Equivalent Properties of Nondegenerate Symmetric Covariant 2-Tensor
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Theorem
Let $V$ and $V^*$ be a finite dimensional vector space and its dual.
Let $q$ be a symmetric covariant 2-tensor on $V$.
Let $\tuple {\epsilon^i}$ be any basis of $V^*$
Then the following properties are equivalent:
- $q$ is nondegenerate.
- $\forall v \in V : v \ne 0 : \exists w \in V : \map q {v, w} \ne 0$
- If $q = q_{ij} \epsilon^i \epsilon^j$, then the matrix $\paren {q_{ij}}$ is invertible.
Proof
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Sources
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Pseudo-Riemannian Metrics