State and prove Sylvester's law of inertia for a real quadratic form.

[You may assume that for each real symmetric matrix A there is an orthogonal matrix $U$, such that $U^{-1} A U$ is diagonal.]

Suppose that $V$ is a real vector space of even dimension $2 m$, that $Q$ is a non-singular quadratic form on $V$ and that $U$ is an $m$-dimensional subspace of $V$ on which $Q$ vanishes. What is the signature of $Q ?$