## Dr. Ahmed G. Abo-Khalil

Electrical Engineering Department

# 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. More precisely, if ƒ : VW 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 vV. This implies it has scale invariance. When the vector spaces involved are over the real numbers, a slightly more 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).

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