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Mathematics for Physics

253 Homogeneity

A property that says scaling an input by a scalar scales the value by a matching power of that scalar that is used to measure vector lengths consistently and to track degree under scaling.

definition [d] (Homogeneity of Degree \(k\)) A property of a function \(f\): there is a fixed \(k\) such that

  • \(f(tx) = t^{k}\, f(x)\)

for all admissible scalars \(t\) and inputs \(x\).

where

  • \(f\) is the function.
  • \(k\) is the degree of homogeneity.
  • \(t\) is a scalar.
  • \(x\) is the input.

Note:

  • when \(k=1\), this is ordinary linear scaling of the output.

definition [d] (Absolute Homogeneity) A property of a norm or seminorm \(\lVert\,\cdot\,\rVert\):

  • \(\lVert \alpha x \rVert = |\alpha|\, \lVert x \rVert\)

for all vectors \(x\) and scalars \(\alpha\).

where

  • \(\lVert\,\cdot\,\rVert\) is the norm or seminorm.
  • \(x\) is a vector.
  • \(\alpha\) is a scalar.
  • \(|\alpha|\) is the absolute value of \(\alpha\).

Note:

  • Kreyszig lists this among the defining axioms of a norm.

definition [d] (Homogeneity of a Polynomial) A property of a polynomial: all of its terms share one common total degree.

where

  • the shared total degree is the degree of homogeneity of the polynomial.

Note:

  • this is the same condition used to call a polynomial homogeneous of degree \(k\).

definition [d] (Homogeneity of a Linear Equation) A property of a linear equation or linear system: the inhomogeneous term is zero, so every scalar multiple of a solution remains a solution.

where

  • the equation is linear in the unknown.
  • the right-hand side vanishes.

253.1 Elementary Example

253.1.1 Simple

Homogeneity of degree \(k\) means scaling the input by \(t\) scales the value by \(t^{k}\).

\[ f(x) = x^{2},\quad k = 2 \]

\[ f(tx) = (tx)^{2} = t^{2} f(x) \]

where

  • \(f\) is the function.
  • \(k\) is the degree of homogeneity.
  • \(t\) is a scalar.

253.1.2 General

A homogeneous polynomial of degree \(2\) in three variables scales by \(t^{2}\).

\[ f(x,y,z) = x^{2} + xy + z^{2} \]

\[ f(tx,ty,tz) = t^{2} f(x,y,z) \]

where

  • \(k = 2\) is the degree.
  • \(x,y,z\) are the input variables.

253.2 References

  1. Kreyszig, E. Introductory Functional Analysis with Applications. Wiley, 1989. — absolute homogeneity of norms; homogeneous linear equations.
  2. Arfken, G. B., Weber, H. J., & Harris, F. E. Mathematical Methods for Physicists, 7th ed. Elsevier Academic Press, 2013. — homogeneity of degree \(k\) for functions; homogeneous linear ODEs.
  3. Dummit, D. S., & Foote, R. M. Abstract Algebra. Wiley, 2004. — homogeneous polynomials.