bilinear form and positive definiteness

Question

Let $B$ a symmetric bilinear form on an $n$ dimensional vector space $E$ with signature $(n-1,1)$. Then there exists a hyperplane $H$ in $E$ in which $B$ is positive definite. How to prove this? Is it true that $B$ has signature $(n-1,1)$ implies the matrix corresponding to $B$ in some fixed basis of $E$ has $n-1$ positive eigenvalues and one negative eigen value. In particular the matrix corresponding to $B$ is invertible.