This is from "Mathematics from Economists" by Simon and Blume:
To determine the definiteness of a quadratic form of $n$ variables, $Q(\mathbf{x})=\mathbf{x}^TA\mathbf{x},$ when restricted to a constraint set given by $m$ linear equations $B\mathbf{x}=\mathbf{0}$, construct the $(n+m)\times(n+m)$ symmetric matrix $H$ by bordering the matrix $A$ above and to the left by the coefficients $B$ of the linear constraints: $$H= \begin{pmatrix} \mathbf{0} & B \\ B^T & A \\ \end{pmatrix}. $$ If ${\det H}$ has the same sign as $(-1)^n$ and if the last $n-m$ leading principal minors alternate in sign, then $Q$ is a negative definite on the constraint set $B\mathbf x=\mathbf{0}$.
Some presentations focus instead on the smallest of the last $n-m$ leading principal submatrices: $H_{2m+1}$, the $2m+1$th order leading pricipal submatrix: To verify negative definiteness, check that $\det H_{2m+1}$ has the sign of $(-1)^{m+1}$ and that the leading pricipal minors of larger order alternate in signs.
Please let me know how checking $(-1)^n$ for $H=H_{n+m}$ is equivalent to checking $(-1)^{m+1}$ for $H_{2m+1}$?