For example, a homogeneous real-valued function of two variables and is a real-valued function that satisfies the condition for some constant and all real numbers The constant is called the degree of homogeneity.
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 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).
The function is homogeneous of degree 2:
For example, suppose and Then
Any linear map is homogeneous of degree 1 since by the definition of linearity
Similarly, any multilinear function is homogeneous of degree since by the definition of multilinearity
Monomials in variables define homogeneous functions For example,
Given a homogeneous polynomial of degree it is possible to get a homogeneous function of degree 1 by raising to the power So for example, for every the following function is homogeneous of degree 1:
For every set of weights the following functions are homogeneous of degree 1:
The resulting function is a polynomial on the vector space
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 to the algebra of homogeneous polynomials on
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 is homogeneous of degree and is homogeneous of degree then is homogeneous of degree away from the zeros of
The natural logarithm scales additively and so is not homogeneous.
This can be demonstrated with the following examples: and This is because there is no such that
Affine functions (the function is an example) do not in general scale multiplicatively.
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 be a vector space over a field and let be a vector space over a field where and will usually be (or possibly just contain as subsets) the real numbers or complex numbers Let be a map.[note 1] Define[note 2] the following terminology:
- Strict positive homogeneity: for all and all positive real
- Nonnegative homogeneity: for all and all non-negative real
- Positive homogeneity: This is usually defined to mean "nonnegative homogeneity" but it is also frequently defined to instead mean "strict positive homogeneity".
- Real homogeneity: for all and all real
- This property is used in the definition of a real linear functional.
- Homogeneity: for all and all scalars
- Conjugate homogeneity: for all and all scalars
- If then typically denotes the complex conjugate of But more generally, could be the image of under some distinguished automorphism of
- 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 with in which case that definition is prefixed with the word "absolute" or "absolutely." For example,
- Absolute real homogeneity: for all and all real
- Absolute homogeneity: for all and all scalars
If is a fixed real number then the above definitions can be further generalized by replacing the condition with (and similarly, by replacing with for conditions using the absolute value, etc.), in which case the homogeneity is said to be "of degree " (where in particular, all of the above definitions are "of degree "). For instance,
- Nonnegative homogeneity of degree : for all and all real
- Real homogeneity of degree : for all and all real
- Homogeneity of degree : for all and all scalars
- Absolute real homogeneity of degree : for all and all real
- Absolute homogeneity of degree : for all and all scalars
A nonzero continuous function that is homogeneous of degree on extends continuously to if and only if
The definitions given above are all specializes of the following more general notion of homogeneity in which 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
If is a monoid with identity element and if then the following notation will be used: let and more generally for any positive integers let be the product of instances of ; that is,
It is common practice (e.g. such as in algebra or calculus) to denote the multiplication operation of a monoid by juxtaposition, meaning that may be written rather than This avoids any need to assign a symbol to a monoid's multiplication operation. When this juxtaposition notation is used then it should be automatically assumed that the monoid's identity element is denoted by
Let be a monoid with identity element whose operation is denoted by juxtaposition and let be a set. A monoid action of on is a map which will also be denoted by juxtaposition, such that and for all and all
Let be a monoid with identity element let and be sets, and suppose that on both and there are defined monoid actions of Let be a non-negative integer and let be a map. Then is said to be homogeneous of degree over if for every and
A function is homogeneous over (resp. absolutely homogeneous over ) if it is homogeneous of degree over (resp. absolutely homogeneous of degree over ).
More generally, it is possible for the symbols to be defined for with being something other than an integer (for example, if is the real numbers and is a non-zero real number then is defined even though is not an integer). If this is the case then will be called homogeneous of degree over if the same equality holds:
The notion of being absolutely homogeneous of degree over is generalized similarly.
Euler's homogeneous function theorem
Continuously differentiable positively homogeneous functions are characterized by the following theorem:
As a consequence, suppose that is differentiable and homogeneous of degree Then its first-order partial derivatives are homogeneous of degree The result follows from Euler's theorem by commuting the operator with the partial derivative.
One can specialize the theorem to the case of a function of a single real variable (), in which case the function satisfies the ordinary differential equation
A continuous function on is homogeneous of degree if and only if
Application to differential equations
The substitution converts the ordinary differential equation
- Note in particular that if then every -valued function on is also -valued.
- For a property such as real homogeneity to even be well-defined, the fields and must both contain the real numbers. We will of course automatically make whatever assumptions on and are necessary in order for the scalar products below to be well-defined.
- In fields like convex analysis, the codomain of is sometimes the set of extended real numbers, in which case the multiplication will be undefined whenever In this case, the conditions "" and "" may not necessarily be used interchangeably. However, if such an satisfies for all and then necessarily and whenever are both real then will hold for all
- Assume that is strictly positively homogeneous and valued in a vector space or a field. Then so subtracting from both sides shows that Writing then for any which shows that is nonnegative homogeneous.
- Blatter, Christian (1979). "20. Mehrdimensionale Differentialrechnung, Aufgaben, 1.". Analysis II (2nd ed.) (in German). Springer Verlag. p. 188. ISBN 3-540-09484-9.