Gram-Schmidt Orthogonalization/Scalar Product Space/Corollary

Theorem

Let $\struct {V, q}$ be an $n$-dimensional scalar product space.

Let $V^*$ be the vector space dual to $V$.

Then there exists a basis $\tuple {\beta^i}$ for $V^*$ with respect to which $q$ has the expression:

$q = \paren {\beta^1}^2 + \ldots + \paren {\beta^r}^2 - \paren {\beta^{r + 1}}^2 - \ldots - \paren {\beta^{r + s}}^2$

where:

$r, s \in \N : r + s = n$.

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