Let $Q$ be a quadratic form on $\mathbb{R}^n$, $G:\mathbb R^n\to\mathbb R^m$ a linear map, and $V=\ker G$. There's a criterion for the positive/negative definiteness of $Q|_V$ involving minors of the "bordered Hessian" of $Q$, roughly the matrix obtained by "bordering" the matrix of $Q$ by the components of the matrix of $G$.
This criterion is used in calculus, for constrained optimization in the context of Lagrange multipliers.
How to prove this criterion? (Or where can I find a proof?)
Wikipedia's page on "Hessian matrix", as well as on "Lagrange multipliers" and "Quadratic optimization", and those of the references cited there that I checked, seem to be of no help.