Homogeneous function: Difference between revisions

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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 the following holds (Eulers theorem of homogeneous functions): If each of the function's arguments is multiplied by the same [[scalar (mathematics)|scalar]], then the function's value is multiplied by some power of this scalar; the power is called the '''degree of homogeneity''', or simply the ''degree''. That is, if {{mvar|k}} is an integer, a function {{mvar|f}} of {{mvar|n}} variables is homogeneous of degree {{mvar|k}} if

:<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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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 functionsMaps===

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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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 ofto {{math|1}}, the function

<math display="inline"> g(s) = f(s \mathbf{x})</math> is well defined. The partial differential equation implies that

<math display=block>