Sylvester's Law of Inertia

Theorem

Let $\struct {V, q}$ be the scalar product space.

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

Suppose there exists a basis $\tuple {\beta^i}$ for $V^*$ such that $q$ is expressible as:

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

where:

$r, s \in \N : r + s = n$

Then $r$ is the maximum dimension among all subspaces on which the restriction of $q$ is positive definite.

Furthermore, $r$ and $s$ are independent of the choice of basis.

Also presented as

Sylvester's Law of Inertia can also be presented as:

The inertia of a Hermitian matrix is invariant under congruence transformations.

Proof

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Source of Name

This entry was named for James Joseph Sylvester.

Historical Note

Sylvester's law of inertia was demonstrated by James Joseph Sylvester in $1852$.

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Pseudo-Riemannian Metrics