Solution
= Solution
Locally write the real analytic <hypersurface> as $\Sigma=\{F=0\}$ with $dF\ne0$. The <conormal bundle> is spanned by $dF$. The surface is a <characteristic hypersurface> at $p$ precisely when the <principal symbol> vanishes on that conormal:
$$
\boxed{\sum_{i,j=1}^n a_{ij}(p)F_{x_i}(p)F_{x_j}(p)=0.}
$$
Only the matrix $(a_{ij}+a_{ji})/2$, which is a <symmetric matrix>, contributes to this expression.