inner product]] of a [[Hilbert space]]).
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{{reflist|group=note}}
{{reflist|group=proof
<ref group=proof name=posHomEquivToNonnegHom>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>
}}
==References==
{{reflist}}
==Sources==
* {{cite book|last=Blatter|first=Christian|title=Analysis II (2nd ed.)|publisher=Springer Verlag|year=1979|language=German|isbn=3-540-09484-9|pages=188|chapter=20. Mehrdimensionale Differentialrechnung, Aufgaben, 1.}}▼
{{sfn whitelist|CITEREFKubrusly2011}}
▲* {{cite book|last=Blatter|first=Christian|title=Analysis II
* {{Kubrusly The Elements of Operator Theory 2nd Edition 2011}} <!--{{sfn|Kubrusly|2011|p=}}-->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!--{{sfn|Schaefer|Wolff|1999|p=}}-->
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