I want to prove the following fact. Let $f$ be a nondegenerate symmetric bilinear function on a vector space $V$ with positive index of inertia equal to 1. And let $f(x,x) > 0$ for some vector $x \in V$. Then restriction of $f$ to any subspace $W$ that contains $v$ is nondegenerate.
I know that if $f|_U$ is nondegenerate then $V = U \oplus U^\perp$ (with respect to $f$). So $W = \langle x \rangle \oplus W'$, where $W'$ is the orthogonal complement of $\langle x \rangle$ in $W$ with respect to $W$. Maybe it's possible to show that $f|_{W'}$ is positive definite, but I'm stuck.