Homogeneous function

Examples

Simple example

The function ${\displaystyle f(x,y)=x^2+y^2}$ is homogeneous of degree 2: $${\displaystyle f(tx,ty)=(tx{)}^2+(ty{)}^2=t^2\left(x^2+y^2\right)=t^{2}f(x,y).}$$

Absolute value and norms

The absolute value of a real number is a positively homogeneous function of degree 1, which is not homogeneous, since ${\displaystyle |sx|=s|x|}$ if ${\displaystyle s>0,}$ and ${\displaystyle |sx|=-s|x|}$ if ${\displaystyle s<0.}$

The absolute value of a complex number is a positively homogeneous function of degree ${\displaystyle 1}$ over the real numbers (that is, when considering the complex numbers as a vector space over the real numbers). It is not homogeneous, over the real numbers as well as over the complex numbers.

More generally, every norm and seminorm is a positively homogeneous function of degree 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 maps

Any linear map ${\displaystyle f:V\to W}$ between vector spaces over a field F is homogeneous of degree 1, by the definition of linearity: $${\displaystyle f(\alpha \mathbf{v})=\alpha f(\mathbf{v})}$$ for all ${\displaystyle \alpha \in F}$ and ${\displaystyle v\in V.}$

Similarly, any multilinear function ${\displaystyle f:V_1\times V_2\times \cdots V_n\to W}$ is homogeneous of degree ${\displaystyle n,}$ by the definition of multilinearity: $${\displaystyle f\left(\alpha {\mathbf{v}}_1,\dots ,\alpha {\mathbf{v}}_n\right)={\alpha }^{n}f({\mathbf{v}}_1,\dots ,{\mathbf{v}}_n)}$$ for all ${\displaystyle \alpha \in F}$ and ${\displaystyle v_1\in V_1,v_2\in V_2,\dots ,v_n\in V_n.}$

Homogeneous polynomials

Monomials in ${\displaystyle n}$ variables define homogeneous functions ${\displaystyle f:{\mathbb{F}}^n\to \mathbb{F}.}$ For example, $${\displaystyle f(x,y,z)=x^5{y}^2{z}^3}$$ is homogeneous of degree 10 since $${\displaystyle f(\alpha x,\alpha y,\alpha z)=(\alpha x{)}^5(\alpha y{)}^2(\alpha z{)}^3={\alpha }^{10}{x}^5{y}^2{z}^3={\alpha }^{10}f(x,y,z).}$$ The degree is the sum of the exponents on the variables; in this example, ${\displaystyle 10=5+2+3.}$

A homogeneous polynomial is a polynomial made up of a sum of monomials of the same degree. For example, $${\displaystyle x^5+2x^3{y}^2+9x{y}^4}$$ is a homogeneous polynomial of degree 5. Homogeneous polynomials also define homogeneous functions.

Given a homogeneous polynomial of degree ${\displaystyle k}$ with real coefficients that takes only positive values, one gets a positively homogeneous function of degree ${\displaystyle k/d}$ by raising it to the power ${\displaystyle 1/d.}$ So for example, the following function is positively homogeneous of degree 1 but not homogeneous: $${\displaystyle {\left(x^2+y^2+z^2\right)}^{\frac{1}{2}}.}$$

Min/max

For every set of weights ${\displaystyle w_1,\dots ,w_n,}$ the following functions are positively homogeneous of degree 1, but not homogeneous:

• ${\displaystyle min\left(\frac{x_1}{w_1},\dots ,\frac{x_n}{w_n}\right)}$ (Leontief utilities)
• ${\displaystyle max\left(\frac{x_1}{w_1},\dots ,\frac{x_n}{w_n}\right)}$

Rational functions

Rational functions formed as the ratio of two homogeneous polynomials are homogeneous functions in their domain, that is, off of the linear cone formed by the zeros of the denominator. Thus, if ${\displaystyle f}$ is homogeneous of degree ${\displaystyle m}$ and ${\displaystyle g}$ is homogeneous of degree ${\displaystyle n,}$ then ${\displaystyle f/g}$ is homogeneous of degree ${\displaystyle m-n}$ away from the zeros of ${\displaystyle g.}$

Non-examples

The homogeneous real functions of a single variable have the form ${\displaystyle x\mapsto c{x}^k}$ for some constant c. So, the affine function ${\displaystyle x\mapsto x+5,}$ the natural logarithm ${\displaystyle x\mapsto \ln (x),}$ and the exponential function ${\displaystyle x\mapsto e^x}$ are not homogeneous.

Euler's theorem

Roughly speaking, Euler's homogeneous function theorem asserts that the positively homogeneous functions of a given degree are exactly the solution of a specific partial differential equation. More precisely:

Euler's homogeneous function theorem—If f is a (partial) function of n real variables that is positively homogeneous of degree k, and continuously differentiable in some open subset of ${\displaystyle {\mathbb{R}}^n,}$ then it satisfies in this open set the partial differential equation $${\displaystyle kf(x_1,\dots ,x_n)=\sum _{i=1}^n{x}_i\frac{\mathrm{\partial }f}{\mathrm{\partial }{x}_i}(x_1,\dots ,x_n).}$$

Conversely, every maximal continuously differentiable solution of this partial differentiable equation is a positively homogeneous function of degree k, defined on a positive cone (here, maximal means that the solution cannot be prolongated to a function with a larger domain).

Proof

For having simpler formulas, we set ${\displaystyle \mathbf{x}=(x_1,\dots ,x_n).}$ The first part results by using the chain rule for differentiating both sides of the equation ${\displaystyle f(s\mathbf{x})=s^{k}f(\mathbf{x})}$ with respect to ${\displaystyle s,}$ and taking the limit of the result when s tends to 1.

The converse is proved by integrating a simple differential equation. Let ${\displaystyle \mathbf{x}}$ be in the interior of the domain of f. For s sufficiently close to 1, the function $g(s)=f(s\mathbf{x})$ is well defined. The partial differential equation implies that $${\displaystyle s{g}^{\prime }(s)=kf(s\mathbf{x})=kg(s).}$$ The solutions of this linear differential equation have the form ${\displaystyle g(s)=g(1)s^k.}$ Therefore, $${\displaystyle f(s\mathbf{x})=g(s)=s^{k}g(1)=s^{k}f(\mathbf{x}),}$$ if s is sufficiently close to 1. If this solution of the partial differential equation would not be defined for all positive s, then the functional equation would allow to prolongate the solution, and the partial differential equation implies that this prolongation is unique. So, the domain of a maximal solution of the partial differential equation is a linear cone, and the solution is positively homogeneous of degree k. ${\displaystyle ◻}$

As a consequence, if ${\displaystyle f:{\mathbb{R}}^n\to \mathbb{R}}$ is continuously differentiable and homogeneous of degree ${\displaystyle k,}$ its first-order partial derivatives ${\displaystyle \mathrm{\partial }f/\mathrm{\partial }{x}_i}$ are homogeneous of degree ${\displaystyle k-1.}$ This results from Euler's theorem by differentiating the partial differential equation with respect to one variable.

In the case of a function of a single real variable (${\displaystyle n=1}$), the theorem implies that a continuously differentiable and positively homogeneous function of degree k has the form ${\displaystyle f(x)=c_{+}{x}^k}$ for ${\displaystyle x>0}$ and ${\displaystyle f(x)=c_{-}{x}^k}$ for ${\displaystyle x<0.}$ The constants ${\displaystyle c_{+}}$ and ${\displaystyle c_{-}}$ are not necessarily the same, as it is the case for the absolute value.

Application to differential equations

The substitution ${\displaystyle v=y/x}$ converts the ordinary differential equation $${\displaystyle I(x,y)\frac{\mathrm{d}y}{\mathrm{d}x}+J(x,y)=0,}$$ where ${\displaystyle I}$ and ${\displaystyle J}$ are homogeneous functions of the same degree, into the separable differential equation $${\displaystyle x\frac{\mathrm{d}v}{\mathrm{d}x}=-\frac{J(1,v)}{I(1,v)}-v.}$$

Generalizations

Homogeneity under a monoid action

The definitions given above are all specialized cases of the following more general notion of homogeneity in which ${\displaystyle X}$ can be any set (rather than a vector space) and the real numbers can be replaced by the more general notion of a monoid.

Let ${\displaystyle M}$ be a monoid with identity element ${\displaystyle 1\in M,}$ let ${\displaystyle X}$ and ${\displaystyle Y}$ be sets, and suppose that on both ${\displaystyle X}$ and ${\displaystyle Y}$ there are defined monoid actions of ${\displaystyle M.}$ Let ${\displaystyle k}$ be a non-negative integer and let ${\displaystyle f:X\to Y}$ be a map. Then ${\displaystyle f}$ is said to be homogeneous of degree ${\displaystyle k}$ over ${\displaystyle M}$ if for every ${\displaystyle x\in X}$ and ${\displaystyle m\in M,}$ $${\displaystyle f(mx)=m^{k}f(x).}$$ If in addition there is a function ${\displaystyle M\to M,}$ denoted by ${\displaystyle m\mapsto |m|,}$ called an absolute value then ${\displaystyle f}$ is said to be absolutely homogeneous of degree ${\displaystyle k}$ over ${\displaystyle M}$ if for every ${\displaystyle x\in X}$ and ${\displaystyle m\in M,}$ $${\displaystyle f(mx)=|m{|}^{k}f(x).}$$

A function is homogeneous over ${\displaystyle M}$ (resp. absolutely homogeneous over ${\displaystyle M}$) if it is homogeneous of degree ${\displaystyle 1}$ over ${\displaystyle M}$ (resp. absolutely homogeneous of degree ${\displaystyle 1}$ over ${\displaystyle M}$).

More generally, it is possible for the symbols ${\displaystyle m^k}$ to be defined for ${\displaystyle m\in M}$ with ${\displaystyle k}$ being something other than an integer (for example, if ${\displaystyle M}$ is the real numbers and ${\displaystyle k}$ is a non-zero real number then ${\displaystyle m^k}$ is defined even though ${\displaystyle k}$ is not an integer). If this is the case then ${\displaystyle f}$ will be called homogeneous of degree ${\displaystyle k}$ over ${\displaystyle M}$ if the same equality holds: $${\displaystyle f(mx)=m^{k}f(x)\text{ for every }x\in X\text{ and }m\in M.}$$

The notion of being absolutely homogeneous of degree ${\displaystyle k}$ over ${\displaystyle M}$ is generalized similarly.

Distributions (generalized functions)

A continuous function ${\displaystyle f}$ on ${\displaystyle {\mathbb{R}}^n}$ is homogeneous of degree ${\displaystyle k}$ if and only if $${\displaystyle {\int }_{{\mathbb{R}}^n}f(tx)\phi (x)dx=t^k{\int }_{{\mathbb{R}}^n}f(x)\phi (x)dx}$$ for all compactly supported test functions ${\displaystyle \phi }$; and nonzero real ${\displaystyle t.}$ Equivalently, making a change of variable ${\displaystyle y=tx,}$ ${\displaystyle f}$ is homogeneous of degree ${\displaystyle k}$ if and only if $${\displaystyle t^{-n}{\int }_{{\mathbb{R}}^n}f(y)\phi \left(\frac{y}{t}\right)dy=t^k{\int }_{{\mathbb{R}}^n}f(y)\phi (y)dy}$$ for all ${\displaystyle t}$ and all test functions ${\displaystyle \phi .}$ The last display makes it possible to define homogeneity of distributions. A distribution ${\displaystyle S}$ is homogeneous of degree ${\displaystyle k}$ if $${\displaystyle t^{-n}⟨S,\phi \circ {\mu }_t⟩=t^k⟨S,\phi ⟩}$$ for all nonzero real ${\displaystyle t}$ and all test functions ${\displaystyle \phi .}$ Here the angle brackets denote the pairing between distributions and test functions, and ${\displaystyle {\mu }_t:{\mathbb{R}}^n\to {\mathbb{R}}^n}$ is the mapping of scalar division by the real number ${\displaystyle t.}$

Glossary of name variants

Let ${\displaystyle f:X\to Y}$ be a map between two vector spaces over a field ${\displaystyle \mathbb{F}}$ (usually the real numbers ${\displaystyle \mathbb{R}}$ or complex numbers ${\displaystyle \mathbb{C}}$). If ${\displaystyle S}$ is a set of scalars, such as ${\displaystyle \mathbb{Z},}$ ${\displaystyle [0,\mathrm{\infty }),}$ or ${\displaystyle \mathbb{R}}$ for example, then ${\displaystyle f}$ is said to be homogeneous over ${\displaystyle S}$ if $f(sx)=sf(x)$ for every ${\displaystyle x\in X}$ and scalar ${\displaystyle s\in S.}$ For instance, every additive map between vector spaces is homogeneous over the rational numbers ${\displaystyle S:=\mathbb{Q}}$ although it might not be homogeneous over the real numbers ${\displaystyle S:=\mathbb{R}.}$

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

1. (Strict) Positive homogeneity: ${\displaystyle f(rx)=rf(x)}$ for all ${\displaystyle x\in X}$ and all positive real ${\displaystyle r>0.}$
   • When the function ${\displaystyle f}$ is valued in a vector space or field, then this property is logically equivalent to nonnegative homogeneity, which by definition means: ${\displaystyle f(rx)=rf(x)}$ for all ${\displaystyle x\in X}$ and all non-negative real ${\displaystyle r\ge 0.}$ It is for this reason that positive homogeneity is often also called nonnegative homogeneity. However, for functions valued in the extended real numbers ${\displaystyle [-\mathrm{\infty },\mathrm{\infty }]=\mathbb{R}\cup \{\pm \mathrm{\infty }\},}$ which appear in fields like convex analysis, the multiplication ${\displaystyle 0\cdot f(x)}$ will be undefined whenever ${\displaystyle f(x)=\pm \mathrm{\infty }}$ and so these statements are not necessarily always interchangeable.
   • This property is used in the definition of a sublinear function.
   • Minkowski functionals are exactly those non-negative extended real-valued functions with this property.
2. Real homogeneity: ${\displaystyle f(rx)=rf(x)}$ for all ${\displaystyle x\in X}$ and all real ${\displaystyle r.}$
   • This property is used in the definition of a real linear functional.
3. Homogeneity: ${\displaystyle f(sx)=sf(x)}$ for all ${\displaystyle x\in X}$ and all scalars ${\displaystyle s\in \mathbb{F}.}$
   • It is emphasized that this definition depends on the scalar field ${\displaystyle \mathbb{F}}$ underlying the domain ${\displaystyle X.}$
   • This property is used in the definition of linear functionals and linear maps.
4. Conjugate homogeneity: ${\displaystyle f(sx)=\overline{s}f(x)}$ for all ${\displaystyle x\in X}$ and all scalars ${\displaystyle s\in \mathbb{F}.}$
   • If ${\displaystyle \mathbb{F}=\mathbb{C}}$ then ${\displaystyle \overline{s}}$ typically denotes the complex conjugate of ${\displaystyle s}$. But more generally, as with semilinear maps for example, ${\displaystyle \overline{s}}$ could be the image of ${\displaystyle s}$ under some distinguished automorphism of ${\displaystyle \mathbb{F}.}$
   • Along with 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).

All of the above definitions can be generalized by replacing the condition ${\displaystyle f(rx)=rf(x)}$ with ${\displaystyle f(rx)=|r|f(x),}$ in which case that definition is prefixed with the word "absolute" or "absolutely." For example,

1. Absolute homogeneity: ${\displaystyle f(sx)=|s|f(x)}$ for all ${\displaystyle x\in X}$ and all scalars ${\displaystyle s\in \mathbb{F}.}$
   • This property is used in the definition of a seminorm and a norm.

If ${\displaystyle k}$ is a fixed real number then the above definitions can be further generalized by replacing the condition ${\displaystyle f(rx)=rf(x)}$ with ${\displaystyle f(rx)=r^{k}f(x)}$ (and similarly, by replacing ${\displaystyle f(rx)=|r|f(x)}$ with ${\displaystyle f(rx)=|r{|}^{k}f(x)}$ for conditions using the absolute value, etc.), in which case the homogeneity is said to be "of degree ${\displaystyle k}$" (where in particular, all of the above definitions are "of degree ${\displaystyle 1}$"). For instance,

1. Real homogeneity of degree ${\displaystyle k}$: ${\displaystyle f(rx)=r^{k}f(x)}$ for all ${\displaystyle x\in X}$ and all real ${\displaystyle r.}$
2. Homogeneity of degree ${\displaystyle k}$: ${\displaystyle f(sx)=s^{k}f(x)}$ for all ${\displaystyle x\in X}$ and all scalars ${\displaystyle s\in \mathbb{F}.}$
3. Absolute real homogeneity of degree ${\displaystyle k}$: ${\displaystyle f(rx)=|r{|}^{k}f(x)}$ for all ${\displaystyle x\in X}$ and all real ${\displaystyle r.}$
4. Absolute homogeneity of degree ${\displaystyle k}$: ${\displaystyle f(sx)=|s{|}^{k}f(x)}$ for all ${\displaystyle x\in X}$ and all scalars ${\displaystyle s\in \mathbb{F}.}$

A nonzero continuous function that is homogeneous of degree ${\displaystyle k}$ on ${\displaystyle {\mathbb{R}}^n\mathrm{\setminus }\{0\}}$ extends continuously to ${\displaystyle {\mathbb{R}}^n}$ if and only if ${\displaystyle k>0.}$