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Mathematics for Physics

183 Nondegenerate

A property of a metric under which only the zero vector is orthogonal to every vector that is used to ensure the metric defines a genuine length.

definition [d] (Nondegenerate = Nonsingular = Invertible) A property of a bilinear form \(g\): the only vector orthogonal to all vectors is zero,

  • \(g(v,w) = 0\) for all \(w\) implies \(v = 0\) .

where

  • \(g\) is a bilinear form.
  • \(v, w\) are vectors in the underlying vector space \(V\).

Note:

  • the same property applies when \(g\) is a metric.
  • equivalent to: \(v \mapsto g(v,\cdot)\) is an isomorphism \(V \rightarrow V^{*}\) when \(\dim V < \infty\).
  • nondegenerate is equated with invertible for bilinear forms.

definition [d] (Nondegenerate = Nonsingular) A property of the component matrix of a bilinear form:

  • \(\det(g_{ij}) \neq 0\) ,

which guarantees that an inverse metric \(g^{ij}\) exists.

where

  • \(g_{ij}\) are the components of the bilinear form.
  • \(g^{ij}\) are the components of the inverse metric.
  • \(\det(g_{ij})\) is the determinant of the component matrix.

Note:

  • the same property applies when \(g_{ij}\) are metric components.
  • nonsingular is used interchangeably for the matrix condition.
  • required for raising and lowering indices.

183.1 Elementary Example

183.1.1 Simple

Nondegeneracy of a metric \(g\) means: if \(g(v,w) = 0\) for every \(w\), then \(v = 0\).

\[ g(e_{i},e_{j}) = \delta_{ij} \]

\[ V = \{ e_{1},\ e_{2},\ e_{3} \} \]

where

  • \(g\) is the metric.
  • \(\delta_{ij}\) is the Kronecker symbol.

183.1.2 General

Equivalently, the metric matrix must be invertible. A diagonal Lorentzian matrix is nondegenerate even though it is not positive-definite.

\[ (g_{ij}) = \operatorname{diag}(-1,1,1) \]

\[ \det(g_{ij}) = -1 \neq 0 \]

where

  • \(\det(g_{ij}) \neq 0\) is the matrix test for nondegeneracy.

183.2 References

  1. Hassani, S. Mathematical Physics, 2nd ed. Springer. — nondegenerate means invertible; \(\det g \neq 0\).
  2. Arfken, G. B., Weber, H. J., & Harris, F. E. Mathematical Methods for Physicists, 7th ed. Elsevier / Academic Press, 2013. — nonsingular metrical matrix; existence of \(g^{ij}\).
  3. Reed, M., & Simon, B. Methods of Modern Mathematical Physics I: Functional Analysis. Academic Press. — nonsingular operators and forms.
  4. Frankel, T. The Geometry of Physics, 3rd ed. Cambridge University Press. — nondegenerate metrics on manifolds.