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Mathematics for Physics

184 Nondegenerate Bilinear Form

A bilinear mapping on pairs of vectors that is nondegenerate that is used to pair vectors with covectors uniquely.

definition [d] (Nondegenerate Bilinear Form) A symmetric bilinear form \(g\) on a finite-dimensional vector space \(V\) that is invertible: the only vector orthogonal to every vector is zero,

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

equivalently: for every nonzero \(v\) there exists \(w\) with \(g(v,w) \neq 0\).

where

  • \(g\) is a symmetric bilinear form on \(V\).
  • \(V\) is a finite-dimensional real vector space.
  • \(v, w\) are vectors in \(V\).

Note:

  • \(V\) may be a tangent space \(T_{P}M\).
  • the form need not be positive-definite; indefinite metrics are allowed.

definition [d] (Nondegenerate Bilinear Form) A bilinear form whose component matrix is nonsingular:

  • \(\det(g_{ij}) \neq 0\) ,

equivalent to \(g(v,w)=0\) for all \(w\) implying \(v=0\).

where

  • \(g\) is a bilinear form.
  • \(g_{ij} = g(e_{i}, e_{j})\) are the components of \(g\) in a basis \(\{e_{i}\}\).
  • \(\{e_{i}\}\) is a basis of \(V\).
  • \(v, w\) are vectors in \(V\).

Note:

  • nondegeneracy guarantees an inverse matrix \(g^{ij}\) and an isomorphism \(V \cong V^{*}\).

184.1 Elementary Example

184.1.1 Simple

A nondegenerate bilinear form \(B\) pairs vectors so only the zero vector is orthogonal to every vector. The identity pairing has this property.

\[ B : V \times V \rightarrow \mathbb{R} \]

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

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

where

  • \(B\) is the bilinear form.
  • \(\delta_{ij}\) equals \(1\) if \(i = j\) and \(0\) otherwise.

184.1.2 General

Nondegeneracy means the component matrix is invertible. Any invertible symmetric \(3 \times 3\) matrix defines such a form.

\[ (B_{ij}) = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 3 \end{pmatrix} \]

\[ \det(B_{ij}) = 6 \neq 0 \]

where

  • \(B_{ij}\) are the components of \(B\).
  • \(\det(B_{ij}) \neq 0\) encodes nondegeneracy in finite dimensions.

184.2 References

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