Homogeneous_function

Homogeneous function

In mathematics, a homogeneous function is a function with multiplicative scaling behaviour: if the argument is multiplied by a factor, then the result is multiplied by some power of this factor.

1 Formal definition
2 Examples
3 Elementary theorems
4 Application to ODEs
5 References
6 External links

Formal definition

Suppose that $f: V \rarr W \qquad\qquad$ is a function between two vector spaces over a field $F \qquad\qquad$ .

We say that $f \qquad\qquad$ is homogeneous of degree $k \qquad\qquad$ if

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

for all nonzero $\alpha \isin F \qquad\qquad$ and $\mathbf{v} \isin V \qquad\qquad$ .

Examples

A linear function $f: V \rarr W \qquad\qquad$ is homogeneous of degree 1, since by the definition of linearity

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

for all $\alpha \isin F \qquad\qquad$ and $\mathbf{v} \isin V \qquad\qquad$ .

A multilinear function $f: V_1 \times \ldots \times V_n \rarr W \qquad\qquad$ 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 $\alpha \isin F \qquad\qquad$ and $\mathbf{v}_1 \isin V_1,\ldots,\mathbf{v}_n \isin V_n \qquad\qquad$ .

It follows from the previous example that the $n$ th Fréchet derivative of a function $f: X \rightarrow Y$ between two Banach spaces $X$ and $Y$ is homogeneous of degree $n$ .

Monomials in $n$ real variables define homogeneous functions $f:\mathbb{R}^n \rarr \mathbb{R}$ . For example,

f (x, y, z) = x 5 y 2 z 3

is homogeneous of degree 10 since

(α x) 5 (α y) 2 (α z) 3 = α 10 x 5 y 2 z 3

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

x 5 + 2 x 3 y 2 + 9 x y 4

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

Elementary theorems

Euler's theorem: Suppose that the function $f:\mathbb{R}^n \rarr \mathbb{R}$ is differentiable and homogeneous of degree $k$ . Then

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

This result is proved as follows. Writing $f=f(x_1,\ldots,x_n)$ and differentiating the equation

$f(\alpha \mathbf{y})=\alpha^k f(\mathbf{y})$

with respect to $α$ , we find by the chain rule that

$\frac{\partial}{\partial x_1}f(\alpha\mathbf{y})\frac{\mathrm{d}}{\mathrm{d}\alpha}(\alpha y_1)+ \cdots \frac{\partial}{\partial x_n}f(\alpha\mathbf{y})\frac{\mathrm{d}}{\mathrm{d}\alpha}(\alpha y_n) = k \alpha ^{k-1} f(\mathbf{y})$ ,

so that

$y_1\frac{\partial}{\partial x_1}f(\alpha\mathbf{y})+ \cdots y_n\frac{\partial}{\partial x_n}f(\alpha\mathbf{y}) = k \alpha^{k-1} f(\mathbf{y})$ .

The above equation can be written in the del notation as

$\mathbf{y} \cdot \nabla f(\alpha \mathbf{y}) = k \alpha^{k-1}f(\mathbf{y}), \qquad\qquad \nabla=(\frac{\partial}{\partial x_1},\ldots,\frac{\partial}{\partial x_n})$ ,

from which the stated result is obtained by setting $α = 1$ .

Suppose that $f:\mathbb{R}^n \rarr \mathbb{R}$ is differentiable and homogeneous of degree $k$ . Then its first-order partial derivatives $\partial f/\partial x_i$ are homogeneous of degree $k-1 \qquad\qquad$ .

This result is proved in the same way as Euler's theorem. Writing $f=f(x_1,\ldots,x_n)$ and differentiating the equation

$f(\alpha \mathbf{y})=\alpha^k f(\mathbf{y})$

with respect to $y i$ , we find by the chain rule that

$\frac{\partial}{\partial x_i}f(\alpha\mathbf{y})\frac{\mathrm{d}}{\mathrm{d}y_i}(\alpha y_i) = \alpha ^k \frac{\partial}{\partial x_i}f(\mathbf{y})\frac{\mathrm{d}}{\mathrm{d}y_i}(y_i)$ ,

so that

$\alpha\frac{\partial}{\partial x_i}f(\alpha\mathbf{y}) = \alpha ^k \frac{\partial}{\partial x_i}f(\mathbf{y})$

and hence

$\frac{\partial}{\partial x_i}f(\alpha\mathbf{y}) = \alpha ^{k-1} \frac{\partial}{\partial x_i}f(\mathbf{y})$ .

Application to ODEs

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

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

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 on PlanetMath