Homogeneous function

In mathematics, a homogeneous function is one with multiplicative scaling behaviour: if all its arguments are multiplied by a factor, then its value is multiplied by some power of this factor.

For example, a homogeneous real-valued function of two variables x and y is a real-valued function that satisfies the condition $f(\alpha x,\alpha y)=\alpha ^{k}f(x,y)$ for some constant k and all real numbers α. The constant k is called the degree of homogeneity.

More generally, if ƒ : V → W is a function between two vector spaces over a field F, and k is an integer, then ƒ is said to be homogeneous of degree k if

f(\alpha \mathbf {v} )=\alpha ^{k}f(\mathbf {v} )

(1)

for all nonzero α ∈ F and v ∈ V. When the vector spaces involved are over the real numbers, a slightly less general form of homogeneity is often used, requiring only that (1) hold for all α > 0.

Homogeneous functions can also be defined for vector spaces with the origin deleted, a fact that is used in the definition of sheaves on projective space in algebraic geometry. More generally, if S ⊂ V is any subset that is invariant under scalar multiplication by elements of the field (a "cone"), then a homogeneous function from S to W can still be defined by (1).

Examples

A homogeneous function is not necessarily continuous, as shown by this example. This is the function f defined by

f(x,y)=x

if

xy>0

and

f(x,y)=0

if

xy\leq 0

. This function is homogeneous of degree 1, i.e.

f(\alpha x,\alpha y)=\alpha f(x,y)

for any real numbers

\alpha ,x,y

. It is discontinuous at

y=0,x\neq 0

.

Example 1

The function $f(x,y)=x^{2}+y^{2}$ is homogeneous of degree 2:

f(tx,ty)=(tx)^{2}+(ty)^{2}=t^{2}\left(x^{2}+y^{2}\right)=t^{2}f(x,y).

For example, suppose x = 2, y = 4 and t = 5. Then

$f(x,y)=2^{2}+4^{2}=4+16=20$ , and
$f(5x,5y)=5^{2}\left(2^{2}+4^{2}\right)=25(20)=500$ .

Linear functions

Any linear map ƒ : V → W is homogeneous of degree 1 since by the definition of linearity

f(\alpha \mathbf {v} )=\alpha f(\mathbf {v} )

for all α ∈ F and v ∈ V.

Similarly, any multilinear function ƒ : V₁ × V₂ × ⋯ × V_n → W is homogeneous of degree n since by the definition of multilinearity

f(\alpha \mathbf {v} _{1},\ldots ,\alpha \mathbf {v} _{n})=\alpha ^{n}f(\mathbf {v} _{1},\ldots ,\mathbf {v} _{n})

for all α ∈ F and v₁ ∈ V₁, v₂ ∈ V₂, ..., v_n ∈ V_n.

It follows that the n-th differential of a function ƒ : X → Y between two Banach spaces X and Y is homogeneous of degree n.

Homogeneous polynomials

Monomials in n variables define homogeneous functions ƒ : Fⁿ → F. For example,

f(x,y,z)=x^{5}y^{2}z^{3}\,

is homogeneous of degree 10 since

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, 10 = 5 + 2 + 3.

A homogeneous polynomial is a polynomial made up of a sum of monomials of the same degree. For example,

x^{5}+2x^{3}y^{2}+9xy^{4}\,

is a homogeneous polynomial of degree 5. Homogeneous polynomials also define homogeneous functions.

Given a homogeneous polynomial of degree k, it is possible to get a homogeneous function of degree 1 by raising to the power 1/k. So for example, for every k the following function is homogeneous of degree 1:

\left(x^{k}+y^{k}+z^{k}\right)^{\frac {1}{k}}

Min/max

For every set of weights $w_{1},\dots ,w_{n}$ , the following functions are homogeneous of degree 1:

$\min \left({\frac {x_{1}}{w_{1}}},\dots ,{\frac {x_{n}}{w_{n}}}\right)$ (Leontief utilities)
$\max \left({\frac {x_{1}}{w_{1}}},\dots ,{\frac {x_{n}}{w_{n}}}\right)$

Polarization

A multilinear function g : V × V × ⋯ × V → F from the n-th Cartesian product of V with itself to the underlying field F gives rise to a homogeneous function ƒ : V → F by evaluating on the diagonal:

f(v)=g(v,v,\dots ,v).

The resulting function ƒ is a polynomial on the vector space V.

Conversely, if F has characteristic zero, then given a homogeneous polynomial ƒ of degree n on V, the polarization of ƒ is a multilinear function g : V × V × ⋯ × V → F on the n-th Cartesian product of V. The polarization is defined by:

g(v_{1},v_{2},\dots ,v_{n})={\frac {1}{n!}}{\frac {\partial }{\partial t_{1}}}{\frac {\partial }{\partial t_{2}}}\cdots {\frac {\partial }{\partial t_{n}}}f(t_{1}v_{1}+\cdots +t_{n}v_{n}).

These two constructions, one of a homogeneous polynomial from a multilinear form and the other of a multilinear form from a homogeneous polynomial, are mutually inverse to one another. In finite dimensions, they establish an isomorphism of graded vector spaces from the symmetric algebra of V^∗ to the algebra of homogeneous polynomials on V.

Rational functions

Rational functions formed as the ratio of two homogeneous polynomials are homogeneous functions off of the affine cone cut out by the zero locus of the denominator. Thus, if f is homogeneous of degree m and g is homogeneous of degree n, then f/g is homogeneous of degree m − n away from the zeros of g.

Non-examples

Logarithms

The natural logarithm $f(x)=\ln x$ scales additively and so is not homogeneous.

This can be demonstrated with the following examples: $f(5x)=\ln 5x=\ln 5+f(x)$ , $f(10x)=\ln 10+f(x)$ , and $f(15x)=\ln 15+f(x)$ . This is because there is no k such that $f(\alpha \cdot x)=\alpha ^{k}\cdot f(x)$ .

Affine functions

Affine functions (the function $f(x)=x+5$ is an example) do not scale multiplicatively.

Positive homogeneity

In the special case of vector spaces over the real numbers, the notion of positive homogeneity often plays a more important role than homogeneity in the above sense.

Let $X$ (resp. $Y$ ) be a vector space over a field $𝔽$ (resp. $𝔾$ ), where $𝔽$ and $𝔾$ will usually be (or possibly just contain) the real numbers $ℝ$ or complex numbers $ℂ$ . Let $f : X \to Y$ be a map.[note 1] We define[note 2] the following terminology:

Strict positive homogeneity: $f (rx) = r f (x)$ for all $x \in X$ and all positive real $r > 0$ .
Nonnegative homogeneity: f (rx) = r f (x) for all x ∈ X and all non-negative real r ≥ 0.
- A non-negative real-valued functions with this property can be characterized as being a Minkowski functional.
- This property is used in the definition of a sublinear function.
Positive homogeneity: This is usually defined to mean "nonnegative homogeneity" but it is also frequently defined to instead mean "strict positive homogeneity".
- This distinction is usually[note 3] irrelevant because for a function valued in a vector space or field, nonnegative homogeneity is the same as strict positive homogeneity: these notions are identical. See this[proof 1] footnote for a proof.
Real homogeneity: f (rx) = r f (x) for all x ∈ X and all real r.
- This property is used in the definition of a real linear functional.
Homogeneity: f (sx) = s f (x) for all x ∈ X and all s ∈ 𝔽.
- It is emphasized that this definition depends on the domain $X$ 's underlying scalar field $𝔽$ .
- This property is used in the definition of linear functionals and linear maps.
Conjugate homogeneity: f (sx) = s f (x) for all x ∈ X and all s ∈ 𝔽.
- If $𝔽 = ℂ$ then $s$ typically denotes the complex conjugate of $s$ . But more generally, $s$ could be the image of $s$ under some distinguished automorphism of $𝔽$ .
- 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 equality $f (rx) = r f (x)$ with $f (rx) = | r | f (x)$ in which case we prefix that definition with the word "absolute" or "absolutely." For example,

Absolute real homogeneity: $f (rx) = | r | f (x)$ for all $x \in X$ and all real $r$ .
Absolute homogeneity: f (sx) = |s| f (x) for all x ∈ X and all s ∈ 𝔽.
- This property is used in the definition of a seminorm and a norm.

If $k$ is a fixed real number then the above definitions can be further generalized by replacing the equality $f (rx) = r f (x)$ with $f (rx) = r k f (x)$ (or with $f (rx) = | r | k f (x)$ for conditions using the absolute value), in which case we say that the homogeneity is "of degree $k$ " (note in particular that all of the above definitions are "of degree $1$ "). For instance,

Nonnegative homogeneity of degree $k$ : $f (rx) = r k f (x)$ for all $x \in X$ and all real $r \geq 0$ .
Real homogeneity of degree $k$ : $f (rx) = r k f (x)$ for all $x \in X$ and all real $r$ .
Absolute real homogeneity of degree $k$ : $f (rx) = | r | k f (x)$ for all $x \in X$ and all real $r$ .
Absolute homogeneity of degree $k$ : $f (sx) = | s | k f (x)$ for all $x \in X$ and all $s \in 𝔽$ .

A (nonzero) continuous function that is homogeneous of degree $k$ on $ℝn \ {0$ } extends continuously to $ℝ n$ if and only if $k > 0$ .

Generalizations

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

Monoids and monoid actions

A monoid is a pair $(M, \cdot)$ consisting of a set $M$ and an associative operator $M \times M \to M$ where there is some element in $S$ called an identity element, which we will denote by $1 \in M$ , such that $1 \cdot m = m = m \cdot 1$ for all $m \in M$ .

Notation: If

(M, \cdot)

is a monoid with identity element

1 \in M

and if

m \in M

, then we will let

m 0 ≝ 1

,

m 1 ≝ m

,

m 2 ≝ m \cdot m

, and more generally for any positive integers

k

, let

m k

be the product of

k

instances of

m

; that is,

m k ≝ m \cdot (m k - 1)

.

Notation: It is common practice (e.g. such as in algebra or calculus) to denote the multiplication operation of a monoid

(M, \cdot)

by juxtaposition, meaning that we may write

m n

rather than

m \cdot n

. This allows us to not even have to assign a symbol to a monoid's multiplication operation. Moreover, when we use this juxtaposition notation then we will automatically assume that the monoid's identity element is denoted by

1

.

Let $M$ be a monoid with identity element $1 \in M$ whose operation is denoted by juxtaposition and let $X$ be a set. A monoid action of $M$ on $X$ is a map $M \times X \to X$ , which we will also denote by juxtaposition, such that $1 x = x = x 1$ and $(m n) x = m (n x)$ for all $x \in X$ and all $m, n \in M$ .

Homogeneity

Let $M$ be a monoid with identity element $1 \in M$ , let $X$ and $Y$ be sets, and suppose that on both $X$ and $Y$ there are defined monoid actions of $M$ . Let $k$ be a non-negative integer and let $f : X \to Y$ be a map. Then we say that $f$ is homogeneous of degree $k$ over $M$ if for every $x \in X$ and $m \in M$ ,

f (m x) = m k f (x)

.

If in addition there is a function $M \to M$ , denoted by $m \mapsto | m |$ , called an absolute value then we say that $f$ is absolutely homogeneous of degree $k$ over $M$ if for every $x \in X$ and $m \in M$ ,

f (m x) = | m | k f (x)

.

If we say that a function is homogeneous over $M$ (resp. absolutely homogeneous over $M$ ) then we mean that it is homogeneous of degree $1$ over $M$ (resp. absolutely homogeneous of degree $1$ over $M$ ).

More generally, note that it is possible for the symbols $m k$ to be defined for $m \in M$ with $k$ being something other than an integer (e.g. if $M$ is the real numbers and $k$ is a non-zero real number then $m k$ is defined even though $k$ is not an integer). In this case, we say that $f$ is homogeneous of degree $k$ over $M$ if the same equality holds:

f (m x) = m k f (x)

for every

x \in X

and

m \in M

.

The notion of being absolutely homogeneous of degree $k$ over $M$ is generalized similarly.

Euler's homogeneous function theorem

Continuously differentiable positively homogeneous functions are characterized by the following theorem:

Euler's homogeneous function theorem. — Suppose that the function $f : ℝn \ {0} → ℝ$ is continuously differentiable. Then $f$ is positively homogeneous of degree $k$ if and only if

\mathbf {x} \cdot \nabla f(\mathbf {x} )=kf(\mathbf {x} ).

Proof

This result follows at once by differentiating both sides of the equation $f (α y) = α k f (y)$ with respect to $α$ , applying the chain rule, and choosing $α$ to be $1$ .

The converse is proved by integrating. Specifically, let $\textstyle g(\alpha )=f(\alpha \mathbf {x} )$ . Since $\textstyle \alpha \mathbf {x} \cdot \nabla f(\alpha \mathbf {x} )=kf(\alpha \mathbf {x} )$ ,

g'(\alpha )=\mathbf {x} \cdot \nabla f(\alpha \mathbf {x} )={\frac {k}{\alpha }}f(\alpha \mathbf {x} )={\frac {k}{\alpha }}g(\alpha ).

Thus, $\textstyle g'(\alpha )-{\frac {k}{\alpha }}g(\alpha )=0$ . This implies $\textstyle g(\alpha )=g(1)\alpha ^{k}$ . Therefore, $\textstyle f(\alpha \mathbf {x} )=g(\alpha )=\alpha ^{k}g(1)=\alpha ^{k}f(\mathbf {x} )$ : $f$ is positively homogeneous of degree $k$ .

As a consequence, suppose that $f : ℝ n \to ℝ$ is differentiable and homogeneous of degree $k$ . Then its first-order partial derivatives $\partial f/\partial x_{i}$ are homogeneous of degree $k - 1$ . The result follows from Euler's theorem by commuting the operator $\mathbf {x} \cdot \nabla$ with the partial derivative.

One can specialize the theorem to the case of a function of a single real variable ( $n = 1$ ), in which case the function satisfies the ordinary differential equation

f'(x)-{\frac {k}{x}}f(x)=0.

This equation may be solved using an integrating factor approach, with solution $\textstyle f(x)=cx^{k}$ , where $c = f (1)$ .

Homogeneous distributions

A continuous function ƒ on $ℝ n$ is homogeneous of degree $k$ if and only if

\int _{\mathbb {R} ^{n}}f(tx)\varphi (x)\,dx=t^{k}\int _{\mathbb {R} ^{n}}f(x)\varphi (x)\,dx

for all compactly supported test functions $\varphi$ ; and nonzero real $t$ . Equivalently, making a change of variable $y = tx$ , ƒ is homogeneous of degree $k$ if and only if

t^{-n}\int _{\mathbb {R} ^{n}}f(y)\varphi \left({\frac {y}{t}}\right)\,dy=t^{k}\int _{\mathbb {R} ^{n}}f(y)\varphi (y)\,dy

for all t and all test functions $\varphi$ . The last display makes it possible to define homogeneity of distributions. A distribution $S$ is homogeneous of degree $k$ if

t^{-n}\langle S,\varphi \circ \mu _{t}\rangle =t^{k}\langle S,\varphi \rangle

for all nonzero real $t$ and all test functions $\varphi$ . Here the angle brackets denote the pairing between distributions and test functions, and $μ t : ℝ n \to ℝ n$ is the mapping of scalar division by the real number $t$ .

Application to differential equations

The substitution $v = y / x$ converts the ordinary differential equation

I(x,y){\frac {\mathrm {d} y}{\mathrm {d} x}}+J(x,y)=0,

where $I$ and $J$ are homogeneous functions of the same degree, into the separable differential equation

x{\frac {\mathrm {d} v}{\mathrm {d} x}}=-{\frac {J(1,v)}{I(1,v)}}-v.

Notes

Note in particular that if $Y = ℂ = 𝔾$ , then every $ℝ$ -valued function on $X$ is also $ℂ$ -valued.
For a property such as real homogeneity to even be well-defined, the fields $𝔽$ and $𝔾$ must both contain the real numbers. We will of course automatically make whatever assumptions on $𝔽$ and $𝔾$ are necessary in order for the scalar products below to be well-defined.
Note that sometimes $f$ 's codomain is the set of extended real numbers (which allows for $\pm\infty$ ), in which case the multiplication $0 \cdot f (x)$ will be undefined whenever $f (x) = \pm\infty$ . In this case, the conditions " $r > 0$ " and " $r \geq 0$ " may not necessarily be used interchangeably.

Assume that $f$ is strictly positively homogeneous and valued in a vector space or a field. Then $f (0) = f (2 \cdot 0) = 2 f (0)$ so subtracting $f (0)$ from both sides shows that $f (0) = 0$ . Writing $r ≝ 0$ , for all $x \in X$ we have $f (r x) = f (0) = 0 = 0 f (x) = r f (x)$ , which shows that $f$ is nonnegative homogeneous.

References

Blatter, Christian (1979). "20. Mehrdimensionale Differentialrechnung, Aufgaben, 1.". Analysis II (2nd ed.) (in German). Springer Verlag. p. 188. ISBN 3-540-09484-9.

External links

"Homogeneous function", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Eric Weisstein. "Euler's Homogeneous Function Theorem". MathWorld.

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[1] Note in particular that if $Y = ℂ = 𝔾$ , then every $ℝ$ -valued function on $X$ is also $ℂ$ -valued.

[2] For a property such as real homogeneity to even be well-defined, the fields $𝔽$ and $𝔾$ must both contain the real numbers. We will of course automatically make whatever assumptions on $𝔽$ and $𝔾$ are necessary in order for the scalar products below to be well-defined.

[3] Note that sometimes $f$ 's codomain is the set of extended real numbers (which allows for $\pm\infty$ ), in which case the multiplication $0 \cdot f (x)$ will be undefined whenever $f (x) = \pm\infty$ . In this case, the conditions " $r > 0$ " and " $r \geq 0$ " may not necessarily be used interchangeably.

[4] Assume that $f$ is strictly positively homogeneous and valued in a vector space or a field. Then $f (0) = f (2 \cdot 0) = 2 f (0)$ so subtracting $f (0)$ from both sides shows that $f (0) = 0$ . Writing $r ≝ 0$ , for all $x \in X$ we have $f (r x) = f (0) = 0 = 0 f (x) = r f (x)$ , which shows that $f$ is nonnegative homogeneous.