Homogeneous function

Example 1

The function f ( x , y ) = x 2 + y 2 is homogeneous of degree 2
f ( t x , t y ) = ( t x ) 2 + ( t y ) 2 = t 2 ( x 2 + y 2 ) = t 2 f ( x , y )
suppose x = 2, y = 4 and t = 5

Linear functions

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

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

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,

is homogeneous of degree 10 since

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,

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:

Min/max

For every set of weights w 1 , … , w n , the following functions are homogeneous of degree 1:

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:

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

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.

Logarithms

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

This can be proved by noting that f ( 5 x ) = ln ⁡ 5 x = ln ⁡ 5 + f ( x ) , f ( 10 x ) = ln ⁡ 10 + f ( x ) , and f ( 15 x ) = ln ⁡ 15 + f ( x ) . Therefore, there is no k such that f ( α ⋅ x ) = α k ⋅ 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. A function ƒ : V {0} → R is positive homogeneous of degree k if

for all α > 0. Here k can be any real number. A (nonzero) continuous function homogeneous of degree k on Rⁿ  {0} extends continuously to Rⁿ if and only if Re{k} > 0.

Positive homogeneous functions are characterized by Euler's homogeneous function theorem. Suppose that the function ƒ : Rⁿ {0} → R is continuously differentiable. Then ƒ is positive homogeneous of degree k if and only if

This result follows at once by differentiating both sides of the equation ƒ(αy) = α^kƒ(y) with respect to α, applying the chain rule, and choosing α to be 1. The converse holds by integrating. Specifically, let g ( α ) = f ( α x ) . Since x ⋅ ∇ f ( α x ) = k f ( α x ) ,

Thus, g ′ ( α ) − k α g ( α ) = 0 . This implies g ( α ) = g ( 1 ) α k . Therefore, f ( α x ) = g ( α ) = α k g ( 1 ) = α k f ( x ) : ƒ is positive homogeneous of degree k.

As a consequence, suppose that ƒ : Rⁿ → R is differentiable and homogeneous of degree k. Then its first-order partial derivatives ∂ f / ∂ x i are homogeneous of degree k − 1. The result follows from Euler's theorem by commuting the operator x ⋅ ∇ with the partial derivative.

One can specialise 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

This equation may be solved using an integrating factor approach, with solution f ( x ) = c x k for some constant real number c.

Homogeneous distributions

A continuous function ƒ on Rⁿ is homogeneous of degree k if and only if

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

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

for all nonzero real t and all test functions φ ;. Here the angle brackets denote the pairing between distributions and test functions, and μ_t : Rⁿ → Rⁿ is the mapping of scalar multiplication by the real number t.

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

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