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Mathematics for Physics

254 Homogeneous

A property under which scaling by a scalar preserves a single fixed degree for an expression that is used to classify polynomials and to simplify equations of one degree.

definition [d] (Homogeneous Polynomial) A property of a polynomial: every term has the same total degree \(k\).

where

  • \(k\) is the common total degree of all terms.
  • the polynomial is then said to be homogeneous of degree \(k\).

Note:

  • a quadratic form is a homogeneous polynomial of degree \(2\).

definition [d] (Homogeneous Function) A property of a function \(f\): scaling the input by \(t\) scales the output by \(t^{k}\),

  • \(f(tx) = t^{k}\, f(x)\) .

where

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

definition [d] (Homogeneous Norm = Absolute Homogeneity) A property of a norm \(\lVert\,\cdot\,\rVert\) on a vector space:

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

for every vector \(x\) and every scalar \(\alpha\).

where

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

Note:

  • this is one of the norm axioms.

definition [d] (Homogeneous Linear Equation) A property of a linear differential equation or linear algebraic system: the right-hand side is identically zero.

where

  • the unknown appears linearly.
  • the forcing term or inhomogeneous term vanishes.

Note:

  • the same usage applies to systems of linear differential equations.

definition [d] (Homogeneous Coordinates) A representation of points of real projective space \(\mathbb{RP}^{n}\) as lines through the origin in \(\mathbb{R}^{n+1}\), written as equivalence classes \([x_{0}:x_{1}:\cdots:x_{n}]\).

where

  • \(\mathbb{RP}^{n}\) is real projective \(n\)-space.
  • \([x_{0}:x_{1}:\cdots:x_{n}]\) are homogeneous coordinates.
  • two tuples represent the same point when one is a nonzero scalar multiple of the other.

254.1 Elementary Example

254.1.1 Simple

A homogeneous expression keeps a single degree under scaling. The monomial \(x^{2}\) is homogeneous of degree \(2\).

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

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

where

  • \(f\) is homogeneous of degree \(2\).
  • \(t\) is a scalar.

254.1.2 General

A sum of degree-\(2\) terms in three variables is homogeneous of degree \(2\).

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

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

where

  • every term has total degree \(2\).

254.2 References

  1. Dummit, D. S., & Foote, R. M. Abstract Algebra. Wiley, 2004. — homogeneous polynomials.
  2. Kreyszig, E. Introductory Functional Analysis with Applications. Wiley, 1989. — absolute homogeneity of a norm; homogeneous linear ODEs and systems.
  3. Arfken, G. B., Weber, H. J., & Harris, F. E. Mathematical Methods for Physicists, 7th ed. Elsevier Academic Press, 2013. — homogeneous functions; homogeneous linear ODEs and PDEs.
  4. Frankel, T. The Geometry of Physics, 3rd ed. Cambridge University Press. — homogeneous coordinates.
  5. Shifrin, T., & Adams, M. Linear Algebra: A Geometric Approach. W. H. Freeman, 2010. — homogeneous coordinates on projective space.